THE ANALYTIC THEORY OF MATRIX ORTHOGONAL POLYNOMIALS

DAVID DAMANIK, ALEXANDER PUSHNITSKI, AND BARRY SIMON

Contents

1. Introduction 2 1.1. Introduction and Overview 2 1.2. Matrix-Valued Measures 6 1.3. Matrix Möbius Transformations 10 1.4. Applications and Examples 14 2. Matrix Orthogonal Polynomials on the Real Line 17 2.1. Preliminaries 17 2.2. Block Jacobi Matrices 23 2.3. The m-Function 29 2.4. Second Kind Polynomials 31 2.5. Solutions to the Difference Equations 32 2.6. Wronskians and the Christoffel–Darboux Formula 33 2.7. The CD Kernel 34 2.8. Christoffel Variational Principle 35 2.9. Zeros 37 2.10. Lower Bounds on p and the Stieltjes–Weyl Formula for m 39 2.11. Wronskians of Vector-Valued Solutions 40 2.12. The Order of Zeros/Poles of m(z) 40 2.13. Resolvent of the Jacobi Matrix 41 3. Matrix Orthogonal Polynomials on the Unit Circle 43 3.1. Definition of MOPUC 43 3.2. The Szegő Recursion 43 3.3. Second Kind Polynomials 47 3.4. Christoffel–Darboux Formulas 51 3.5. Zeros of MOPUC 53 3.6. Bernstein–Szegő Approximation 56 3.7. Verblunsky’s Theorem 57 3.8. Matrix POPUC 57 3.9. Matrix-Valued Carathéodory and Schur Functions 57 3.10. Coefficient Stripping, the Schur Algorithm, and

Date: October 22, 2007.

2000 Mathematics Subject Classification. 42C05, 47B36, 30C10.

Key words and phrases. Orthogonal Polynomials, Matrix-Valued Measures, Block Jacobi Matrices, Block CMV Matrices.

D.D. was supported in part by NSF grants DMS–0500910 and DMS-0653720.

B.S. was supported in part by NSF grants DMS–0140592 and DMS-0652919 and U.S.–Israel Binational Science Foundation (BSF) Grant No. 2002068.

Geronimus’ Theorem 59

3.11. The CMV Matrix 62 3.12. The Resolvent of the CMV Matrix 65 3.13. Khrushchev Theory 72 4. The Szegő Mapping and the Geronimus Relations 73 5. Regular MOPRL 77 5.1. Upper Bound and Definition 77 5.2. Density of Zeros 77 5.3. General Asymptotics 78 5.4. Weak Convergence of the CD Kernel and Consequences 79 5.5. Widom’s Theorem 80 5.6. A Conjecture 81 References 81

1. Introduction

1.1. Introduction and Overview. Orthogonal polynomials on the real line (OPRL) were developed in the nineteenth century and orthogonal polynomials on the unit circle (OPUC) were initially developed around 1920 by Szegő. Their matrix analogues are of much more recent vintage. They were originally developed in the MOPUC case indirectly in the study of prediction theory [116, 117, 129, 131, 132, 138, 196] in the period 1940–1960. The connection to OPUC in the scalar case was discovered by Krein [131]. Much of the theory since is in the electrical engineering literature [36, 37, 38, 39, 40, 41, 120, 121, 122, 123, 203]; see also [84, 86, 87, 88, 142].

The corresponding real line theory (MOPRL) is still more recent: Following early work of Krein [133] and Berezan’ski [9] on block Jacobi matrices, mainly as applied to self-adjoint extensions, there was a seminal paper of Aptekarev–Nikishin [4] and a flurry of papers since the 1990s [10, 11, 12, 14, 16, 17, 19, 20, 21, 22, 29, 35, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 83, 85, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 137, 139, 140, 143, 144, 145, 148, 149, 150, 155, 156, 157, 154, 161, 162, 179, 186, 198, 200, 201, 202, 204]; see also [7].

There is very little on the subject in monographs — the more classical ones (e.g., [23, 82, 93, 184]) predate most of the subject; see, however, Atkinson [5, Section 6.6]. Ismail [118] has no discussion and Simon [167, 168] has a single section! Because of the use of MOPRL in [33], we became interested in the subject and, in particular, we needed some basic results for that paper which we couldn’t find in the literature or which, at least, weren’t very accessible. Thus, we decided to produce this comprehensive review that we hope others will find useful.

As with the scalar case, the subject breaks into two parts, conveniently called the analytic theory (general structure results) and the algebraic theory (the set of non-trivial examples). This survey deals entirely with the analytic theory. We note, however, that one of the striking developments in recent years has been the discovery that there are rich classes of genuinely new MOPRL, even at the classical level of Bochner’s theorem; see [20, 55, 70, 72, 102, 109, 110, 111, 112, 113, 156, 161] and the forthcoming monograph [63] for further discussion of this algebraic side.

In this introduction, we will focus mainly on the MOPRL case. For scalar OPRL, a key issue is the passage from measure to monic OPRL, then to normalized OPRL, and finally to Jacobi parameters. There are no choices in going from measure to monic OP, P_{n}(x). They are determined by

P_{n}(x) = x^{n} + lower order,  ⟨x^{j}, P_{n}⟩ = 0 j = 1, …, n − 1. (1.1)

However, the basic condition on the orthonormal polynomials, namely,

⟨p_{n}, p_{m}⟩ = δ_{nm}  (1.2)

does not uniquely determine the p_{n}(x). The standard choice is

p_{n}(x) = \frac{P_{n}(x)}{∥P_{n}∥}.

However, if θ_{0}, θ_{1}, … are arbitrary real numbers, then

p̃_{n}(x) = \frac{e^{iθ_{n}}P_{n}(x)}{∥P_{n}∥}  (1.3)

also obey (1.2). If the recursion coefficients (aka Jacobi parameters), are defined via

xp_{n} = a_{n+1}p_{n+1} + b_{n+1}p_{n} + a_{n}p_{n−1}  (1.4)

then the choice (1.3) leads to

b̃_{n} = b_{n},  ã_{n} = e^{iθ_{n}}a_{n}e^{−iθ_{n−1}}.  (1.5)

The standard choice is, of course, most natural here; for example, if

p_{n}(x) = κ_{n}x^{n} + lower order  (1.6)

then a_{n} > 0 implies κ_{n} > 0. It would be crazy to make any other choice.

For MOPRL, these choices are less clear. As we will explain in Section 1.2, there are now two matrix-valued “inner products” formally written as

⟨⟨f, g⟩⟩_{R} = ∫ f(x)^{†} dμ(x)g(x)  (1.7)
⟨⟨f, g⟩⟩_{L} = ∫ g(x) dμ(x)f^{†}(x)  (1.8)

where now μ is a matrix-valued measure and ^{†} denotes the adjoint, and corresponding two sets of monic OPRL: P_{n}^{R}(x) and P_{n}^{L}(x). The orthonormal polynomials are required to obey

⟨⟨p_{n}^{R}, p_{m}^{R}⟩⟩_{R} = δ_{nm}1.  (1.9)

The analogue of (1.3) is

p̃_{n}^{R}(x) = P_{n}^{R}(x)⟨⟨P_{n}^{R}, P_{n}^{R}⟩⟩^{−1/2}σ_{n}  (1.10)

for a unitary σ_{n}. For the immediately following, use p_{n}^{R} to be the choice σ_{n} ≡ 1. For any such choice, we have a recursion relation,

xp_{n}^{R}(x) = p_{n+1}^{R}(x)A_{n+1}^{†} + p_{n}^{R}(x)B_{n+1} + p_{n−1}^{R}(x)A_{n}  (1.11)

with the analogue of (1.5) (comparing σ_{n} ≡ 1 to general σ_{n})

B̃_{n} = σ_{n}^{†}B_{n}σ_{n}  Ã_{n} = σ_{n−1}^{†}A_{n}σ_{n}.  (1.12)

The obvious analogue of the scalar case is to pick σ_{n} ≡ 1, which makes κ_{n} in

p_{n}^{R}(x) = κ_{n}x^{n} + lower order  (1.13)

obey κ_{n} > 0. Note that (1.11) implies

κ_{n} = κ_{n+1}A_{n+1}^{†}  (1.14)

or, inductively,

κ_{n} = (A_{n}^{†} … A_{1}^{†})^{−1}.  (1.15)

In general, this choice does not lead to A_{n} positive or even Hermitian. Alternatively, one can pick σ_{n} so Ã_{n} is positive. Besides these two “obvious” choices, κ_{n} > 0 or A_{n} > 0, there is a third that A_{n} be lower triangular that, as we will see in Section 1.4, is natural. Thus, in the study of MOPRL one needs to talk about equivalent sets of p_{n}^{R} and of Jacobi parameters, and this is a major theme of Chapter 2. Interestingly enough for MOPUC, the commonly picked choice equivalent to A_{n} > 0 (namely, ρ_{n} > 0) seems to suffice for applications. So we do not discuss equivalence classes for MOPUC.

Associated to a set of matrix Jacobi parameters is a block Jacobi matrix, that is, a matrix which when written in l × l blocks is tridiagonal; see (2.29) below.

In Chapter 2, we discuss the basics of MOPRL while Chapter 3 discusses MOPUC. Chapter 4 discusses the Szegő mapping connection of MOPUC and MOPRL. Finally, Chapter 5 discusses the extension of the theory of regular OPs [180] to MOPRL.

While this is mainly a survey, it does have numerous new results, of which we mention:

(a) The clarification of equivalent Jacobi parameters and several new theorems (Theorems 2.8 and 2.9).

(b) A new result (Theorem 2.28) on the order of poles or zeros of m(z) in terms of eigenvalues of J and the once stripped J^{(1)}.

(c) Formulas for the resolvent in the MOPRL (Theorem 2.29) and MOPUC (Theorem 3.24) cases.

(d) A theorem on zeros of det(Φ_{n}^{R}) (Theorem 3.7) and eigenvalues of a cutoff CMV matrix (Theorem 3.10).

(e) A new proof of the Geronimus relations (Theorem 4.2).

(f) Discussion of regular MOPRL (Chapter 5). There are numerous open questions and conjectures in this paper, of

which we mention:

(1) We prove that type 1 and type 3 Jacobi parameters in the Nevai class have A_{n} → 1 but do not know if this is true for type 2 and, if so, how to prove it.

(2) Determine which monic matrix polynomials, Φ, can occur as monic MOPUC. We know det(Φ(z)) must have all of its zeros in the unit disk in ℂ, but unlike the scalar case where this is sufficient, we do not know necessary and sufficient conditions.

(3) Generalize Khrushchev theory [125, 126, 101] to MOPUC; see Section 3.13.

(4) Provide a proof of Geronimus relations for MOPUC that uses the theory of canonical moments [43]; see the discussion at the start of Chapter 4.

(5) Prove Conjecture 5.9 extending a result of Stahl– Totik [180] from OPRL to MOPRL.

It is a pleasure to thank Alexander Aptekarev, Christian Berg, Antonio Durán, Jeff Geronimo, Fritz Gesztesy, Alberto Grünbaum, Paco Marcellán, Ken McLaughlin, Hermann Schulz-Baldes, and Walter Van Assche for useful correspondence.

1.2. Matrix-Valued Measures. Let M_{l} denote the ring of all l × l complex-valued matrices; we denote by α^{†} the Hermitian conjugate of α ∈ M_{l}. (Because of the use of ^{∗} for Szegő dual in the theory of OPUC, we do not use it for adjoint.) For α ∈ M_{l}, we denote by ∥α∥ its Euclidean norm (i.e., the norm of α as a linear operator on ℂ^{l} with the usual Euclidean norm). Consider the set P of all polynomials in z ∈ ℂ with coefficients from M_{l}. The set P can be considered either as a right or as a left module over M_{l}; clearly, conjugation makes the left and right structures isomorphic. For n = 0, 1, …, P_{n} will denote those polynomials in P of degree at most n. The set V denotes the set of all polynomials in z ∈ ℂ with coefficients from ℂ^{l}. The standard inner product in ℂ^{l} is denoted by ⟨·, ·⟩_{ℂ^{l}}.

A matrix-valued measure, μ, on ℝ (or ℂ) is the assignment of a positive semi-definite l × l matrix μ(X) to every Borel set X which is countably additive. We will usually normalize it by requiring

μ(ℝ) = 1  (1.16)

(or μ(ℂ) = 1) where 1 is the l ×l identity matrix. (We use 1 in general for an identity operator, whether in M_{l} or in the operators on some other Hilbert space, and 0 for the zero operator or matrix.) Normally, our measures for MOPRL will have compact support and, of course, our measures for MOPUC will be supported on all or part of ∂𝔻 (𝔻 is the unit disk in ℂ).

Associated to any such measures is a scalar measure

μ_{tr}(X) = Tr(μ(X))  (1.17)

the trace (normalized by Tr(1) = l). μ_{tr} is normalized by μ_{tr}(ℝ) = l.

Applying the Radon–Nikodym theorem to the matrix elements of μ, we see there is a positive semi-definite matrix function M_{ij}(x) so

dμ_{ij}(x) = M_{ij}(x) dμ_{tr}(x).  (1.18)

Clearly, by (1.17),

Tr(M(x)) = 1  (1.19)

for dμ_{tr}-a.e. x. Conversely, any scalar measure with μ_{tr}(ℝ) = l and positive semi-definite matrix-valued function M obeying (1.19) define a matrix-valued measure normalized by (1.16).

Given l × l matrix-valued functions f, g, we define the l × l matrix ⟨⟨f, g⟩⟩_{R} by

⟨⟨f, g⟩⟩_{R} = ∫ f(x)^{†}M(x)g(x) dμ_{tr}(x)  (1.20)

that is, its (j, k) entry is

∑_{nm} ∫ \overline{f_{nj}(x)} M_{nm}(x)g_{mk}(x) dμ_{tr}(x).  (1.21)

Since f^{†}Mf ≥ 0, we see that

⟨⟨f, f⟩⟩_{R} ≥ 0.  (1.22)

One might be tempted to think of ⟨⟨f, f⟩⟩_{R}^{1/2} as some kind of norm, but that is doubtful. Even if μ is supported at a single point, x_{0}, with M = l^{−1}1, this “norm” is essentially the absolute value of A = f(x_{0}), which is known not to obey the triangle inequality! (See [169, Sect. I.1] for an example.)

However, if one looks at

∥f∥_{R} = (Tr⟨⟨f, f⟩⟩_{R})^{1/2}  (1.23)

one does have a norm (or, at least, a semi-norm). Indeed,

⟨f, g⟩_{R} = Tr⟨⟨f, g⟩⟩_{R}  (1.24)

is a sesquilinear form which is positive semi-definite, so (1.23) is the semi-norm corresponding to an inner product and, of course, one has a Cauchy–Schwarz inequality

|Tr⟨⟨f, g⟩⟩_{R}| ≤ ∥f∥_{R}∥g∥_{R}.  (1.25)

We have not specified which f’s and g’s can be used in (1.20). We have in mind mainly polynomials in x in the real case and Laurent polynomials in z in the ∂𝔻 case although, obviously, continuous functions are okay. Indeed, it suffices that f (and g) be measurable and obey

∫ Tr(f^{†}(x)f(x)) dμ_{tr}(x) < ∞  (1.26)

for the integrals in (1.21) to converge. The set of equivalence classes under f ∼ g if ∥f − g∥_{R} = 0 defines a Hilbert space, H, and ⟨f, g⟩_{R} is the inner product on this space.

Instead of (1.20), we use the suggestive shorthand

⟨⟨f, g⟩⟩_{R} = ∫ f(x)^{†} dμ(x)g(x).  (1.27)

The use of R here comes from “right” for if α ∈ M_{l},

⟨⟨f, gα⟩⟩_{R} = ⟨⟨f, g⟩⟩_{R}α  (1.28)
⟨⟨fα, g⟩⟩_{R} = α^{†}⟨⟨f, g⟩⟩_{R}  (1.29)

but, in general, ⟨⟨f, αg⟩⟩_{R} is not related to ⟨⟨f, g⟩⟩_{R}.

While (Tr⟨⟨f, f⟩⟩_{R})^{1/2} is a natural analogue of the norm in the scalar case, it will sometimes be useful to instead consider

[det⟨⟨f, f⟩⟩_{R}]^{1/2}.  (1.30)

Indeed, this is a stronger “norm” in that det > 0 ⇒ Tr > 0 but not vice-versa.

When dμ is a “direct sum,” that is, each M(x) is diagonal, one can appreciate the difference. In that case, dμ = dμ_{1} ⊕ ⋯ ⊕ dμ_{l} and the MOPRL are direct sums (i.e., diagonal matrices) of scalar OPRL

P_{n}^{R}(x, dμ) = P_{n}(x, dμ_{1}) ⊕ ⋯ ⊕ P_{n}(x, dμ_{l}).  (1.31)

Then

∥P_{n}^{R}∥_{R} = ( ∑_{j=1}^{l}∥P_{n}(·, dμ_{j})∥_{L^{2}(dμ_{j})}^{2})^{1/2}  (1.32)

while

(det⟨⟨P_{n}^{R}, P_{n}^{R}⟩⟩_{R})^{1/2} = ∏_{j=1}^{l}∥P_{n}(·, dμ_{j})∥_{L^{2}(dμ_{j})}.  (1.33)

In particular, in terms of extending the theory of regular measures [180], ∥P_{n}^{R}∥_{R}^{1/n} is only sensitive to max∥P_{n}(·, dμ_{j})∥_{L^{2}(dμ_{j})}^{1/2} while (det⟨⟨P_{n}^{R}, P_{n}^{R}⟩⟩_{R})^{1/2} is sensitive to them all. Thus, det will be needed for that theory (see Chapter 5).

There will also be a left inner product and, correspondingly, two sets of MOPRL and MOPUC. We discuss this further in Sections 2.1 and 3.1.

Occasionally, for ℂ^{l} vector-valued functions f and g, we will want to consider the scalar

∑_{k,j} ∫ \overline{f_{k}(x)} M_{kj}(x)g_{j}(x) dμ_{tr}(x)  (1.34)

which we will denote

∫ d⟨f(x), μ(x)g(x)⟩_{ℂ^{l}}.  (1.35)

We next turn to module Fourier expansions. A set {φ_{j}}_{j=1}^{N} in H (N may be infinite) is called orthonormal if and only if

⟨⟨φ_{j}, φ_{k}⟩⟩_{R} = δ_{jk}1.  (1.36)

This natural terminology is an abuse of notation since (1.36) implies orthogonality in ⟨·, ·⟩_{R} but not normalization, and is much stronger than orthogonality in ⟨·, ·⟩_{R}.

Suppose for a moment that N < ∞. For any a_{1}, …, a_{N} ∈ M_{l}, we can form ∑_{j=1}^{N} φ_{j}a_{j} and, by the right multiplication relations (1.28), (1.29), and (1.36), we have

⟨⟨∑_{j=1}^{N} φ_{j}a_{j}, ∑_{j=1}^{N} φ_{j}b_{j}⟩⟩_{R} = ∑_{j=1}^{N} a_{j}^{†}b_{j}.  (1.37)

We will denote the set of all such ∑_{j=1}^{N} φ_{j}a_{j} by H_{(φ_{j})}—it is a vector subspace of H of dimension (over ℂ) Nl^{2}.

Define for f ∈ H,

π_{(φ_{j})}(f) = ∑_{j=1}^{N} φ_{j}⟨⟨φ_{j}, f⟩⟩_{R}.  (1.38)

It is easy to see it is the orthogonal projection in the scalar inner product ⟨·, ·⟩_{R} from H to H_{(φ_{j})}.

By the standard Hilbert space calculation, taking care to only multiply on the right, one finds the Pythagorean theorem,

⟨⟨f, f⟩⟩_{R} = ⟨⟨f − π_{(φ_{j})}f, f − π_{(φ_{j})}f⟩⟩_{R} + ∑_{j=1}^{N}⟨⟨φ_{j}, f⟩⟩_{R}^{†}⟨⟨φ_{j}, f⟩⟩_{R}. (1.39)

As usual, this proves for infinite N that

∑_{j=1}^{N}⟨⟨φ_{j}, f⟩⟩_{R}^{†}⟨⟨φ_{j}, f⟩⟩_{R} ≤ ⟨⟨f, f⟩⟩_{R}  (1.40)

and the convergence of

∑_{j=1}^{N} φ_{j}⟨⟨φ_{j}, f⟩⟩_{R} ≡ π_{(φ_{j})}(f)  (1.41)

allowing the definition of π_{(φ_{j})} and of H_{(φ_{j})} ≡ Ran π_{(φ_{j})} for N = ∞.

An orthonormal set is called complete if H_{(φ_{j})} = H. In that case, equality holds in (1.40) and π_{(φ_{j})}(f) = f.

For orthonormal bases, we have the Parseval relation from (1.39)

⟨⟨f, f⟩⟩_{R} = ∑_{j=1}^{∞}⟨⟨φ_{j}, f⟩⟩_{R}^{†}⟨⟨φ_{j}, f⟩⟩_{R}  (1.42)

and

∥f∥_{R}^{2} = ∑_{j=1}^{∞} Tr(⟨⟨φ_{j}, f⟩⟩_{R}^{†}⟨⟨φ_{j}, f⟩⟩_{R}).  (1.43)

1.3. Matrix Möbius Transformations. Without an understanding of matrix Möbius transformations, the form of the MOPUC Geronimus theorem we will prove in Section 3.10 will seem strange-looking. To set the stage, recall that scalar fractional linear transformations (FLT) are associated to matrices T = ( \begin{matrix}a & b \\ c & d \end{matrix}) with det T ≠ 0 via

f_{T} (z) = \frac{az + b}{cz + d} .  (1.44)

Without loss, one can restrict to

det(T) = 1.  (1.45)

Indeed, T ↦ f_{T} is a 2 to 1 map of 𝕊𝕃(2, ℂ) to maps of ℂ ∪ {∞} to itself. One advantage of the matrix formalism is that the map is a matrix homomorphism, that is,

f_{T ∘S} = f_{T} ∘ f_{S}  (1.46)

which shows that the group of FLTs is 𝕊𝕃(2, ℂ)/{1, −1}.

While (1.46) can be checked by direct calculation, a more instructive way is to look at the complex projective line. u, v ∈ ℂ^{2} \ {0} are called equivalent if there is λ ∈ ℂ \ {0} so that u = λv. Let [·] denote equivalence classes. Except for [\binom{1}{0}], every equivalence class contains exactly one point of the form \binom{z}{1} with z ∈ ℂ. If [\binom{1}{0}] is associated with ∞, the set of equivalence classes is naturally associated with ℂ ∪ {∞}. f_{T} then obeys

[T \binom{z}{1}] = [\binom{f_{T} (z)}{1}]  (1.47)

from which (1.46) is immediate.

By Möbius transformations we will mean those FLTs that map 𝔻 onto itself. Let

J = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )  (1.48)

Then [u] = [\binom{z}{1}] with |z| = 1 (resp. |z| < 1) if and only if ⟨u, Ju⟩ = 0 (resp. ⟨u, Ju⟩ < 0). From this, it is not hard to show that if det(T) = 1, then f_{T} maps 𝔻 invertibly onto 𝔻 if and only if

T^{†}JT = J.  (1.49)

If T has the form ( \begin{matrix}a & b \\ c & d \end{matrix}), this is equivalent to

|a|^{2} − |c|^{2} = 1,  |b|^{2} − |d|^{2} = −1,  āb − c̄d = 0.  (1.50)

The set of T ’s obeying det(T) = 1 and (1.49) is called 𝕊𝕌(1, 1). It is studied extensively in [168, Sect. 10.4].

The self-adjoint elements of 𝕊𝕌(1, 1) are parametrized by α ∈ 𝔻 via ρ = (1 − |α|^{2})^{1/2}

T_{α} = \frac{1}{ρ} (\begin{matrix}1 & α \\ ᾱ & 1 \end{matrix})  (1.51)

associated to

f_{T_{α}}(z) = \frac{z + α}{1 + ᾱz} .  (1.52)

Notice that

T_{α}^{−1} = T_{−α}  (1.53)

and that

∀z ∈ 𝔻, ∃! α such that T_{α}(0) = z

namely, α = z.

It is a basic theorem that every holomorphic bijection of 𝔻 to 𝔻 is an f_{T} for some T in 𝕊𝕌(1, 1) (unique up to ±1).

With this in place, we can turn to the matrix case. Let M_{l} be the space of l × l complex matrices with the Euclidean norm induced by the vector norm ⟨·, ·⟩_{ℂ^{l}}^{1/2}. Let

𝔻_{l} = {A ∈ M_{l} : ∥A∥ < 1}.  (1.54)

We are interested in holomorphic bijections of 𝔻_{l} to itself, especially via a suitable notion of FLT. There is a huge (and diffuse) literature on the subject, starting with its use in analytic number theory. It has also been studied in connection with electrical engineering filters and indefinite matrix Hilbert spaces. Among the huge literature, we mention [1, 3, 78, 99, 114, 166]. Especially relevant to MOPUC is the book of Bakonyi–Constantinescu [6].

Consider M_{l} ⊕ M_{l} = M_{l}[2] as a right module over M_{l}. The M_{l}- projective line is defined by saying [ \begin{matrix}X \\ Y \end{matrix}] ∼ [ \begin{matrix}X′ \\ Y′ \end{matrix} ], both in M_{l}[2] \ \{0\}, if and only if there exists Λ ∈ M_{l}, Λ invertible so that

X = X′Λ,  Y = Y′Λ.  (1.55)

Let T be a map of M_{l}[2] of the form

T = (\begin{matrix}A & B \\ C & D \end{matrix})  (1.56)

acting on M_{l}[2] by

T [\begin{matrix}X \\ Y \end{matrix}] = [\begin{matrix}AX + BY \\ CX + DY \end{matrix} ] .  (1.57)

Because this acts on the left and Λ equivalence on the right, T maps equivalence classes to themselves. In particular, if CX +D is invertible, T maps the equivalence class of [ \begin{matrix}X \\ 1 \end{matrix} ] to the equivalence class of [ \begin{matrix}f_{T} [X] \\ 1 \end{matrix} ], where

f_{T} [X] = (AX + B)(CX + D)^{−1}.  (1.58)

So long as CX + D remains invertible, (1.46) remains true. Let J be the 2l × 2l matrix in l × l block form

J = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .  (1.59)

Note that (with [ \begin{matrix}X \\ 1 \end{matrix} ]^{†} = [X^{†}1])

[\begin{matrix}X \\ 1 \end{matrix}]^{†} J [\begin{matrix}X \\ 1 \end{matrix}] ≤ 0 ⇔ X^{†}X ≤ 1 ⇔ ∥X∥ ≤ 1.  (1.60)

Therefore, if we define 𝕊𝕌(l, l) to be those T ’s with det T = 1 and

T^{†}JT = J  (1.61)

then

T ∈ 𝕊𝕌(l, l) ⇒ f_{T} [𝔻_{l}] = 𝔻_{l} as a bijection.  (1.62)

If T has the form (1.56), then (1.61) is equivalent to

A^{†}A − C^{†}C = D^{†}D − B^{†}B = 1,  (1.63)
A^{†}B = C^{†}D  (1.64)

(the fourth relation B^{†}A = D^{†}C is equivalent to (1.64)).

This depends on

Proposition 1.1. If T = ( \begin{matrix}A & B \\ C & D \end{matrix}) obeys (1.61) and ∥X∥ < 1, then CX + D is invertible.

Proof. (1.61) implies that

T^{−1} = JT^{†}J  (1.65)
= ( \begin{matrix}A^{†} & −C^{†} \\ −B^{†} & D^{†} \end{matrix} )  (1.66)

Clearly, (1.61) also implies T^{−1} ∈ 𝕊𝕌(l, l). Thus, by (1.63) for T^{−1},

DD^{†} − CC^{†} = 1.  (1.67)

This implies first that DD^{†} ≥ 1, so D is invertible, and second that

∥D^{−1}C∥ ≤ 1.  (1.68)

Thus, ∥X∥ < 1 implies ∥D^{−1}CX∥ < 1 so 1+D^{−1}CX is invertible, and thus so is D(1 + D^{−1}CX). □

It is a basic result of Cartan [18] (see Helgason [114] and the discussion therein) that Theorem 1.2. A holomorphic bijection, g, of 𝔻_{l} to itself is either of the form

g(X) = f_{T} (X)  (1.69)

for some T ∈ 𝕊𝕌(l, l) or

g(X) = f_{T} (X^{t}).  (1.70)

Given α ∈ M_{l} with ∥α∥ < 1, define

ρ^{L} = (1 − α^{†}α)^{1/2},  ρ^{R} = (1 − αα^{†})^{1/2}.  (1.71)

Lemma 1.3. We have

αρ^{L} = ρ^{R}α,  α^{†}ρ^{R} = ρ^{L}α^{†},  (1.72)
α(ρ^{L})^{−1} = (ρ^{R})^{−1}α,  α^{†}(ρ^{R})^{−1} = (ρ^{L})^{−1}α^{†}.  (1.73)

Proof. Let f be analytic in 𝔻 with f(z) = ∑_{n=0}^{∞} c_{n}z^{n} its Taylor series at z = 0. Since ∥α^{†}α∥ < 1, we have

f(α^{†}α) = ∑_{n=0}^{∞} c_{n}(α^{†}α)^{n}  (1.74)

norm convergent, so α(α^{†}α)^{n} = (αα^{†})^{n}α implies

αf(α^{†}α) = f(αα^{†})α  (1.75)

which implies the first halves of (1.72) and (1.73). The other halves follow by taking adjoints. □

Theorem 1.4. There is a one-one correspondence between α’s in M_{l} obeying ∥α∥ < 1 and positive self-adjoint elements of 𝕊𝕌(l, l) via

T_{α} = (\begin{matrix}(ρ^{R})^{−1} & (ρ^{R})^{−1}α \\ (ρ^{L})^{−1}α^{†} & (ρ^{L})^{−1} \end{matrix})  (1.76)

Proof. A straightforward calculation using Lemma 1.3 proves that T_{α} is self-adjoint and T_{α}^{†}JT_{α} = J. Conversely, if T is self-adjoint, T = ( \begin{matrix}A & B \\ C & D \end{matrix}) and in 𝕊𝕌(l, l), then T^{†} = T ⇒ A^{†} = A, B^{†} = C, so (1.63) becomes

AA^{†} − BB^{†} = 1  (1.77)

so if

α = A^{−1}B  (1.78)

then (1.77) becomes

A^{−1}(A^{−1})^{†} + αα^{†} = 1.  (1.79)

Since T ≥ 0, A ≥ 0 so (1.79) implies A = (ρ^{R})^{−1}, and then (1.78) implies B = (ρ^{R})^{−1}α.

By Lemma 1.3,

C = B^{†} = α^{†}(ρ^{R})^{−1} = (ρ^{L})^{−1}α^{†}  (1.80)

and then (by D = D^{†}, C^{†} = B, and (1.63)) DD^{†}−CC^{†} = 1 plus D > 0 implies D = (ρ^{L})^{−1}. □

Corollary 1.5. For each α ∈ 𝔻_{l}, the map

f_{T_{α}}(X) = (ρ^{R})^{−1}(X + α)(1 + α^{†}X)^{−1}(ρ^{L})  (1.81)

takes 𝔻_{l} to 𝔻_{l}. Its inverse is given by

f_{T_{α}}^{−1}(X) = f_{T_{−α}}(X) = (ρ^{R})^{−1}(X − α)(1 − α^{†}X)^{−1}(ρ^{L}).  (1.82)

There is an alternate form for the right side of (1.81).

Proposition 1.6. The following identity holds true for any X, ∥X∥ ≤ 1:

ρ^{R}(1 + Xα^{†})^{−1}(X + α)(ρ^{L})^{−1} = (ρ^{R})^{−1}(X + α)(1 + α^{†}X)^{−1}ρ^{L}. (1.83)

x-ref? Proof. We have TK. Sasha to insert proof here

□

1.4. Applications and Examples. There are a number of simple examples which show that beyond their intrinsic mathematical interest, MOPRL and MOPUC have wide application.

(a) Jacobi matrices on a strip. Let Λ ⊂ ℤ^{ν} be a subset (perhaps infinite) of the ν-dimensional lattice ℤ^{ν} and let l^{2}(Λ) be square summable sequences indexed by Λ. Suppose a real symmetric matrix α_{ij} is given for all i, j ∈ Λ with α_{ij} = 0 unless |i − j| = 1 (nearest neighbors). Let β_{i} be a real sequence indexed by i ∈ Λ. Suppose

sup_{i,j} |α_{ij}| + sup_{i} |β_{i}| < ∞.  (1.84)

Define a bounded operator, J, on l^{2}(Λ) by

(Ju)_{i} = ∑_{j} α_{ij}u_{j} + β_{i}u_{i}.  (1.85)

The sum is finite with at most 2ν elements.

The special case Λ = {1, 2, … } with b_{i} = β_{i}, a_{i} = α_{i,i+1} > 0 corresponds precisely to classical semi-infinite tridiagonal Jacobi matrices.

Now consider the situation where Λ′ ⊂ ℤ^{ν−1} is a finite set with l elements and

Λ = {j ∈ ℤ^{ν} : j_{1} ∈ {1, 2, … }; (j_{2}, … j_{ν}) ∈ Λ′}  (1.86)

a “strip” with cross-section Λ′. J then has a block l × l matrix Jacobi form where (γ, δ ∈ Λ′)

(B_{i})_{γδ} = b_{(i,γ)},  (γ = δ),  (1.87)
= a_{(i,γ)(i,δ)},  (γ ≠ δ),  (1.88)
(A_{i})_{γδ} = a_{(i,γ)(i+1,δ)}.  (1.89)

The nearest neighbor condition says (A_{i})_{γδ} = 0 if γ ≠ δ. If

a_{(i,γ)(i+1,γ)} > 0  (1.90)

for all i, γ, then A_{i} is invertible and we have a block Jacobi matrix of the kind described in Section 2.2 below.

By allowing general A_{i}, B_{i}, we obtain an obvious generalization of this model—an interpretation of general MOPRL.

Schrödinger operators on strips have been studied in part as approximations to ℤ^{ν}; see [31, 95, 130, 134, 151, 164]. From this point of view, it is also natural to allow periodic boundary conditions in the vertical directions. Furthermore, there is closely related work on Schrödinger (and other) operators with matrix-valued potentials; see, for example, [8, 24, 25, 26, 27, 28, 30, 96, 97, 165].

(b) Two-sided Jacobi matrices. This example goes back at least to Nikishin [153]. Consider the case ν = 2, Λ′ = {0, 1} ⊂ ℤ, and Λ as above. Suppose (1.90) holds, and in addition,

a_{(1,0)(1,1)} > 0,  (1.91)
a_{(i,0)(i,1)} = 0, i = 2, 3, …  (1.92)

Then there are no links between the rungs of the “ladder,” {1, 2, … }× {0, 1} except at the end and the ladder can be unfolded to ℤ! Thus, a two-sided Jacobi matrix can be viewed as a special kind of one-sided 2 × 2 matrix Jacobi operator.

It is known that for two-sided Jacobi matrices, the spectral theory is determined by the 2 × 2 matrix

dμ = (\begin{matrix}dμ_{00} & dμ_{01} \\ dμ_{10} & dμ_{11} \end{matrix})  (1.93)

where dμ_{kl} is the measure with

⟨δ_{k}, (J − λ)^{−1}δ_{l}⟩ = ∫ \frac{dμ_{kl}(x)}{x − λ}  (1.94)

but also that it is very difficult to specify exactly which dμ correspond to two-sided Jacobi matrices.

This difficulty is illuminated by the theory of MOPRL. By Favard’s theorem (see Theorem 2.11), every such dμ (given by (1.93) and positive definite and non-trivial in a sense we will describe in Lemma 2.1) yields a unique block Jacobi matrix with A_{j} > 0 (positive definite). This dμ comes from a two-sided Jacobi matrix if and only if

(a) B_{j} is diagonal for j = 2, 3, . . ..

(b) A_{j} is diagonal for j = 1, 2, . . ..

(c) B_{j} has strictly positive off-diagonal elements. These are very complicated indirect conditions on dμ!

(c) Banded matrices. Classical Jacobi matrices are semi-infinite symmetric tridiagonal matrices, that is,

J_{km} = 0 if |k − m| > 1  (1.95)

with

J_{km} > 0 if |k − m| = 1.  (1.96)

A natural generalization are (2l + 1)-diagonal symmetric matrices, that is,

J_{km} = 0 if |k − m| > l,  (1.97)
J_{km} > 0 if |k − m| = l.  (1.98)

Such a matrix can be partitioned into l×l blocks, which is tridiagonal in block. The conditions (1.97) and (1.98) are equivalent to A_{k} ∈ L, the set of lower triangular matrices; and conversely, A_{k} ∈ L, with A_{k}, B_{k} real (and B_{k} symmetric) correspond precisely to such banded matrices. This is why we introduce type 3 MOPRL.

Banded matrices correspond to certain higher-order difference equations. Unlike the second-order equation (which leads to tridiagonal matrices) where every equation with positive coefficients is equivalent via a variable rescaling to a symmetric matrix, only certain higher-order difference equations correspond to symmetric block Jacobi matrices.

(d) Magic formula. In [33], Damanik, Killip, and Simon studied perturbations of Jacobi and CMV matrices with periodic Jacobi parameters (or Verblunsky coefficients). They proved that if Δ is the discriminant of a two-sided periodic J_{0}, then a bounded two-sided J has Δ(J) = S^{p} + S^{−p} ((Su)_{n} ≡ u_{n+1}) if and only if J lies in the isospectral torus of J_{0}. They call this the magic formula.

This allows the study of perturbations of the isospectral torus by studying Δ(J) which is a polynomial in J of degree p, and so a 2p + 1 banded matrix. Thus, the study of perturbations of periodic problems is connected to perturbations of S^{p} + S^{−p} as block Jacobi matrices.

Indeed, it was this connection that stimulated our interest in MOPRL, and [33] uses some of our results here.

(e) Vector-valued prediction theory. As noted in Section 1.1, both prediction theory and filtering theory use OPUC and have natural MOPUC settings that motivated much of the MOPUC literature.

2. Matrix Orthogonal Polynomials on the Real Line

2.1. Preliminaries. OPRL are the most basic and developed of orthogonal polynomials, and so this chapter on the matrix analogue is the most important of this survey. We present the basic formulas, assuming enough familiarity with the scalar case (see [23, 82, 167, 176, 184, 185]) that we do not need to explain why the objects we define are important.

2.1.1. Polynomials, Inner Products, Norms. Let dμ be an l × l matrixvalued Hermitian positive semi-definite finite measure on ℝ with compact support, normalized by μ(ℝ) = 1 ∈ M_{l}. Define (as in (1.20))

⟨⟨f, g⟩⟩_{R} = ∫ f(x)^{†} dμ(x) g(x), ∥f∥_{R} = (Tr⟨⟨f, f⟩⟩_{R})^{1/2}, f, g ∈ P,
⟨⟨f, g⟩⟩_{L} = ∫ g(x) dμ(x) f(x)^{†}, ∥f∥_{L} = (Tr⟨⟨f, f⟩⟩_{L})^{1/2}, f, g ∈ P.

Clearly, we have

⟨⟨f, g⟩⟩_{R}^{†} = ⟨⟨g, f⟩⟩_{R},  ⟨⟨f, g⟩⟩_{L}^{†} = ⟨⟨g, f⟩⟩_{L},  (2.1)
⟨⟨f, g⟩⟩_{L} = ⟨⟨g^{†}, f^{†}⟩⟩_{R},  ∥f∥_{L} = ∥f^{†}∥_{R}.  (2.2)

As noted in Section 1.2, we have the left and right analogues of the Cauchy inequality

|Tr⟨⟨f, g⟩⟩_{R}| ≤ ∥f∥_{R}∥g∥_{R}, |Tr⟨⟨f, g⟩⟩_{L}| ≤ ∥f∥_{L} ∥g∥_{L}.

Thus, ∥·∥_{R} and ∥·∥_{L} are semi-norms in P. Indeed, as noted in Section 1.2, they are associated to an inner product. The sets {f : ∥f∥_{R} = 0} and {f : ∥f∥_{L} = 0} are linear subspaces. Let P_{R} be the completion of P/{f : ∥f∥_{R} = 0} (viewed as a right module over M_{l}) with respect to the norm ∥·∥_{R}. Similarly, let P_{L} be the completion of P/{f : ∥f∥_{L} = 0} (viewed as a left module) with respect to the norm ∥·∥_{L}.

The set V defined in Section 1.2 is a linear space. Let us introduce a semi-norm in V by

|||f||| = {∫ d⟨f(x), μ(x)f(x)⟩_{ℂ^{l}}}^{1/2}.  (2.3)

Let V_{0} ⊂ V be the linear subspace of all polynomials such that |||f||| = 0 and let V_{∞} be the completion of the quotient space V/V_{0} with respect to the norm ||| · |||.

Lemma 2.1. The following are equivalent:

(1) ∥f∥_{R} > 0 for every non-zero f ∈ P.

(2) For all n, the dimension in P_{R} of the set of all polynomials of degree at most n is (n + 1)l^{2}.

(3) ∥f∥_{L} > 0 for every non-zero f ∈ P.

(4) For all n, the dimension in P_{L} of the set of all polynomials of degree at most n is (n + 1)l^{2}.

(5) For every non-zero v ∈ V, we have that |||v||| ≠ 0.

(6) For all n, the dimension in V_{∞} of all vector-valued polynomials of degree at most n is (n + 1)l.

The measure dμ is called non-trivial if these equivalent conditions hold.

Remark. If l = 1, these are equivalent to the usual non- triviality condition, that is, supp(μ) is infinite. For l > 1, we cannot define triviality in this simple way, as can be seen by looking at the direct sum of a trivial and non-trivial measure. In that case, the measure is not non-trivial in the above sense but its support is infinite.

Proof. The equivalences (1) ⇔ (2), (3) ⇔ (4), and (5) ⇔ (6) are immediate. The equivalence (1) ⇔ (3) follows from (2.2). Let us prove the equivalence (1) ⇔ (5). Assume that (1) holds and let v ∈ V be non-zero. Let f ∈ M_{l} denote the matrix that has v as its leftmost column and that has zero columns otherwise. Then, 0 ≠ ∥f∥_{R}^{2} = Tr⟨⟨f, f⟩⟩_{R} = |||v|||^{2} and hence (5) holds. Now assume that (1) fails and let f ∈ P be non-zero with ∥f∥_{R} = 0. Then, at least one of the column vectors of f is non-zero. Suppose for simplicity that this is the first column and denote this column vector by v. Let t ∈ M_{l} be the matrix t_{ij} = δ_{i1}δ_{j1}; then we have

∥f∥_{R} = 0 ⇒ ⟨⟨f, f⟩⟩_{R} = 0 ⇒ 0 = Tr(t^{∗}⟨⟨f, f⟩⟩_{R}t) = |||v|||^{2}

and hence (5) fails. □

Throughout the rest of this chapter, we assume the measure dμ to be non-trivial.

2.1.2. Monic Orthogonal Polynomials.

Lemma 2.2. Let dμ be a non-trivial measure.

(i) There exists a unique monic polynomial P_{n}^{R} of degree n, which minimizes the norm ∥P_{n}^{R}∥_{R}.

(ii) The polynomial P_{n}^{R} can be equivalently defined as the monic polynomial of degree n which satisfies

⟨⟨P_{n}^{R}, f⟩⟩_{R} = 0 for any f ∈ P, deg f < n.  (2.4)

(iii) There exists a unique monic polynomial P_{n}^{L} of degree n, which

minimizes the norm ∥P_{n}^{L}∥_{L}. (iv) The polynomial P_{n}^{L} can be equivalently defined as the monic poly-

nomial of degree n which satisfies

⟨⟨P_{n}^{L}, f⟩⟩_{L} = 0 for any f ∈ P, deg f < n.  (2.5)

(v) One has P_{n}^{L}(x) = P_{n}^{R}(x)^{†} for all x ∈ ℝ and

⟨⟨P_{n}^{R}, P_{n}^{R}⟩⟩_{R} = ⟨⟨P_{n}^{L}, P_{n}^{L}⟩⟩_{L}.  (2.6)

Proof. As noted, P has an inner product ⟨·, ·⟩_{R}, so there is an orthogonal projection π_{n}^{(R)} onto P_{n} discussed in Section 1.2. Then

P_{n}^{R}(x) = x^{n} − π_{n−1}^{(R)}(x^{n}).  (2.7)

As usual, in inner product spaces, this uniquely minimizes x^{n} − Q over all Q ∈ P_{n−1}. It clearly obeys

Tr(⟨⟨P_{n}^{R}, f⟩⟩_{R}) = 0  (2.8)

for all f ∈ P_{n−1}. But then for any matrix α,

Tr(⟨⟨P_{n}^{R}, f⟩⟩_{R}α) = Tr(⟨⟨P_{n}^{R}, fα⟩⟩_{R}) = 0

so (2.4) holds.

This proves (i) and (ii). (iii) and (iv) are similar. To prove (v), note that P_{n}^{L}(x) = P_{n}^{R}(x)^{†} follows from the criteria (2.4), (2.5). The identity (2.6) follows from (2.2). □

Lemma 2.3. Let μ be non-trivial. For any monic polynomial P, we have det⟨⟨P, P ⟩⟩_{R} ≠ 0 and det⟨⟨P, P ⟩⟩_{L} ≠ 0.

Proof. Let P be a monic polynomial of degree n such that ⟨⟨P, P ⟩⟩_{R} has a non-trivial kernel. Then one can find α ∈ M_{l}, α ≠ 0, such that α^{†}⟨⟨P, P ⟩⟩_{R}α = 0. It follows that ∥P α∥_{R} = 0. But since P is monic, the leading coefficient of P α is α, so P α ≠ 0, which contradicts the non-triviality assumption. A similar argument works for ⟨⟨P, P ⟩⟩_{L}. □

By the orthogonality of Q_{n} − P_{n}^{R} to P_{n}^{R} for any monic polynomial Q_{n} of degree n, we have

⟨⟨Q_{n}, Q_{n}⟩⟩_{R} = ⟨⟨Q − P_{n}^{R}, Q − P_{n}^{R}⟩⟩_{R} + ⟨⟨P_{n}^{R}, P_{n}^{R}⟩⟩_{R}  (2.9)

and, in particular,

⟨⟨P_{n}^{R}, P_{n}^{R}⟩⟩_{R} ≤ ⟨⟨Q_{n}, Q_{n}⟩⟩_{R}  (2.10)

with (by non-triviality) equality if and only if Q_{n} = P_{n}^{R}. Since Tr and det are strictly monotone on strictly positive matrices, we have the following variational principles ((2.11) restates (i) of Lemma 2.2):

Theorem 2.4. For any monic Q_{n} of degree n, we have

∥Q_{n}∥_{R} ≥ ∥P_{n}^{R}∥_{R},  (2.11)
det⟨⟨Q_{n}, Q_{n}⟩⟩_{R} ≥ det⟨⟨P_{n}^{R}, P_{n}^{R}⟩⟩_{R}  (2.12)

with equality if and only if P_{n}^{R} = Q_{n}.

2.1.3. Expansion.

Theorem 2.5. Let dμ be non-trivial.

(i) We have

⟨⟨P_{k}^{R}, P_{n}^{R}⟩⟩_{R} = γ_{n}δ_{kn}  (2.13)

for some positive invertible matrices γ_{n}. (ii) {P_{k}^{R}}_{k=0}^{n} are a right-module basis for P_{n}; indeed, any f ∈ P_{n} has

a unique expansion,

f = ∑_{j=0}^{n} P_{j}^{R}f_{j}^{R}.  (2.14)

Indeed, essentially by (1.38),

f_{j}^{R} = γ_{j}^{−1}⟨⟨P_{j}^{R}, f⟩⟩_{R}.  (2.15)

Remark. There are similar formulas for ⟨⟨·, ·⟩⟩_{L}. By (2.6),

⟨⟨P_{k}^{L}, P_{n}^{L}⟩⟩_{L} = γ_{n}δ_{kn}  (2.16)

(same γ_{n}, which is why we use γ_{n} and not γ_{n}^{R}).

Proof. (i) (2.13) for n < k is immediate from (2.5) and for n > k by symmetry. γ_{n} ≥ 0 follows from (1.22). By Lemma 2.3, det(γ_{n}) ≠ 0, so γ_{n} is invertible.

(ii) Map (M_{l})^{n+1} to P_{n} by

⟨α_{0}, …, α_{n}⟩ ↦ ∑_{j=0}^{n} P_{j}^{R}α_{j} ≡ X(α_{0}, …, α_{n}).

By (2.13),

α_{j} = γ_{j}^{−1}⟨⟨P_{j}^{R}, X(α_{0}, …, α_{n})⟩⟩

so that map is one-one. By dimension counting, it is onto. □

2.1.4. Recurrence Relations for Monic Orthogonal Polynomials. Denote by ζ_{n}^{R} (resp. ζ_{n}^{L}) the coefficient of x^{n−1} in P_{n}^{R}(x) (resp. P_{n}^{L}(x)), that is,

P_{n}^{R}(x) = x^{n}1 + ζ_{n}^{R}x^{n−1} + lower order terms,
P_{n}^{L}(x) = x^{n}1 + ζ_{n}^{L}x^{n−1} + lower order terms.

Since P_{n}^{R}(x)^{†} = P_{n}^{L}(x), we have (ζ_{n}^{R})^{†} = ζ_{n}^{L}. Using the parameters γ_{n} of (2.13) and ζ_{n}^{R}, ζ_{n}^{L} one can write down recurrence relations for P_{n}^{R}(x), P_{n}^{L}(x).

Lemma 2.6. (i) We have a commutation relation

γ_{n−1}(ζ_{n}^{R} − ζ_{n−1}^{R}) = (ζ_{n}^{L} − ζ_{n−1}^{L})γ_{n−1}.  (2.17)

(ii) We have the recurrence relations

xP_{n}^{R}(x) = P_{n+1}^{R}(x) + P_{n}^{R}(x)(ζ_{n}^{R} − ζ_{n+1}^{R}) + P_{n−1}(x)γ_{n−1}^{−1}γ_{n}, (2.18)
xP_{n}^{L}(x) = P_{n+1}^{L}(x) + (ζ_{n}^{L} − ζ_{n+1}^{L})P_{n}^{L}(x) + γ_{n}γ_{n−1}^{−1}P_{n−1}^{L}(x). (2.19)

Proof. (i) We have

P_{n}^{R}(x) − xP_{n−1}^{R}(x) = (ζ_{n}^{R} − ζ_{n−1}^{R})x^{n−1} + lower order terms

and so

(ζ_{n}^{L} − ζ_{n−1}^{L})γ_{n−1} = (ζ_{n}^{R} − ζ_{n−1}^{R})^{†}⟨⟨P_{n−1}^{R}, P_{n−1}^{R}⟩⟩_{R}
= (ζ_{n}^{R} − ζ_{n−1}^{R})^{†}⟨⟨x^{n−1}, P_{n−1}^{R}⟩⟩_{R}
= ⟨⟨P_{n}^{R} − xP_{n−1}^{R}, P_{n−1}^{R}⟩⟩_{R}
= ⟨⟨P_{n}^{R}, P_{n−1}^{R}⟩⟩_{R} − ⟨⟨xP_{n−1}^{R}, P_{n−1}^{R}⟩⟩_{R}
= −⟨⟨xP_{n−1}^{R}, P_{n−1}^{R}⟩⟩_{R}
= −⟨⟨P_{n−1}^{R}, xP_{n−1}^{R}⟩⟩_{R}
= ⟨⟨P_{n−1}^{R}, P_{n}^{R} − xP_{n−1}^{R}⟩⟩_{R}
= ⟨⟨P_{n−1}^{R}, x^{n−1}(ζ_{n}^{R} − ζ_{n−1}^{R})⟩⟩_{R}
= ⟨⟨P_{n−1}^{R}, x^{n−1}⟩⟩_{R}(ζ_{n}^{R} − ζ_{n−1}^{R})
= γ_{n−1}(ζ_{n}^{R} − ζ_{n−1}^{R}).

(ii) By Theorem 2.5,

xP_{n}^{R}(x) = P_{n+1}^{R}(x)C_{n+1} + P_{n}^{R}(x)C_{n} + P_{n−1}^{R}(x)C_{n−1} + ⋯ + P_{0}^{R}C_{0}

with some matrices C_{0}, …, C_{n+1}. It is straightforward that C_{n+1} = 1 and C_{n} = ζ_{n}^{R} − ζ_{n+1}^{R}. By the orthogonality property (2.4), we find C_{0} = ⋯ = C_{n−2} = 0. Finally, it is easy to calculate C_{n−1}:

γ_{n} = ⟨⟨P_{n}^{R}, xP_{n−1}^{R}⟩⟩_{R} = ⟨⟨xP_{n}^{R}, P_{n−1}^{R}⟩⟩_{R}

= ⟨⟨P_{n+1}^{R} + P_{n}^{R}(ζ_{n}^{R} − ζ_{n+1}^{R}) + P_{n−1}^{R}C_{n−1}, P_{n−1}^{R}⟩⟩_{R}
= C_{n−1}^{†}γ_{n−1}

and so, taking adjoints and using self-adjointness of γ_{j}, C_{n−1} = γ_{n−1}^{−1}γ_{n}. This proves (2.18); the other relation (2.19) is obtained by conjugation.

□

2.1.5. Normalized Orthogonal Polynomials. We call p_{n}^{R} ∈ P a right orthonormal polynomial if deg p_{n}^{R} ≤ n and

⟨⟨p_{n}^{R}, f⟩⟩_{R} = 0 for every f ∈ P with deg f < n,  (2.20)
⟨⟨p_{n}^{R}, p_{n}^{R}⟩⟩_{R} = 1.  (2.21)

Similarly, we call p_{n}^{L} ∈ P a left orthonormal polynomial if deg p_{n}^{L} ≤ n and

⟨⟨p_{n}^{L}, f⟩⟩_{L} = 0 for every f ∈ P with deg f < n,  (2.22)
⟨⟨p_{n}^{L}, p_{n}^{L}⟩⟩_{L} = 1.  (2.23)

Lemma 2.7. Any orthonormal polynomial has the form

p_{n}^{R}(x) = P_{n}^{R}(x)⟨⟨P_{n}^{R}, P_{n}^{R}⟩⟩_{R}^{−1/2}σ_{n}, p_{n}^{L}(x) = τ_{n}⟨⟨P_{n}^{L}, P_{n}^{L}⟩⟩_{L}^{−1/2}P_{n}^{L}(x)

(2.24)

where σ_{n}, τ_{n} ∈ M_{l} are unitaries. In particular, deg p_{n}^{R} = deg p_{n}^{L} = n.

Proof. Let K_{n} be the coefficient of x^{n} in p_{n}^{R}. Consider the polynomial q(x) = P_{n}^{R}(x)K_{n}−p_{n}^{R}(x), where P_{n}^{R} is the monic orthogonal polynomial from Lemma 2.2. Then deg q < n and so from (2.4) and (2.20), it follows that ⟨⟨q, q⟩⟩_{R} = 0 and so q(x) vanishes identically. Thus, we have

1 = ⟨⟨p_{n}^{R}, p_{n}^{R}⟩⟩_{R} = K_{n}^{†}⟨⟨P_{n}^{R}, P_{n}^{R}⟩⟩_{R}K_{n}  (2.25)

and so det(K_{n}) ≠ 0. From (2.25) we get (K_{n}^{†})^{−1}K_{n}^{−1} = ⟨⟨P_{n}^{R}, P_{n}^{R}⟩⟩_{R}, and so K_{n}K_{n}^{†} = ⟨⟨P_{n}^{R}, P_{n}^{R}⟩⟩_{R}^{−1}. From here we get K_{n} = ⟨⟨P_{n}^{R}, P_{n}^{R}⟩⟩_{R}^{−1/2}σ_{n} with a unitary σ_{n}. The proof for p_{n}^{L} is similar. □

By Theorem 2.5, the polynomials p_{n}^{R} form a right orthonormal module basis in P_{R}. Thus, for any f ∈ P_{R}, we have

f(x) = ∑_{m=0}^{∞} p_{m}^{R}f_{m},  f_{m} = ⟨⟨p_{m}^{R}, f⟩⟩_{R}  (2.26)

and the Parseval identity

∑_{m=0}^{∞} Tr(f_{m}f_{m}^{†}) = ∥f∥_{R}^{2}  (2.27)

holds true. Obviously, since f is a polynomial, there are only finitely many non-zero terms in (2.26) and (2.27).

2.2. Block Jacobi Matrices. The study of block Jacobi matrices goes back at least to Krein [133].

2.2.1. Block Jacobi Matrices as Matrix Representations. Suppose that a sequence of unitary matrices 1 = σ_{0}, σ_{1}, σ_{2}, … is fixed, and p_{n}^{R} are defined according to (2.24). As noted above, p_{n}^{R} form a right orthonormal basis in P_{R}.

The map f(x) ↦ xf(x) can be considered as a right homomorphism in P_{R}. Consider the matrix J_{nm} of this homomorphism with respect to the basis p_{n}^{R}, that is,

J_{nm} = ⟨⟨p_{n−1}^{R}, xp_{m−1}^{R}⟩⟩_{R}.  (2.28)

Following Killip–Simon [128] and Simon [167, 168, 176], our Jacobi matrices are indexed with n = 1, 2, … but, of course, p_{n} has n = 0, 1, 2, . . .. That is why (2.28) has n − 1 and m − 1.

As in the scalar case, using the orthogonality properties of p_{n}^{R}, we get that J_{nm} = 0 if |n − m| > 1. Denote

B_{n} = J_{nn} = ⟨⟨p_{n−1}^{R}, xp_{n−1}^{R}⟩⟩_{R}

and

A_{n} = J_{n,n+1} = J_{n+1,n}^{†} = ⟨⟨p_{n−1}^{R}, xp_{n}^{R}⟩⟩_{R}.

Then we have

J = (\begin{matrix}B_{1} & A_{1} & 0 & ⋯ \\ A_{1}^{†} & B_{2} & A_{2} & ⋯ \\ 0 & A_{2}^{†} & B_{3} & ⋯ \\ ⋮ & ⋮ & ⋮ & ... \end{matrix})  (2.29)

Applying (2.26) to f(x) = xp_{n}^{R}(x), we get the recurrence relation

xp_{n}^{R}(x) = p_{n+1}^{R}(x)A_{n+1}^{†}+p_{n}^{R}(x)B_{n+1}+p_{n−1}^{R}(x)A_{n}, n = 1, 2, … (2.30)

If we set p_{−1}^{R}(x) = 0 and A_{0} = 1, the relation (2.30) also holds for n = 0. By (2.2), we can always pick p_{n}^{L} so that for x real, p_{n}^{L}(x) = p_{n}^{R}(x)^{†}, and thus for complex z,

p_{n}^{L}(z) = p_{n}^{R}(z̄)^{†}  (2.31)

by analytic continuation. By conjugating (2.30), we get

xp_{n}^{L}(x) = A_{n+1}p_{n+1}^{L}(x) + B_{n+1}p_{n}^{L}(x) + A_{n}^{†}p_{n−1}^{L}(x), n = 0, 1, 2, … (2.32)

Comparing this with the recurrence relations (2.18), (2.19), we get

A_{n} = σ_{n−1}^{†}γ_{n−1}^{−1/2}γ_{n}^{1/2}σ_{n}, B_{n} = σ_{n−1}^{†}γ_{n−1}^{1/2}(ζ_{n−1}^{R} − ζ_{n}^{R})γ_{n−1}^{−1/2}σ_{n−1}.

(2.33)

In particular, det A_{n} ≠ 0 for all n.

Notice that since σ_{n} is unitary, |det(σ_{n})| = 1, so (2.33) implies det(γ_{n}^{1/2}) = det(γ_{n−1}^{1/2})|det(A_{n})| which, by induction, implies that

det⟨⟨P_{n}^{R}, P_{n}^{R}⟩⟩ = |det(A_{1} … A_{n})|^{2}  (2.34)

Any block matrix of the form (2.29) with B_{n} = B_{n}^{†} and det A_{n} ≠ 0 for all n will be called a block Jacobi matrix corresponding to the Jacobi parameters A_{n} and B_{n}.

2.2.2. Basic Properties of Block Jacobi Matrices. Suppose we are given a block Jacobi matrix J corresponding to Jacobi parameters A_{n} and B_{n}, where B_{n} = B_{n}^{†} and det A_{n} ≠ 0 for each n.

Consider the Hilbert space H_{v} = ℓ^{2}(ℤ_{+}, ℂ^{l}) (here ℤ_{+} = {1, 2, 3, … }) with inner product

⟨f, g⟩_{H_{v}} = ∑_{n=1}^{∞}⟨f_{n}, g_{n}⟩_{ℂ^{l}}

and orthonormal basis {e_{k,j}}_{k∈ℤ_{+},1≤j≤l}, where

(e_{k,j})_{n} = δ_{k,n}v_{j}

and {v_{j}}_{1≤j≤l} is the standard basis of ℂ^{l}. J acts on H_{v} via

(Jf)_{n} = A_{n−1}^{†}f_{n−1} + B_{n}f_{n} + A_{n}f_{n+1}, f ∈ H_{v}  (2.35)

(with f_{0} = 0) and defines a symmetric operator on this space. Note that using invertibility of the A_{n}’s, induction shows

span{e_{k,j} : 1 ≤ k ≤ n, 1 ≤ j ≤ l} = span{J^{k−1}e_{1,j} : 1 ≤ k ≤ n, 1 ≤ j ≤ l}

(2.36)

for every n ≥ 1. We want to emphasize that elements of H_{v} and H are vector-valued and matrix-valued, respectively. For this reason, we will be interested in both matrix- and vector-valued solutions of the basic difference equations.

We will consider only bounded block Jacobi matrices, that is, those corresponding to Jacobi parameters satisfying

sup_{n} Tr(A_{n}^{†}A_{n} + B_{n}^{†}B_{n}) < ∞.  (2.37)

Equivalently,

sup_{n}(∥A_{n}∥ + ∥B_{n}∥) < ∞.  (2.38)

In this case, J is a bounded self-adjoint operator. This is equivalent to μ having compact support.

We call two Jacobi matrices J and J̃ equivalent if there exists a sequence of unitaries u_{n} ∈ M_{l}, n ≥ 1, with u_{1} = 1 such that J̃_{nm} = u_{n}^{†}J_{nm}u_{m}. From Lemma 2.7 it is clear that if p_{n}^{R}, p̃_{n}^{R} are two sequences of normalized orthogonal polynomials, corresponding to the same measure (but having different normalization), then the Jacobi matrices J_{nm} = ⟨⟨p_{n−1}^{R}, xp_{m−1}^{R}⟩⟩_{R} and J̃_{nm} = ⟨⟨p̃_{n−1}^{R}, xp̃_{m−1}^{R}⟩⟩_{R} are equivalent (u_{n} = σ_{n−1}^{†}σ̃_{n−1}). Thus,

B̃_{n} = u_{n}^{†}B_{n}u_{n},  Ã_{n} = u_{n}^{†}A_{n}u_{n+1}.  (2.39)

Therefore, we have a map

Φ : μ ↦{J : J_{mn} = ⟨⟨p_{n−1}^{R}, xp_{m−1}^{R}⟩⟩_{R}, p_{n}^{R}

correspond to dμ for some normalization}

(2.40)

from the set of all Hermitian positive semi-definite non-trivial compactly supported measures to the set of all equivalence classes of bounded block Jacobi matrices. Below, we will see how to invert this map.

2.2.3. Special Representatives of the Equivalence Classes. Let J be a block Jacobi matrix with the Jacobi parameters A_{n}, B_{n}. We say that J is:

• of type 1, if A_{n} > 0 for all n;
• of type 2, if A_{1}A_{2} … A_{n} > 0 for all n;
• of type 3, if A_{n} ∈ L for all n.

Here, L is the class of all lower triangular matrices with strictly positive elements on the diagonal. Type 3 is of interest because they correspond precisely to bounded Hermitian matrices with 2l + 1 non-vanishing diagonals with the extreme diagonals strictly positive; see Section 1.4(c). Type 2 is the case where the leading coefficients of p_{n}^{R} are strictly positive definite.

Theorem 2.8. (i) Each equivalence class of block Jacobi matrices

contains exactly one element each of type 1, type 2, or type 3. (ii) Let J be a block Jacobi matrix corresponding to a sequence of poly-

nomials p_{n}^{R} as in (2.24). Then J is of type 2 if and only if σ_{n} = 1 for all n.

Proof. The proof is based on the following two well-known facts:

(a) For any t ∈ M_{l} with det(t) ≠ 0, there exists a unique unitary u ∈ M_{l} such that tu is Hermitian positive semi-definite: tu ≥ 0.

(b) For any t ∈ M_{l} with det(t) ≠ 0, there exists a unique unitary u ∈ M_{l} such that tu ∈ L.

We first prove that every equivalence class of block Jacobi matrices contains at least one element of type 1. For a given sequence A_{n}, let us construct a sequence u_{1} = 1, u_{2}, u_{3}, … of unitaries such that u_{n}^{†}A_{n}u_{n+1} ≥ 0. By the existence part of (a), we find u_{2} such that A_{1}u_{2} ≥ 0, then find u_{3} such that u_{2}^{†}A_{2}u_{3} ≥ 0, etc. This, together with (2.39), proves the statement. In order to prove the uniqueness part, suppose we have A_{n} ≥ 0 and u_{n}^{†}A_{n}u_{n+1} ≥ 0 for all n. Then, by the uniqueness part of (a), A_{1} ≥ 0 and A_{1}u_{2} ≥ 0 imply u_{2} = 1; next, A_{2} ≥ 0 and u_{2}^{†}A_{2}u_{3} = A_{2}u_{3} ≥ 0 imply u_{3} = 1, etc.

The statement (i) concerning type 3 can be proven in the same way, using (b) instead of (a).

The statement (i) concerning type 2 can be proven similarly. Existence: find u_{2} such that A_{1}u_{2} ≥ 0, then u_{3} such that (A_{1}u_{2})(u_{2}^{†}A_{2}u_{3}) = A_{1}A_{2}u_{3} ≥ 0, etc. Uniqueness: if A_{1} … A_{n} ≥ 0 and A_{1} … A_{n}u_{n+1} ≥ 0, then u_{n+1} = 1.

By (2.33), we have A_{1}A_{2} … A_{n} = γ_{n}^{1/2}σ_{n} and the statement (ii) follows from the positivity of γ_{n}. □

We say that a block Jacobi matrix J belongs to the Nevai class if

B_{n} → 0 and A_{n}^{†}A_{n} → 1 as n → ∞.

It is clear that J is in the Nevai class if and only if all equivalent Jacobi matrices belong to the Nevai class.

Theorem 2.9. If J belongs to the Nevai class and is of type 1 or type 3, then A_{n} → 1 as n → ∞.

Proof. If J is of type 1, then A_{n}^{†}A_{n} = A_{n}^{2} → 1 clearly implies A_{n} → 1 since square root is continuous on positive Hermitian matrices.

Suppose J is of type 3. We shall prove that A_{n} → 1 by considering the rows of the matrix A_{n} one by one, starting from the lth row. Denote (A_{n})_{jk} = a_{j,k}^{(n)}. We have

(A_{n}^{†}A_{n})_{ll} = (a_{l,l}^{(n)})^{2} → 1, and so a_{l,l}^{(n)} → 1.

Then, for any k < l, we have

(A_{n}^{†}A_{n})_{lk} = a_{l,l}^{(n)}a_{l,k}^{(n)} → 0, and so a_{l,k}^{(n)} → 0.

Next, consider the (l − 1)st row. We have

(A_{n}^{†}A_{n})_{l−1,l−1} = (a_{l−1,l−1}^{(n)})^{2} + |a_{l,l−1}^{(n)}|^{2} → 1

and so, using the previous step, a_{l−1,l−1}^{(n)} → 1 as n → ∞. Then for all k < l − 1, we have

(A_{n}^{†}A_{n})_{l−1,k} = \overline{a_{l−1,l−1}^{(n)}} a_{l−1,k}^{(n)} + \overline{a_{l,l−1}^{(n)}} a_{l,k}^{(n)} → 0

and so, using the previous steps, a_{l−1,k} → 0. Continuing this way, we get a_{j,k}^{(n)} → δ_{j,k} as required. □

It is an interesting open question if this result also applies to the type 2 case.

2.2.4. Favard’s Theorem. Here we construct an inverse of the mapping Φ (defined by (2.40)). Thus, Φ sets up a bijection between non-trivial measures of compact support and equivalence classes of bounded block Jacobi matrices.

Before proceeding to do this, let us prove:

Lemma 2.10. The mapping Φ is injective.

Proof. Let μ and μ̃ be two Hermitian positive semi-definite non-trivial compactly supported measures. Suppose that Φ(μ) = Φ(μ̃).

Let p_{n}^{R} and p̃_{n}^{R} be normalized orthogonal polynomials corresponding to μ and μ̃. Suppose that the normalization both for p_{n}^{R} and for p̃_{n}^{R} has been chosen such that σ_{n} = 1 (see (2.24)), that is, type 2. From Lemma 2.8 and the assumption Φ(μ) = Φ(μ̃) it follows that the corresponding Jacobi matrices coincide, that is, ⟨⟨p_{n}^{R}, xp_{m}^{R}⟩⟩_{R} = ⟨⟨p̃_{n}^{R}, xp̃_{m}^{R}⟩⟩_{R} for all n and m. Together with the recurrence relation (2.30) this yields p_{n}^{R} = p̃_{n}^{R} for all n.

For any n ≥ 0, we can represent x^{n} as

x^{n} = ∑_{k=0}^{n} p_{k}^{R}(x)C_{k}^{(n)} = ∑_{k=0}^{n} p̃_{k}^{R}(x)C̃_{k}^{(n)}.

The coefficients C_{k}^{(n)} and C̃_{k}^{(n)} are completely determined by the coefficients of the polynomials p_{n}^{R} and p̃_{n}^{R} and so C_{k}^{(n)} = C̃_{k}^{(n)} for all n and k.

For the moments of the measure μ, we have

∫ x^{n}dμ(x) = ⟨⟨1, x^{n}⟩⟩_{R} = ∑_{k=0}^{n}⟨⟨1, p_{k}^{R}C_{k}^{(n)}⟩⟩_{R} = ⟨⟨1, 1⟩⟩_{R} C_{0}^{(n)} = C_{0}^{(n)}.

Since the same calculation is valid for the measure μ̃, we get

∫ x^{n}dμ(x) = ∫ x^{n}dμ̃(x)

for all n. It follows that

∫ f(x)dμ(x)g(x) = ∫ f(x)dμ̃(x)g(x)

for all matrix-valued polynomials f and g, and so the measures μ and μ̃ coincide. □

We can now construct the inverse of the map Φ. Let a block Jacobi matrix J be given. By a version of the spectral theorem for self-adjoint operators with finite multiplicity (see, e.g., [2, Sect. 72]), there exists a matrix-valued measure dμ with

⟨e_{1,j}, f(J)e_{1,k}⟩_{H_{v}} = ∫ f(x) dμ_{j,k}(x)  (2.41)

and an isometry

R : H_{v} → L^{2}(ℝ, dμ; ℂ^{l})

such that (recall that {v_{j}} is the standard basis in ℂ^{l})

[Re_{1,j}](x) = v_{j}, 1 ≤ j ≤ l,  (2.42)

and, for any g ∈ H_{v}, we have

(RJg)(x) = x(Rg)(x).  (2.43)

If the Jacobi matrices J and J̃ are equivalent, then we have J̃ = U^{∗}JU for some U = ⊕_{n=1}^{∞}u_{n}, u_{1} = 1. Thus,

⟨e_{1,j}, f(J̃)e_{1,k}⟩_{H_{v}} = ⟨Ue_{1,j}, f(J)Ue_{1,k}⟩_{H_{v}} = ⟨e_{1,j}, f(J)e_{1,k}⟩_{H_{v}}

and so the measures corresponding to J and J̃ coincide. Thus, we have a map

Ψ: {J̃: J̃ is equivalent to J} ↦ μ  (2.44)

from the set of all equivalence classes of bounded block Jacobi matrices to the set of all Hermitian positive semi-definite compactly supported measures.

Theorem 2.11. (i) All measures in the image of the map Ψ are non-

degenerate. (ii) Φ ∘ Ψ = id. (iii) Ψ ∘ Φ = id.

Proof. (i) To put things in the right context, we first recall that ∥·∥_{H_{v}} is a norm (rather than a semi-norm), whereas ||| · ||| on V (cf. (2.3)) is, in general, a semi-norm. Using the assumption that det(A_{k}) ≠ 0 for all k (which is included in our definition of a Jacobi matrix), we will prove that ||| · ||| is in fact a norm. More precisely, we will prove that |||p||| > 0 for any polynomial p ∈ V; by Lemma 2.1 this will imply that μ is non-degenerate.

Let p ∈ V be a non-zero polynomial, deg p = n. Notice that (2.42) and (2.43) give

[RJ^{k}e_{1,j}](x) = x^{k}v_{j}  (2.45)

for every k ≥ 0 and 1 ≤ j ≤ l. This shows that p can be represented as p = Rg, where g = ∑_{k=0}^{n} J^{k}f_{k}, and f_{0}, …, f_{n} are vectors in H_{v} such that ⟨f_{i}, e_{j,k}⟩_{H_{v}} = 0 for all i = 0, …, n, j ≥ 2, k = 1, …, l (i.e., the only non-zero components of f_{j} are in the first ℂ^{l} in H_{v}). Assumption deg p = n means f_{n} ≠ 0.

Since R is isometric, we have |||p||| = ∥g∥_{H_{v}}, and so we have to prove that g ≠ 0. Indeed, suppose that g = 0. Using the assumption det(A_{k}) ≠ 0 and the tri-diagonal nature of J, we see that ∑_{k=0}^{n} J^{k}f_{k} = 0 yields f_{n} = 0, contrary to our assumption.

(ii) Consider the elements Re_{n,k} ∈ L^{2}(ℝ, dμ; ℂ^{l}). First note that, by (2.36) and (2.45), Re_{n,k} is a polynomial of degree at most n − 1. Next, by the unitarity of R, we have

⟨Re_{n,k}, Re_{m,j}⟩_{L^{2}(ℝ,dμ;ℂ^{l})} = δ_{m,n}δ_{k,j}.  (2.46)

Let us construct matrix-valued polynomials q_{n}(x), using Re_{n,1}, Re_{n,2}, …, Re_{n,l} as columns of q_{n−1}(x):

[q_{n−1}(x)]_{j,k} = [Re_{n,k}(x)]_{j}.

We have deg q_{n} ≤ n and ⟨⟨q_{m}, q_{n}⟩⟩_{R} = δ_{m,n}1; the last relation is just a reformulation of (2.46). Hence the q_{n}’s are right normalized orthogonal polynomials with respect to the measure dμ. We find

J_{nm} = [⟨e_{n,j}, Je_{m,k}⟩_{H_{v}}]_{1≤j,k≤l}
= [⟨Re_{n,j}, RJe_{m,k}⟩_{L^{2}(ℝ,dμ;ℂ^{l})}]_{1≤j,k≤l}
= [⟨Re_{n,j}, xRe_{m,k}⟩_{L^{2}(ℝ,dμ;ℂ^{l})}]_{1≤j,k≤l}
= [⟨[q_{n−1}(x)]_{·,j}, x[q_{m−1}(x)]_{·,k}⟩_{L^{2}(ℝ,dμ;ℂ^{l})}]_{1≤j,k≤l}
= ⟨⟨q_{n−1}, xq_{m−1}⟩⟩_{R}

as required.

(iii) Follows from (ii) and from Lemma 2.10. □

2.3. The m-Function.

2.3.1. The Definition of the m-Function. We denote the Borel transform of dμ by m:

m(z) = ∫ \frac{dμ(x)}{x − z} ,  Im z > 0.  (2.47)

It is a matrix-valued Herglotz function, that is, it is analytic and obeys Im m(z) > 0. For information on matrix-valued Herglotz functions, see [98] and references therein. Extensions to operator-valued Herglotz functions can be found in [94].

Lemma 2.12. Suppose dμ is given, p_{n}^{R} are right normalized orthogonal polynomials, and J is the associated block Jacobi matrix. Then,

m(z) = ⟨⟨p_{0}^{R}, (x − z)^{−1}p_{0}^{R}⟩⟩_{R}  (2.48)

and

m(z) = ⟨e_{1,·}, (J − z)^{−1}e_{1,·}⟩_{H_{v}}.  (2.49)

Proof. Since p_{0}^{R} = 1, (2.48) is just a way of rewriting the definition of m. The second identity, (2.49), is a consequence of (2.41) and Theorem 2.11(iii). □

2.3.2. Coefficient Stripping. If J is a block Jacobi matrix corresponding to invertible A_{n}’s and Hermitian B_{n}’s, we denote the k-times stripped block Jacobi matrix, corresponding to {A_{k+n}, B_{k+n}}_{n≥1}, by J^{(k)}. That is,

J^{(k)} = (\begin{matrix}B_{k+1} & A_{k+1} & 0 & ⋯ \\ A_{k+1}^{†} & B_{k+2} & A_{k+2} & ⋯ \\ 0 & A_{k+2}^{†} & B_{k+3} & ⋯ \\ ⋮ & ⋮ & ⋮ & ... \end{matrix})

The m-function corresponding to J^{(k)} will be denoted by m^{(k)}. Note that, in particular, J^{(0)} = J and m^{(0)} = m.

Proposition 2.13. Let J be a block Jacobi matrix with σ_{ess}(J) ⊆ [a, b]. Then, for every ε > 0, there is k_{0} ≥ 0 such that for k ≥ k_{0}, we have that σ(J^{(k)}) ⊆ [a − ε, b + ε].

Proof. This is an immediate consequence of (the proof of) [42, Lemma 1]. □

Proposition 2.14 (Due to Aptekarev–Nikishin [4]). We have that

m^{(k)}(z)^{−1} = B_{k+1} − z − A_{k+1}m^{(k+1)}(z)A_{k+1}^{†}

for Im z > 0 and k ≥ 0.

Proof. It suffices to handle the case k = 0. Given (2.49), this is a special case of a general formula for 2 × 2 block operator matrices, due to Schur [163], that reads

(\begin{matrix}A & B \\ C & D \end{matrix})^{−1} = (\begin{matrix}(A − BD^{−1}C)^{−1} & −A^{−1}B(D − CA^{−1}B)^{−1} \\ −D^{−1}C(A − BD^{−1}C)^{−1} & (D − CA^{−1}B)^{−1} \end{matrix})

and which is readily verified. Here A = B_{1} − z, B = A_{1}, C = A_{1}^{†}, and D = J^{(1)} − z. □

2.4. Second Kind Polynomials. Define the second kind polynomials by q_{−1}^{R}(z) = −1,

q_{n}^{R}(z) = ∫_{ℝ} dμ(x)\frac{p_{n}^{R}(z) − p_{n}^{R}(x)}{z − x}, n = 0, 1, 2, …

As is easy to see, for n ≥ 1, q_{n}^{R} is a polynomial of degree n − 1. For future reference, let us display the first several polynomials p_{n}^{R} and q_{n}^{R}:

p_{−1}^{R}(x) = 0, p_{0}^{R}(x) = 1, p_{1}^{R}(x) = (x − B_{1})A_{1}^{−1},  (2.50)
q_{−1}^{R}(x) = −1, q_{0}^{R}(x) = 0, q_{1}^{R}(x) = A_{1}^{−1}.  (2.51)

The polynomials q_{n}^{R} satisfy the equation (same form as (2.30))

xq_{n}^{R}(x) = q_{n+1}^{R}(x)A_{n+1}^{†} + q_{n}^{R}(x)B_{n+1} + q_{n−1}^{R}(x)A_{n}, n = 0, 1, 2, …

(2.52)

For n = 0, this can be checked by a direct substitution of (2.51). For n ≥ 1, as in the scalar case, this can be checked by taking (2.30) for x and for z, subtracting, dividing by x − z, integrating over dμ, and taking into account the orthogonality relation

∫ dμ(x)p_{n}^{R}(x) = 0, n ≥ 1.

Finally, let us define

ψ_{n}^{R}(z) = q_{n}^{R}(z) + m(z)p_{n}^{R}(z).

According to the definition of q_{n}^{R}, we have

ψ_{n}^{R}(z) = ⟨⟨f_{z}, p_{n}^{R}⟩⟩_{R}, f_{z}(x) = (x − z̄)^{−1}.

By the Parseval identity, this shows that for all Im z > 0, the sequence ψ_{n}^{R}(z) is in ℓ^{2}, that is,

∑_{n=0}^{∞} Tr(ψ_{n}^{R}(z)^{†}ψ_{n}^{R}(z)) < ∞.  (2.53)

In the same way, we define q_{−1}^{L}(z) = −1,

q_{n}^{L}(z) = ∫_{ℝ} \frac{p_{n}^{L}(z) − p_{n}^{L}(x)}{z − x}dμ(x) , n = 0, 1, 2, …

and ψ_{n}^{L}(z) = q_{n}^{L}(z) + p_{n}^{L}(z)m(z).

2.5. Solutions to the Difference Equations. For Im z > 0, consider the solutions to the equations

zu_{n}(z) = ∑_{m=1}^{∞} u_{m}(z)J_{mn}, n = 2, 3, …  (2.54)

zv_{n}(z) = ∑_{m=1}^{∞} J_{nm}v_{m}(z), n = 2, 3, …  (2.55)

Clearly, u_{n}(z) solves (2.54) if and only if v_{n}(z) = (u_{n}(z̄))^{†} solves (2.55). In the above, we normally intend z as a fixed parameter but it then can be used as a variable. That is, z is fixed and u_{n}(z) is a fixed sequence, not a z-dependent function. A statement like v_{n}(z) = (u_{n}(z̄))^{†} means if u_{n} is a sequence solving (2.54) for z = z̄_{0}, then v_{n} obeys (2.55) for z = z_{0}. Of course, if u_{n}(z) is a function of z in a region, we can apply our estimates to all z in the region. For any solution {u_{n}(z)}_{n=1}^{∞} of (2.54), let us define

u_{0}(z) = zu_{1}(z) − u_{1}(z)B_{1} − u_{2}(z)A_{1}^{†}.  (2.56)

With this definition, the equation (2.54) for n = 1 is equivalent to u_{0}(z) = 0. In the same way, we set

v_{0}(z) = zv_{1}(z) − B_{1}v_{1}(z) − A_{1}v_{2}(z).

Lemma 2.15. Let Im z > 0 and suppose {u_{n}(z)}_{n=0}^{∞} solves (2.54) (for n ≥ 2) and (2.56) and belongs to ℓ^{2}. Then

(Im z) ∑_{n=1}^{∞} Tr(u_{n}(z)^{†}u_{n}(z)) = − Im Tr(u_{1}(z)u_{0}(z)^{†}).  (2.57)

In particular, u_{n}(z) = αp_{n−1}^{R}(z) is in ℓ^{2} only if α = 0.

Proof. Denote s_{n} = Tr(u_{n}(z)A_{n−1}^{†}u_{n−1}(z)^{†}). Here A_{0} = 1. Multiplying (2.54) for n ≥ 2 and (2.56) for n = 1 by u_{n}(z)^{†} on the right, taking traces, and summing over n, we get

z ∑_{n=1}^{N} Tr(u_{n}(z)u_{n}(z)^{†}) = ∑_{n=1}^{N} s_{n+1} + ∑_{n=1}^{N} Tr(u_{n}(z)B_{n+1}u_{n}(z)^{†}) + ∑_{n=1}^{N} \overline{s_{n}} .

Taking imaginary parts and letting N → ∞, we obtain (2.57) since the middle sum is real and the outer sums cancel up to boundary terms. Applying (2.57) to u_{n}(z) = αp_{n−1}^{R}(z), we get zero in the right-hand side:

(Im z) ∑_{n=1}^{∞} Tr(αp_{n−1}^{R}(z)p_{n−1}^{R}(z)^{†}α^{†}) = 0

and hence α = 0 since p_{0}^{R} = 1. □

Theorem 2.16. Let Im z > 0.

(i) Any solution {u_{n}(z)}_{n=0}^{∞} of (2.54) (for n ≥ 2) can be represented

as

u_{n}(z) = ap_{n−1}^{R}(z) + bq_{n−1}^{R}(z)  (2.58)

for suitable a, b ∈ M_{l}. In fact, a = u_{1}(z) and b = −u_{0}(z). (ii) A sequence (2.58) satisfies (2.54) for all n ≥ 1 if and only if b = 0. (iii) A sequence (2.58) belongs to ℓ^{2} if and only if u_{n}(z) = cψ_{n−1}^{R}(z) for

some c ∈ M_{l}. Equivalently, a sequence (2.58) belongs to ℓ^{2} if and only if u_{1}(z) + u_{0}(z)m(z) = 0.

Proof. (i) Let u_{n}(z) be a solution to (2.54). Consider

ũ_{n}(z) = u_{n}(z) − u_{1}(z)p_{n−1}^{R}(z) + u_{0}(z)q_{n−1}^{R}(z).

Then ũ_{n}(z) also solves (2.54) and ũ_{0}(z) = ũ_{1}(z) = 0. It follows that ũ_{n}(z) = 0 for all n. This proves (i).

(ii) A direct substitution of (2.58) into (2.54) for n = 1 yields the statement.

(iii) We already know that cψ_{n−1}^{R} is an ℓ^{2} solution. Suppose that u_{n}(z) is an ℓ^{2} solution to (2.54). Rewrite (2.58) as

u_{n}(z) = (a − bm(z))p_{n−1}^{R}(z) + bψ_{n−1}^{R}(z).

Since ψ_{n}^{R} is in ℓ^{2} and cp_{n}^{R} is not in ℓ^{2}, we get a = bm(z), which is equivalent to u_{1}(z) + u_{0}(z)m(z) = 0 or to u_{n}(z) = bψ_{n−1}^{R}(z). □

By conjugation, we obtain:

Theorem 2.17. Let Im z > 0.

(i) Any solution {v_{n}(z)}_{n=0}^{∞} of (2.55) (for n ≥ 2) can be represented

as

v_{n}(z) = p_{n−1}^{L}(z)a + q_{n−1}^{L}(z)b.  (2.59)

In fact, a = v_{1}(z) and b = −v_{0}(z). (ii) A sequence (2.59) satisfies (2.55) for all n ≥ 1 if and only if b = 0. (iii) A sequence (2.59) belongs to ℓ^{2} if and only if v_{n}(z) = ψ_{n−1}^{L}(z)c for

some c ∈ M_{l}. Equivalently, a sequence (2.59) belongs to ℓ^{2} if and only if v_{1}(z) + m(z)v_{0}(z) = 0.

2.6. Wronskians and the Christoffel–Darboux Formula. For any two M_{l}-valued sequences u_{n}, v_{n}, define the Wronskian by

W_{n}(u, v) = u_{n}A_{n}v_{n+1} − u_{n+1}A_{n}^{†}v_{n}.  (2.60)

Note that W_{n}(u, v) = −W_{n}(v^{†}, u^{†})^{†}. If u_{n}(z) and v_{n}(z) are solutions to (2.54) and (2.55), then by a direct calculation, we see that W_{n}(u(z), v(z)) is independent of n. Put differently, if both u_{n}(z) and v_{n}(z) are solutions to (2.54), then W_{n}(u(z), v(z̄)^{†}) is independent of n.

Or, if both u_{n}(z) and v_{n}(z) are solutions to (2.55), then W_{n}(u(z̄)^{†}, v(z)) is independent of n. In particular, by a direct evaluation for n = 0, we get

W_{n}(p_{·−1}^{R}(z), p_{·−1}^{R}(z̄)^{†}) = W_{n}(q_{·−1}^{R}(z), q_{·−1}^{R}(z̄)^{†}) = 0,
W_{n}(p_{·−1}^{L}(z̄)^{†}, p_{·−1}^{L}(z)) = W_{n}(q_{·−1}^{L}(z̄)^{†}, q_{·−1}^{L}(z)) = 0,
W_{n}(p_{·−1}^{R}(z), q_{·−1}^{R}(z̄)^{†}) = W_{n}(p_{·−1}^{L}(z̄)^{†}, q_{·−1}^{L}(z)) = 1.

Let both u(z) and v(z) be solutions to (2.54) of the type (2.58), namely,

u_{n}(z) = ap_{n−1}^{R}(z) + bq_{n−1}^{R}(z), v_{n}(z) = cp_{n−1}^{R}(z) + dq_{n−1}^{R}(z).

Then the above calculation implies

W_{n}(u(z), v(z̄)^{†}) = ad^{†} − bc^{†}.

Theorem 2.18 (CD Formula). For any x, y ∈ ℂ and n ≥ 1, one has

(x − y) ∑_{k=0}^{n} p_{k}^{R}(x)p_{k}^{L}(y) = −W_{n+1}(p_{·−1}^{R}(x), p_{·−1}^{L}(y)).  (2.61)

Proof. Multiplying (2.30) by p_{n}^{L}(y) on the right and (2.32) (with y in place of x) by p_{n}^{R}(x) on the left and subtracting, we get

(x − y)p_{n}^{R}(x)p_{n}^{R}(y) = W_{n}(p_{·−1}^{R}(x), p_{·−1}^{L}(y)) − W_{n+1}(p_{·−1}^{R}(x), p_{·−1}^{L}(y)).

Summing over n and noting that W_{0}(p^{R}(x), p^{L}(y)) = 0, we get the required statement. □

2.7. The CD Kernel. The CD kernel is defined for z, w ∈ ℂ by

K_{n}(z, w) = ∑_{k=0}^{n} p_{k}^{R}(z)p_{k}^{R}(w̄)^{†}  (2.62)

= ∑_{k=0}^{n} p_{k}^{L}(z̄)^{†}p_{k}^{L}(w).  (2.63)

(2.63) follows from (2.62) and (2.31). Notice that K is independent of the choices σ_{n}, τ_{n} in (2.24) and that (2.61) can be written

(z − w̄)K_{n}(z, w) = −W_{n+1}(p_{·−1}^{R}(z), p_{·−1}^{R}(w̄)^{†}).  (2.64)

The independence of K_{n} of σ, τ can be understood by noting that if f_{m} is given by (2.26), then

∫ K_{n}(z, w) dμ(w)f(w) = ∑_{m=0}^{n} p_{m}^{R}(z)f_{m}  (2.65)

so K is the kernel of the orthogonal projection onto polynomials of degree up to n, and so intrinsic. Similarly, if f_{m}^{(L)} = ⟨⟨f, p_{m}^{L}⟩⟩_{L}, so

f(x) = ∑_{m=0}^{∞} f_{m}^{(L)}p_{m}^{L}(x)  (2.66)

then, by (2.63),

∫ f(z) dμ(z)K_{n}(z, w) = ∑_{m=0}^{n} f_{m}^{(L)}p_{m}^{L}(w).  (2.67)

One has

∫ K_{n}(z, w) dμ(w)K_{n}(w, ζ) = K_{n}(z, ζ)  (2.68)

as can be checked directly and which is an expression of the fact that the map in (2.65) is a projection, and so its own square.

We will let π_{n} be the map of L^{2}(dμ) to itself given by (2.65) (or by (2.67)). (2.64) can then be viewed (for z, w ∈ ℝ) as an expression of the integral kernel of the commutator [J, π_{n}], which leads to another proof of it [175].

Let J_{n;F} be the finite nl × nl matrix obtained from J by taking the top leftmost n^{2} blocks. It is the matrix of π_{n−1}M_{x}π_{n−1} where M_{x} is multiplication by x in the {p_{j}^{R}}_{j=0}^{n−1} basis. For y ∈ ℂ and γ ∈ ℂ^{l}, let φ_{n,γ}(y) be the vector whose components are

(φ_{n,γ}(y))_{j} = p_{j−1}^{L}(y)γ  (2.69)

for j = 1, 2, …, n. We claim that

[(J_{n;F} − y)φ_{n,γ}(y)]_{j} = −δ_{jn}A_{n}p_{n}^{L}(y)γ  (2.70)

as follows immediately from (2.32).

This is intimately related to (2.61) and (2.64). For recalling J is the matrix in p^{R} basis, φ_{n,γ}(y) corresponds to the function

∑_{j=0}^{n−1} p_{j}^{R}(x)(φ_{n,γ}(y))_{j−1} = K_{n}(x, y)γ.

As we will see in the next two sections, (2.70) has important consequences.

2.8. Christoffel Variational Principle. There is a matrix version of the Christoffel variational principle (see Nevai [152] for a discussion of uses in the scalar case; this matrix case is discussed by Duran–Polo [76]):

Theorem 2.19. For any non-trivial l × l matrix- valued measure, dμ, on ℝ, we have that for any n, any x_{0} ∈ ℝ, and matrix polynomials Q_{n}(x) of degree at most n with

Q_{n}(x_{0}) = 1  (2.71)

we have that

⟨⟨Q_{n}, Q_{n}⟩⟩_{R} ≥ K_{n}(x_{0}, x_{0})^{−1}  (2.72)

with equality if and only if

Q_{n}(x) = K_{n}(x, x_{0})K_{n}(x_{0}, x_{0})^{−1}  (2.73)

Remark. (2.72) also holds for ⟨⟨·, ·⟩⟩_{L} but the minimizer is then K_{n}(x_{0}, x_{0})^{−1}K_{n}(x, x_{0}).

Proof. Let Q_{n}^{(0)} denote the right-hand side of (2.73). Then for any polynomial R_{n} of degree at most n, we have

⟨⟨Q_{n}^{(0)}, R_{n}⟩⟩_{R} = K_{n}(x_{0}, x_{0})^{−1}R_{n}(x_{0})  (2.74)

because of (2.65). Since Q_{n}(x_{0}) = Q_{n}^{(0)}(x_{0}) = 1, we conclude

⟨⟨Q_{n} − Q_{n}^{(0)}, Q_{n} − Q_{n}^{(0)}⟩⟩_{R} = ⟨⟨Q_{n}, Q_{n}⟩⟩_{R} − K_{n}(x_{0}, x_{0})^{−1}  (2.75)

from which (2.72) is immediate and, given the supposed non- triviality, the uniqueness of minimizer. □

With this, one easily gets an extension of a result of Máté–Nevai [146] to MOPRL. (They had it for scalar OPUC. For OPUC, it is in Máté–Nevai–Totik [147] on [−1, 1] and in Totik [187] for general OPRL. The result below can be proven using polynomial mappings à la Totik [188] or Jost solutions à la Simon [174].)

Theorem 2.20. Let dμ be a non-trivial l × l matrix-valued measure on ℝ with compact support, E. Let I = (a, b) be an open interval with I ⊂ E. Then for Lebesgue a.e. x ∈ I,

lim sup(n + 1)K_{n}(x, x)^{−1} ≤ w(x)  (2.76)

Remark. This is intended in terms of expectations in any fixed vector.

We state this explicitly since we will need it in Section 5.4, but we note that the more detailed results of Máté–Nevai–Totik [147], Lubinsky [141], Simon [173], and Totik [189] also extend.

2.9. Zeros. We next look at zeros of det(P_{n}^{L}(z)), which we will prove soon is also det(P_{n}^{R}(z)). Following [66, 177], we will identify these zeros with eigenvalues of J_{n;F}. It is not obvious a priori that these zeros are real and, unlike the scalar situation, where the classical arguments on the zeros rely on orthogonality, we do not know how to get reality just from that (but see the remark after Theorem 2.25).

Lemma 2.21. Let C(z) be an l × l matrix-valued function analytic near z = 0. Let

k = dim(ker(C(0))).  (2.77)

Then det(C(z)) has a zero at z = 0 of order at least k.

Remarks. 1. Even in the 1 × 1 case, where k = 1, the zeros can clearly be of higher order than k since c_{11}(z) can have a zero of any order!

2. The temptation to take products of eigenvalues will lead at best to a complicated proof as the cases C(z) = ( \begin{matrix}0 & z \\ 1 & 0 \end{matrix} ) and C(z) = ( \begin{matrix}0 & z^{2} \\ 1 & 0 \end{matrix} ) illustrate.

Proof. Let e_{1} … e_{l} be an orthonormal basis with e_{1} … e_{k} ∈ ker(C(0)). By Hadamard’s inequality (see Bhatia [13]),

|det(C(z))| ≤ ∥C(z)e_{1}∥ … ∥C(z)e_{l}∥
≤ C|z|^{k}

since ∥C(z)e_{j}∥ ≤ C|z| if j = 1, …, k and ∥C(z)e_{j}∥ ≤ d for j = k + 1, …, l. □

The following goes back at least to [34]; see also [66, 165, 177, 178].

Theorem 2.22. We have that

det_{ℂ^{l}}(P_{n}^{L}(z)) = det_{ℂ^{nl}}(z − J_{n;F}).  (2.78)

Proof. By (2.70), if γ is such that p_{n}^{L}(y)γ = 0, then φ_{n,γ}(y) is an eigenvector for J_{n;F} with eigenvalue y. Conversely, if φ is an eigenvector and γ is defined as that vector in ℂ^{l} whose components are the first l components of φ, then a simple inductive argument shows φ = φ_{n,γ}(y) and then, by (2.70) and the fact that A_{n} is invertible, we see that p_{n}^{L}(y)γ = 0. This shows that for any y,

dim ker(P_{n}^{L}(y)) = dim ker(J_{n;F} − y).  (2.79)

By Lemma 2.21, if y is any eigenvalue of J_{n;F} of multiplicity k, then det(P_{n}^{L}(z)) has a zero of order at least k at y. Now let us consider the polynomials in z on the left and right in (2.78). Since J_{n;F} is Hermitian, the sum of the multiplicities of the zeros on the right is nl. Since the polynomial on the left is of degree nl, by a counting argument it has the same set of zeros with the same multiplicities as the polynomial on the right. Since both polynomials are monic, they are equal. □

Corollary 2.23. All the zeros of det(P_{n}^{L}(z)) are real. Moreover,

det(P_{n}^{R}(z)) = det(P_{n}^{L}(z)).  (2.80)

Proof. Since J_{n;F} is Hermitian, all its eigenvalues are real, so (2.78) implies the zeros of det(P_{n}^{L}(z)) are real. Thus, since the polynomial is monic,

\overline{det(P_{n}^{L}(z̄))} = det(P_{n}^{L}(z)).  (2.81)

By Lemma 2.2(v), we have

P_{n}^{R}(z) = P_{n}^{L}(z̄)^{†}  (2.82)

since both sides are analytic and agree if z is real. Thus,

det(P_{n}^{R}(z)) = det(P_{n}^{L}(z̄)^{†}) = \overline{det(P_{n}^{L}(z̄))}

proving (2.80). □

The following appeared before in [178]; see also [165].

Corollary 2.24. Let {x_{n,j}}_{j=1}^{nl} be the zeros of det(P_{n}^{L}(x)) counting multiplicity ordered by

x_{n,1} ≤ x_{n,2} ≤ ⋯ ≤ x_{n,nl}.  (2.83)

Then

x_{n+1,j} ≤ x_{n,j} ≤ x_{n+1,j+l}.  (2.84)

Remarks. 1. This is interlacing if l = 1 and replaces it for general l.

2. Using A_{n} invertible, one can show the inequalities in (2.84) are strict.

Proof. The min-max principle [159] says that

x_{n,j} = max_{\begin{matrix}L⊂ℂ^{nl} \\ dim(L)≤jl \end{matrix}} sup_{\begin{matrix}f∈L^{⊥} \\ ∥f=1∥ \end{matrix}}⟨f, J_{n;F} f⟩_{ℂ^{nl}}  (2.85)

If P: ℂ^{(n+1)l} → ℂ^{nl} is the natural projection, then ⟨P f, J_{n+1;F} P f⟩_{ℂ^{(n+1)l}} = ⟨P f, J_{n;F} P f⟩_{ℂ^{nl}} and as L runs through all subspaces of ℂ^{(n+1)l} dimension at most jl, P [L] runs through all subspaces of dimension at most jl in ℂ^{nl}, so (2.85) implies x_{n+1,j} ≤ x_{n,j}. Using the same argument on −J_{n;F} and −J_{n+1;F} shows x_{j}(−J_{n;F}) ≥ x_{j}(−J_{n+1;F}). But x_{j}(−J_{n;F}) = −x_{nl+1−j}(J_{n;F}) and x_{j}(−J_{n+1;F}) = −x_{(n+1)l+1−j}(J_{n+1;F}). That yields the other inequality. □

2.10. Lower Bounds on p and the Stieltjes–Weyl Formula for m. Next, we want to exploit (2.70) to prove uniform lower bounds on ∥p_{n}^{L}(y)γ∥ when y ∉ cvh(σ(J)), the convex hull of the support of J, and thereby uniform bounds ∥p_{n}^{L}(y)^{−1}∥. We will then use that to prove that for z ∉ σ(J), we have

m(z) = lim_{n→∞} −p_{n}^{L}(z)^{−1}q_{n}^{L}(z)  (2.86)

the matrix analogue of a formula that spectral theorists associate with Weyl’s definition of the m-function [193], although for the discrete case, it goes back at least to Stieltjes [181].

We begin by mimicking an argument from [172]. Let H = cvh(σ(J)) = [c − D, c + D] with

D = \frac{1}{2} diam(H).  (2.87)

By the definition of A_{n},

∥A_{n}∥ = ∥⟨⟨p_{n−1}^{R}, (x − c)p_{n}^{R}⟩⟩_{R}∥ ≤ D.  (2.88)

Suppose y ∉ H and let

d = dist(y, H).  (2.89)

By the spectral theorem, for any vector φ ∈ H_{v},

|⟨φ, (J − y)φ⟩_{H_{v}}| ≥ d∥φ∥^{2}.  (2.90)

By (2.70), with φ = φ_{n,γ}(y),

|⟨φ, (J − y)φ⟩| ≤ ∥A_{n}∥ ∥p_{n}^{L}(y)γ∥ ∥p_{n−1}^{L}(y)γ∥  (2.91)

while

∥φ∥^{2} = ∑_{j=0}^{n−1} ∥p_{j}^{L}(y)γ∥^{2}.  (2.92)

As in [172, Prop. 2.2], we get:

Theorem 2.25. If y ∉ H, for any γ,

∥p_{n}^{L}(y)γ∥ ≥ (\frac{d}{D})(1 + (\frac{d}{D})^{2})^{(n−1)/2}∥γ∥.  (2.93)

In particular,

∥p_{n}^{L}(y)^{−1}∥ ≤ \frac{D}{d} .  (2.94)

Remark. (2.93) implies det(p_{n}^{L}(y)) ≠ 0 if Im y > 0, providing another proof that its zeros are real.

By the discussion after (2.52), if Im z > 0, q_{n}^{L}(z) + p_{n}^{L}(z)m(z) is in ℓ^{2}, so goes to zero. Since p_{n}^{L}(z)^{−1} is bounded, we obtain:

Corollary 2.26. For any z ∈ ℂ_{+} = {z | Im z > 0},

m(z) = lim_{n→∞} −p_{n}^{L}(z)^{−1}q_{n}^{L}(z).  (2.95)

Remark. This holds for z ∉ H.

Taking adjoints using (2.82) and m(z)^{†} = m(z̄), we see that

m(z) = lim_{n→∞} −q_{n}^{R}(z)p_{n}^{R}(z)^{−1}.  (2.96)

2.11. Wronskians of Vector-Valued Solutions. Let α, β be two vector-valued solutions (ℂ^{l}) of (2.55) for n = 2, 3, . . .. Define their scalar Wronskian as (Euclidean inner product on ℂ^{l})

W_{n}(α, β) = ⟨α_{n}, A_{n}β_{n+1}⟩ − ⟨A_{n}α_{n+1}, β_{n}⟩  (2.97)

for n = 2, 3, . . .. One can obtain two matrix solutions by using α or β for one column and 0 for the other columns. The scalar Wronskian is just a matrix element of the resulting matrix Wronskian, so W_{n} is constant (as can also be seen by direct calculation). Here is an application:

Theorem 2.27. Let z_{0} ∈ ℝ \ σ_{ess}(J). For k = 0, 1, let q_{k} be the multiplicity of z_{0} as an eigenvalue of J^{(k)}. Then, q_{0} + q_{1} ≤ l.

Proof. If β̃ is an eigenfunction for J^{(1)} and we define β by

β_{n} = \begin{cases}0, & n = 1, \\ β̃_{n−1}, & n ≥ 2, \end{cases}  (2.98)

then β solves (2.55) for n ≥ 2. If α is an eigenfunction for J = J^{(0)}, it also solves (2.55). Since α_{n} → 0, β_{n} → 0, and A_{n} is bounded, W_{n}(α, β) → 0 as n → ∞ and so it is identically zero. But since β_{1} = 0,

0 = W_{1}(α, β) = ⟨α_{1}, A_{1}β_{2}⟩ = ⟨α_{1}, A_{1}β̃_{1}⟩.  (2.99)

Let V^{(k)} be the set of values of eigenfunctions of J^{(k)} at n = 1. (2.99) says

V^{(0)} ⊂ [A_{1}V^{(1)}]^{⊥}.  (2.100)

Since q_{k} = dim(V^{(k)}) and A_{1} is non-singular, (2.100) implies that q_{0} ≤ l − q_{1}. □

2.12. The Order of Zeros/Poles of m(z).

Theorem 2.28. Let z_{0} ∈ ℝ \ σ_{ess}(J). For k = 0, 1, let q_{k} be the multiplicity of z_{0} as an eigenvalue of J^{(k)}. If q_{1}−q_{0} ≥ 0, then det(m(z)) has a zero at z = z_{0} of order q_{1} − q_{0}. If q_{1} − q_{0} < 0, then det(m(z)) has a pole at z = z_{0} of order q_{0} − q_{1}.

Remarks. 1. To say det(m(z)) has a zero at z = z_{0} of order 0 means it is finite and non-vanishing at z_{0}!

2. Where dμ is a direct sum of scalar measures, so is m(z), and det(m(z)) is then a product. In the scalar case, m(z) has a pole at z_{0} if J^{(0)} has z_{0} as eigenvalue and a zero at z_{0} if J^{(1)} has z_{0} as eigenvalue. In the direct sum case, we see there can be cancellations, which helps explain why q_{1} − q_{0} occurs.

3. Formally, one can understand this theorem as follows. Cramer’s rule suggests det(m(z)) = det(J^{(1)} − z)/ det(J^{(0)} − z). Even though det(J^{(k)} − z) is not well-defined in the infinite case, we expect a cancellation of zeros of order q_{1} and q_{0}. For z_{0} ∉ H, the convex hull of σ(J^{(0)}), one can use (2.95) to prove the theorem following this intuition. Spurious zeros in gaps of σ(J^{(0)}) make this strategy difficult in gaps.

4. Unlike in Lemma 2.21, we can write m as a product of eigenvalues and analyze that directly because m(x) is self-adjoint for x real, which sidesteps some of the problems associated with non-trivial Jordan normal forms.

5. This proof gives another demonstration that q_{0} + q_{1} ≤ l.

Proof. m(z) has a simple pole at z = z_{0} with a residue which is rank q_{0} and strictly negative definite on its range. Let

f(z) = (z − z_{0})m(z).

f is analytic near z_{0} and self-adjoint for z real and near z_{0}. Thus, by eigenvalue perturbation theory [124, 159], f(z) has l eigenvalues ρ_{1}(z), …, ρ_{l}(z) analytic near z_{0} with ρ_{1}, ρ_{2}, …, ρ_{q_{0}} non-zero at z_{0} and ρ_{q_{0}+1}, …, ρ_{l} zero at z_{0}.

Thus, m(z) has l eigenvalues near z_{0}, λ_{j}(z) = ρ_{j}(z)/(z − z_{0}), where λ_{1}, …, λ_{q_{0}} have simple poles and the others are regular.

By Proposition 2.14, m(z)^{−1} has a simple pole at z_{0} with residue of rank q_{1} (because A_{1} is non-singular), so m(z)^{−1} has q_{1} eigenvalues with poles. That means q_{1} of λ_{q_{0}+1}, …, λ_{l} have simple zeros at z_{0} and the others are non-zero. Thus, det(m(z)) = ∏_{j=1}^{l} λ_{j}(z) has a pole/zero of order q_{0} − q_{1}. □

2.13. Resolvent of the Jacobi Matrix. Consider the matrix

G_{nm}(z) = ⟨⟨p_{n−1}^{R}, (x − z)^{−1}p_{m−1}^{R}⟩⟩_{R}.

Theorem 2.29. One has

G_{nm}(z) = \begin{cases}ψ_{n−1}^{L}(z)p_{m−1}^{R}(z), & if n ≥ m, \\ p_{n−1}^{L}(z)ψ_{m−1}^{R}(z), & if n ≤ m. \end{cases}  (2.101)

Proof. We have

∑_{m=1}^{∞} G_{km}(z)J_{mn} = zG_{kn}(z), n ≠ k,

∑_{m=1}^{∞} J_{nm}G_{mk}(z) = zG_{nk}(z), n ≠ k.

Fix k ≥ 0 and let u_{m}(z) = G_{km}(z). Then u_{m}(z) satisfies the equation (2.54) for n ≠ k, and so we can use Theorem 2.16 to describe this solution. First suppose k > 1. As u_{m} is an ℓ^{2} solution and u_{m} satisfies (2.54) for n = 1, we have

G_{km}(z) = \begin{cases}a_{k}(z)p_{m−1}^{R}(z), & m ≤ k, \\ b_{k}(z)ψ_{m−1}^{R}(z), & m ≥ k. \end{cases}  (2.102)

If k = 1, (2.102) also holds true. For m ≥ k, this follows by the same argument, and for m = k = 1, this is a trivial statement. Next, similarly, let us consider v_{m}(z) = G_{mk}(z). Then v_{m}(z) solves (2.55) and so, using Theorem 2.17, we obtain

G_{mk}(z) = \begin{cases}p_{m−1}^{L}(z)c_{k}(z), & m ≤ k, \\ ψ_{m−1}^{L}(z)d_{k}(z), & m ≥ k. \end{cases}  (2.103)

Comparing (2.102) and (2.103), we find

a_{k}(z)p_{m−1}^{R}(z) = ψ_{k−1}^{L}(z)d_{m}(z),
b_{k}(z)ψ_{m−1}^{R}(z) = p_{k−1}^{L}(z)c_{m}(z).

As p_{0}^{R} = p_{0}^{L} = 1, it follows that

a_{k}(z) = ψ_{k−1}^{L}(z)d_{1}(z), c_{m}(z) = b_{1}(z)ψ_{m−1}^{R}(z)

and so we obtain

G_{nm}(z) = \begin{cases}ψ_{n−1}^{L}(z)d_{1}(z)p_{m−1}^{R}(z) & if n ≥ m, \\ p_{n−1}^{L}(z)b_{1}(z)ψ_{m−1}^{R}(z) & if n ≤ m. \end{cases}  (2.104)

It remains to prove that

b_{1}(z) = d_{1}(z) = 1.  (2.105)

Consider the case m = n = 1. By the definition of the resolvent,

G_{11}(z) = ⟨⟨p_{0}^{R}, (J − z)^{−1}p_{0}^{R}⟩⟩_{R} = ∫ \frac{dμ(x)}{x − z} = m(z).

On the other hand, by (2.104),

G_{11}(z) = ψ_{0}^{L}(z)d_{1}(z)p_{0}^{R}(z) = m(z)d_{1}(z),

G_{11}(z) = p_{0}^{L}(z)b_{1}(z)ψ_{0}^{R}(z) = b_{1}(z)m(z),

which proves (2.105). □

3. Matrix Orthogonal Polynomials on the Unit Circle

3.1. Definition of MOPUC. In this chapter, μ is an l × l matrixvalued measure on ∂𝔻. ⟨⟨·, ·⟩⟩_{R} and ⟨⟨·, ·⟩⟩_{L} are defined as in the MOPRL case. Non-triviality is defined as for MOPRL. We will always assume μ is non-trivial. We define monic matrix polynomials Φ_{n}^{R}, Φ_{n}^{L} by applying Gram–Schmidt to {1, z1, … }, that is, Φ_{n}^{R} is the unique matrix polynomial z^{n}1+ lower order with

⟨⟨z^{k}1, Φ_{n}^{R}⟩⟩_{R} = 0 k = 0, 1, …, n − 1.  (3.1)

We will define the normalized MOPUC shortly. We will only consider the analogue of what we called type 1 for MOPRL because only those appear to be useful. Unlike in the scalar case, the monic polynomials do not appear much because it is for the normalized, but not monic, polynomials that the left and right Verblunsky coefficients are the same.

3.2. The Szegő Recursion. Szegő [184] included the scalar Szegő recursion for the first time. It seems likely that Geronimus had it independently shortly after Szegő. Not knowing of the connection with this work, Levinson [138] rederived the recursion but with matrix coefficients! So the results of this section go back to 1947.

For a matrix polynomial P_{n} of degree n, we define the reversed polynomial P_{n}^{∗} by

P_{n}^{∗}(z) = z^{n}P_{n}(1/z̄)^{†}.  (3.2)

Notice that

(P_{n}^{∗})^{∗} = P_{n}  (3.3)

and for any α ∈ M_{l},

(αP_{n})^{∗} = P_{n}^{∗}α^{†},  (P_{n}α)^{∗} = α^{†}P_{n}^{∗}.  (3.4)

Lemma 3.1. We have

⟨⟨f, g⟩⟩_{L} = ⟨⟨g, f⟩⟩_{L}^{†} ,  ⟨⟨f, g⟩⟩_{R} = ⟨⟨g, f⟩⟩_{R}^{†}  (3.5)

and

⟨⟨f^{∗}, g^{∗}⟩⟩_{L} = ⟨⟨f, g⟩⟩_{R}^{†} , ⟨⟨f^{∗}, g^{∗}⟩⟩_{R} = ⟨⟨f, g⟩⟩_{L}^{†} .  (3.6)

Proof. The first and second identities follow immediately from the definition. The third identity is derived as follows:

⟨⟨f^{∗}, g^{∗}⟩⟩_{L} = ∫ e^{inθ}g(θ)^{†} dμ(θ) (e^{inθ}f(θ)^{†})^{†}

= ∫ e^{inθ}g(θ)^{†} dμ(θ) e^{−inθ}f(θ)
= ∫ g(θ)^{†} dμ(θ) f(θ)
= ⟨⟨g, f⟩⟩_{R}
= ⟨⟨f, g⟩⟩_{R}^{†}.

The proof of the last identity is analogous. □

Lemma 3.2. If P_{n} has degree n and is left-orthogonal with respect to z1, … z^{n}1, then P_{n} = c(Φ_{n}^{R})^{∗} for some suitable matrix c.

Proof. By assumption,

0 = ⟨⟨P_{n}, z^{j}1⟩⟩_{L} = ⟨⟨(z^{j}1)^{∗}, P_{n}^{∗}⟩⟩_{R} = ⟨⟨z^{n−j}1, P_{n}^{∗}⟩⟩_{R} for 1 ≤ j ≤ n.

Thus, P_{n}^{∗} is right-orthogonal with respect to 1, z1, …, z^{n−1}1 and hence it is a right-multiple of Φ_{n}^{R}. Consequently, P_{n} is a left- multiple of (Φ_{n}^{R})^{∗}. □

Let us define normalized orthogonal matrix polynomials by

φ_{0}^{L} = φ_{0}^{R} = 1, φ_{n}^{L} = κ_{n}^{L}Φ_{n}^{L} and φ_{n}^{R} = Φ_{n}^{R}κ_{n}^{R}

where the κ’s are defined according to the normalization condition

⟨⟨φ_{n}^{R}, φ_{m}^{R}⟩⟩_{R} = δ_{nm}1 ⟨⟨φ_{n}^{L}, φ_{m}^{L}⟩⟩_{L} = δ_{nm}1

along with (a type 1 condition)

κ_{n+1}^{L}(κ_{n}^{L})^{−1} > 0 and (κ_{n}^{R})^{−1}κ_{n+1}^{R} > 0.  (3.7)

Notice that κ_{n}^{L} are determined by the normalization condition up to multiplication on the left by unitaries; these unitaries can always be uniquely chosen so as to satisfy (3.7).

Now define

ρ_{n}^{L} = κ_{n}^{L}(κ_{n+1}^{L})^{−1} and ρ_{n}^{R} = (κ_{n+1}^{R})^{−1}κ_{n}^{R}.

Notice that as inverses of positives matrices, ρ_{n}^{L} > 0 and ρ_{n}^{R} > 0. In particular, we have that

κ_{n}^{L} = (ρ_{n−1}^{L} … ρ_{0}^{L})^{−1} and κ_{n}^{R} = (ρ_{0}^{R} … ρ_{n−1}^{R})^{−1}.

Theorem 3.3 (Szegő Recursion). (a) For suitable matrices α_{n}^{L,R}, one has

zφ_{n}^{L} − ρ_{n}^{L}φ_{n+1}^{L} = (α_{n}^{L})^{†}φ_{n}^{R,∗},  (3.8)
zφ_{n}^{R} − φ_{n+1}^{R}ρ_{n}^{R} = φ_{n}^{L,∗}(α_{n}^{R})^{†}.  (3.9)

(b) The matrices α_{n}^{L} and α_{n}^{R} are equal and will henceforth be denoted by α_{n}.

(c) ρ_{n}^{L} = (1 − α_{n}^{†}α_{n})^{1/2} and ρ_{n}^{R} = (1 − α_{n}α_{n}^{†})^{1/2}.

Proof. (a) The matrix polynomial zφ_{n}^{L} has leading term z^{n+1}κ_{n}^{L}. On the other hand, the matrix polynomial ρ_{n}^{L}φ_{n+1}^{L} has leading term z^{n+1}ρ_{n}^{L}κ_{n+1}^{L}. By definition of ρ_{n}^{L}, these terms are equal. Consequently, zφ_{n}^{L} − ρ_{n}^{L}φ_{n+1}^{L} is a matrix polynomial of degree at most n. Notice that it is left-orthogonal with respect to z1, … z^{n}1 since

⟨⟨zφ_{n}^{L} − ρ_{n}^{L}φ_{n+1}^{L}, z^{j}1⟩⟩_{L} = ⟨⟨φ_{n}^{L}, z^{j−1}1⟩⟩_{L} − ⟨⟨ρ_{n}^{L}φ_{n+1}^{L}, z^{j}1⟩⟩_{L} = 0 − 0 = 0.

Now apply Lemma 3.2. The other claim is proved in the same way.

(b) By part (a) and identities established earlier,

(α_{n}^{L})^{†} = 0 + (α_{n}^{L})^{†}1

= ⟨⟨φ_{n}^{R,∗}, ρ_{n}^{L}φ_{n+1}^{L}⟩⟩_{L} + (α_{n}^{L})^{†}⟨⟨φ_{n}^{R}, φ_{n}^{R}⟩⟩_{R}

= ⟨⟨φ_{n}^{R,∗}, ρ_{n}^{L}φ_{n+1}^{L}⟩⟩_{L} + (α_{n}^{L})^{†}⟨⟨φ_{n}^{R,∗}, φ_{n}^{R,∗}⟩⟩_{L} by (3.6)

= ⟨⟨φ_{n}^{R,∗}, ρ_{n}^{L}φ_{n+1}^{L} + (α_{n}^{L})^{†}φ_{n}^{R,∗}⟩⟩_{L}

= ⟨⟨φ_{n}^{R,∗}, zφ_{n}^{L}⟩⟩_{L}

= ⟨⟨zφ_{n}^{R}, φ_{n}^{L,∗}⟩⟩_{R}^{†} (using the (n + 1)-degree *)

= ⟨⟨φ_{n+1}^{R}ρ_{n}^{R} + φ_{n}^{L,∗}(α_{n}^{R})^{†}, φ_{n}^{L,∗}⟩⟩_{R}^{†}

= ⟨⟨φ_{n+1}^{R}ρ_{n}^{R}, φ_{n}^{L,∗}⟩⟩_{R}^{†} + ⟨⟨φ_{n}^{L,∗}(α_{n}^{R})^{†}, φ_{n}^{L,∗}⟩⟩_{R}^{†}

= 0 + ⟨⟨φ_{n}^{L,∗}, φ_{n}^{L,∗}(α_{n}^{R})^{†}⟩⟩_{R}

= ⟨⟨φ_{n}^{L,∗}, φ_{n}^{L,∗}⟩⟩_{R} (α_{n}^{R})^{†}

= ⟨⟨φ_{n}^{L}, φ_{n}^{L}⟩⟩_{L} (α_{n}^{R})^{†}

= (α_{n}^{R})^{†}.

(c) Using parts (a) and (b), we see that

1 = ⟨⟨zφ_{n}^{L}, zφ_{n}^{L}⟩⟩_{L}

= ⟨⟨ρ_{n}^{L}φ_{n+1}^{L} + α_{n}^{†}φ_{n}^{R,∗}, ρ_{n}^{L}φ_{n+1}^{L} + α_{n}^{†}φ_{n}^{R,∗}⟩⟩_{L}

= ρ_{n}^{L}⟨⟨φ_{n+1}^{L}, φ_{n+1}^{L}⟩⟩_{L}(ρ_{n}^{L})^{†} + α_{n}^{†}⟨⟨φ_{n}^{R,∗}, φ_{n}^{R,∗}⟩⟩_{L} α_{n}

= (ρ_{n}^{L})^{2} + α_{n}^{†}⟨⟨φ_{n}^{R}, φ_{n}^{R}⟩⟩_{R} α_{n}

= (ρ_{n}^{L})^{2} + α_{n}^{†}α_{n}.

A similar calculation yields the other claim. □

The matrices α_{n} will henceforth be called the Verblunsky coefficients associated with the measure dμ. Since ρ_{n}^{L} is invertible, we have

∥α_{n}∥ < 1.  (3.10)

We will eventually see (Theorem 3.12) that any set of α_{n}’s obeying (3.10) occurs as the set of Verblunsky coefficients for a unique nontrivial measure.

Note that the Szegő recursion for the monic orthogonal polynomials is

\begin{aligned}zΦ_{n}^{L} − Φ_{n+1}^{L} & = (κ_{n}^{L})^{−1}α_{n}^{†}(κ_{n}^{R})^{†}Φ_{n}^{R,∗}, \\ zΦ_{n}^{R} − Φ_{n+1}^{R} & = Φ_{n}^{L,∗}(κ_{n}^{L})^{†}α_{n}^{†}(κ_{n}^{R})^{−1}, \end{aligned}  (3.11)

so the coefficients in the L and R equations are not equal and depend on all the α_{j}, j = 1, …, n.

Let us write the Szegő recursion in matrix form, starting with leftorthogonal polynomials. By Theorem 3.3,

φ_{n+1}^{L} = (ρ_{n}^{L})^{−1}[zφ_{n}^{L} − α_{n}^{†}φ_{n}^{R,∗}],
φ_{n+1}^{R} = [zφ_{n}^{R} − φ_{n}^{L,∗}α_{n}^{†}](ρ_{n}^{R})^{−1}.

which implies that

φ_{n+1}^{L} = z(ρ_{n}^{L})^{−1}φ_{n}^{L} − (ρ_{n}^{L})^{−1}α_{n}^{†}φ_{n}^{R,∗},  (3.12)
φ_{n+1}^{R,∗} = (ρ_{n}^{R})^{−1}φ_{n}^{R,∗} − z(ρ_{n}^{R})^{−1}α_{n}φ_{n}^{L}.  (3.13)

In other words,

(\begin{matrix}φ_{n+1}^{L} \\ φ_{n+1}^{R,∗} \end{matrix}) = A^{L}(α_{n}, z) (\begin{matrix}φ_{n}^{L} \\ φ_{n}^{R,∗} \end{matrix})  (3.14)

where

A^{L}(α, z) = (\begin{matrix}z(ρ^{L})^{−1} & −(ρ^{L})^{−1}α^{†} \\ −z(ρ^{R})^{−1}α & (ρ^{R})^{−1} \end{matrix})

and ρ^{L} = (1 − α^{†}α)^{1/2}, ρ^{R} = (1 − αα^{†})^{1/2}. Note that, for z ≠ 0, the inverse of A^{L}(α, z) is given by

A^{L}(α, z)^{−1} = (\begin{matrix}z^{−1}(ρ^{L})^{−1} & z^{−1}(ρ^{L})^{−1}α^{†} \\ (ρ^{R})^{−1}α & (ρ^{R})^{−1} \end{matrix})

which gives rise to the inverse Szegő recursion (first emphasized in the scalar and matrix cases by Delsarte el al. [37])

φ_{n}^{L} = z^{−1}(ρ_{n}^{L})^{−1}φ_{n+1}^{L} + z^{−1}(ρ_{n}^{L})^{−1}α_{n}^{†}φ_{n+1}^{R,∗},
φ_{n}^{R,∗} = (ρ_{n}^{R})^{−1}α_{n}φ_{n+1}^{L} + (ρ_{n}^{R})^{−1}φ_{n+1}^{R,∗}.

For right-orthogonal polynomials, we find the following matrix formulas. By Theorem 3.3,

φ_{n+1}^{R} = zφ_{n}^{R}(ρ_{n}^{R})^{−1} − φ_{n}^{L,∗}α_{n}^{†}(ρ_{n}^{R})^{−1},  (3.15)
φ_{n+1}^{L,∗} = φ_{n}^{L,∗}(ρ_{n}^{L})^{−1} − zφ_{n}^{R}α_{n}(ρ_{n}^{L})^{−1}.  (3.16)

In other words,

(φ_{n+1}^{R} φ_{n+1}^{L,∗}) = (φ_{n}^{R} φ_{n}^{L,∗}) A^{R}(α_{n}, z)

where

A^{R}(α, z) = (\begin{matrix}z(ρ^{R})^{−1} & −zα(ρ^{L})^{−1} \\ −α^{†}(ρ^{R})^{−1} & (ρ^{L})^{−1} \end{matrix})

For z ≠ 0, the inverse of A^{R}(α, z) is given by

A^{R}(α, z)^{−1} = (\begin{matrix}z^{−1}(ρ^{R})^{−1} & (ρ^{R})^{−1}α \\ z^{−1}(ρ^{L})^{−1}α^{†} & (ρ^{L})^{−1} \end{matrix})

and hence

φ_{n}^{R} = z^{−1}φ_{n+1}^{R}(ρ_{n}^{R})^{−1} + z^{−1}φ_{n+1}^{L,∗}(ρ_{n}^{L})^{−1}α_{n}^{†},  (3.17)
φ_{n}^{L,∗} = φ_{n+1}^{R}(ρ_{n}^{R})^{−1}α_{n} + φ_{n+1}^{L,∗}(ρ_{n}^{L})^{−1}.  (3.18)

3.3. Second Kind Polynomials. In the scalar case, second kind polynomials go back to Geronimus [89, 91, 92]. For n ≥ 1, let us introduce the second kind polynomials ψ_{n}^{L,R} by

ψ_{n}^{L}(z) = ∫ \frac{e^{iθ} + z}{e^{iθ} − z} (φ_{n}^{L}(e^{iθ}) − φ_{n}^{L}(z)) dμ(θ),  (3.19)
ψ_{n}^{R}(z) = ∫ \frac{e^{iθ} + z}{e^{iθ} − z} dμ(θ) (φ_{n}^{R}(e^{iθ}) − φ_{n}^{R}(z)).  (3.20)

For n = 0, let us set ψ_{0}^{L}(z) = ψ_{0}^{R}(z) = 1. For future reference, let us display the first polynomials of each series:

φ_{1}^{L}(z) = (ρ_{0}^{L})^{−1}(z − α_{0}^{†}),  φ_{1}^{R}(z) = (z − α_{0}^{†})(ρ_{0}^{R})^{−1},  (3.21)
ψ_{1}^{L}(z) = (ρ_{0}^{L})^{−1}(z + α_{0}^{†}),  ψ_{1}^{R}(z) = (z + α_{0}^{†})(ρ_{0}^{R})^{−1}.  (3.22)

We will also need formulas for ψ_{n}^{L,∗} and ψ_{n}^{R,∗}, n ≥ 1. These formulas follow directly from the above definition and from

\overline{(\frac{e^{iθ} + 1/z̄}{e^{iθ} − 1/z̄})} = −\frac{e^{iθ} + z}{e^{iθ} − z} .

Indeed, we have

ψ_{n}^{L,∗}(z) = z^{n} ∫ \frac{e^{iθ} + z}{e^{iθ} − z} dμ(θ) (φ_{n}^{L}(1/z̄)^{†} − φ_{n}^{L}(e^{iθ})^{†}),  (3.23)

ψ_{n}^{R,∗}(z) = z^{n} ∫ \frac{e^{iθ} + z}{e^{iθ} − z} (φ_{n}^{R}(1/z̄)^{†} − φ_{n}^{R}(e^{iθ})^{†}) dμ(θ).  (3.24)

Proposition 3.4. The second kind polynomials obey the recurrence relations

ψ_{n+1}^{L}(z) = (ρ_{n}^{L})^{−1}(zψ_{n}^{L}(z) + α_{n}^{†}ψ_{n}^{R,∗}(z)),  (3.25)
ψ_{n+1}^{R,∗}(z) = (ρ_{n}^{R})^{−1}(zα_{n}ψ_{n}^{L}(z) + ψ_{n}^{R,∗}(z)),  (3.26)

and

ψ_{n+1}^{R}(z) = (zψ_{n}^{R}(z) + ψ_{n}^{L,∗}(z)α_{n}^{†})(ρ_{n}^{R})^{−1},  (3.27)
ψ_{n+1}^{L,∗}(z) = (ψ_{n}^{L,∗}(z) + zψ_{n}^{R}(z)α_{n})(ρ_{n}^{L})^{−1}  (3.28)

for n ≥ 0.

Proof. 1. Let us check (3.25) for n ≥ 1. Denote the right-hand side of (3.25) by ψ̃_{n+1}^{L}(z). Using the recurrence relations for φ_{n}^{L}, φ_{n}^{R,∗} and the definition (3.19) of ψ_{n}^{L}, we find

ψ_{n+1}^{L}(z) − ψ̃_{n+1}^{L}(z) = ∫ \frac{e^{iθ} + z}{e^{iθ} − z} A_{n}(θ, z) dμ(θ)

where

A_{n}(θ, z) = φ_{n+1}^{L}(e^{iθ}) − φ_{n+1}^{L}(z)
− (ρ_{n}^{L})^{−1}[zφ_{n}^{L}(e^{iθ}) − zφ_{n}^{L}(z) + α_{n}^{†}z^{n}φ_{n}^{R}(1/z̄)^{†} − α_{n}^{†}z^{n}φ_{n}^{R}(e^{iθ})^{†}]
= (ρ_{n}^{L})^{−1}[e^{iθ}φ_{n}^{L}(e^{iθ}) − α_{n}^{†}φ_{n}^{R,∗}(e^{iθ}) − zφ_{n}^{L}(z) + α_{n}^{†}φ_{n}^{R,∗}(z)
− zφ_{n}^{L}(e^{iθ}) + zφ_{n}^{L}(z) − α_{n}^{†}φ_{n}^{R,∗}(z) + α_{n}^{†}z^{n}φ_{n}^{R}(e^{iθ})^{†}]
= (ρ_{n}^{L})^{−1}[(e^{iθ} − z)φ_{n}^{L}(e^{iθ}) + α_{n}^{†}(z^{n}e^{−inθ} − 1)φ_{n}^{R,∗}(e^{iθ})].

Using the orthogonality relations

∫ φ_{n}^{L}(e^{iθ}) dμ(θ)e^{−imθ} = ∫ φ_{n}^{R,∗}(e^{iθ}) dμ(θ)e^{−i(m+1)θ} = 0,  (3.29)

m = 0, 1, …, n − 1, and the formula

\frac{e^{inθ} − z^{n}}{e^{iθ} − z} = e^{i(n−1)θ} + e^{i(n−2)θ}z + ⋯ + z^{n−1}

we obtain

ρ_{n}^{L} ∫ \frac{e^{iθ} + z}{e^{iθ} − z} A_{n}(θ, z) dμ(θ) = ∫ (e^{iθ}φ_{n}^{L}(e^{iθ}) − α_{n}^{†}φ_{n}^{R,∗}(e^{iθ})) dμ(θ)
= ρ_{n}^{L} ∫ φ_{n+1}^{L}(e^{iθ}) dμ(θ) = 0.

2. Let us check (3.26) for n ≥ 1. Denote the right-hand side of (3.26) by ψ̃_{n+1}^{R,∗}(z). Similarly to the argument above, we find

ψ_{n+1}^{R,∗}(z) − ψ̃_{n+1}^{R,∗}(z) = ∫ \frac{e^{iθ} + z}{e^{iθ} − z} B_{n}(θ, z) dμ(θ),

where

B_{n}(θ, z) = z^{n+1}φ_{n+1}^{R}(1/z̄)^{†} − z^{n+1}φ_{n+1}^{R}(e^{iθ})^{†}
− (ρ_{n}^{R})^{−1}[zα_{n}φ_{n}^{L}(e^{iθ}) − zα_{n}φ_{n}^{L}(z) + z^{n}φ_{n}^{R}(1/z̄)^{†} − z^{n}φ_{n}^{R}(e^{iθ})^{†}]

= (ρ_{n}^{R})^{−1}[φ_{n}^{R,∗}(z) − α_{n}zφ_{n}^{L}(z) − z^{n+1}e^{−i(n+1)θ}φ_{n}^{R,∗}(e^{iθ}) + z^{n+1}e^{−inθ}α_{n}φ_{n}^{L}(e^{iθ})

− zα_{n}φ_{n}^{L}(e^{iθ}) + zα_{n}φ_{n}^{L}(z) − φ_{n}^{R,∗}(z) + z^{n}e^{−inθ}φ_{n}^{R,∗}(e^{iθ})]

= (ρ_{n}^{R})^{−1}[zα_{n}(z^{n}e^{−inθ} − 1)φ_{n}^{L}(e^{iθ}) + z^{n}e^{−inθ}(1 − ze^{−iθ})φ_{n}^{R,∗}(e^{iθ})].

Using the orthogonality relations (3.29), we get

ρ_{n}^{R} ∫ \frac{e^{iθ} + z}{e^{iθ} − z} B_{n}(θ, z) dμ(θ) =

= z^{n+1} ∫ (e^{−i(n+1)θ}φ_{n}^{R,∗}(e^{iθ}) − α_{n}e^{−inθ}φ_{n}^{L}(e^{iθ})) dμ(θ)

= z^{n+1}ρ_{n}^{R} ∫ e^{−i(n+1)θ}φ_{n+1}^{R,∗}(e^{iθ}) dμ(θ) = 0.

3. Relations (3.25) and (3.26) can be checked for n = 0 by a direct substitution of (3.21) and (3.22).

4. We obtain (3.27) and (3.28) from (3.25) and (3.26) by applying the ∗-operation. □

Writing the above recursion in matrix form, we get

(\begin{matrix}ψ_{n+1}^{L} \\ ψ_{n+1}^{R,∗} \end{matrix}) = A^{L}(−α_{n}, z) (\begin{matrix}ψ_{n}^{L} \\ ψ_{n}^{R,∗} \end{matrix}) , (\begin{matrix}ψ_{0}^{L} \\ ψ_{0}^{R,∗} \end{matrix}) = (\begin{matrix}1 \\ 1 \end{matrix})

for left-orthogonal polynomials and

(ψ_{n+1}^{R} ψ_{n+1}^{L,∗}) = (ψ_{n}^{R} ψ_{n}^{L,∗}) A^{R}(−α_{n}, z), (ψ_{0}^{R} ψ_{0}^{L,∗}) = (1 1) .

for right-orthogonal polynomials.

Equivalently,

(\begin{matrix}ψ_{n+1}^{L} \\ −ψ_{n+1}^{R,∗} \end{matrix}) = A^{L}(α_{n}, z) (\begin{matrix}ψ_{n}^{L} \\ −ψ_{n}^{R,∗} \end{matrix}) ,  (\begin{matrix}ψ_{0}^{L} \\ −ψ_{0}^{R,∗} \end{matrix}) = (\begin{matrix}1 \\ −1 \end{matrix}) (3.30)

and

(ψ_{n+1}^{R} −ψ_{n+1}^{L,∗}) = (ψ_{n}^{R} −ψ_{n}^{L,∗}) A^{R}(α_{n}, z), (ψ_{0}^{R} −ψ_{0}^{L,∗}) = (1 −1) .

In particular, we see that the second kind polynomials ψ_{n}^{L,R} correspond to Verblunsky coefficients {−α_{n}}. We have the following Wronskian-type relations:

Proposition 3.5. For n ≥ 0 and z ∈ ℂ, we have

2z^{n}1 = φ_{n}^{L}(z)ψ_{n}^{L,∗}(z) + ψ_{n}^{L}(z)φ_{n}^{L,∗}(z),  (3.31)
2z^{n}1 = ψ_{n}^{R,∗}(z)φ_{n}^{R}(z) + φ_{n}^{R,∗}(z)ψ_{n}^{R}(z).  (3.32)
0 = φ_{n}^{L}(z)ψ_{n}^{R}(z) − ψ_{n}^{L}(z)φ_{n}^{R}(z),  (3.33)
0 = ψ_{n}^{R,∗}(z)φ_{n}^{L,∗}(z) − φ_{n}^{R,∗}(z)ψ_{n}^{L,∗}(z).  (3.34)

Proof. We prove this by induction. The four identities clearly hold for n = 0. Suppose (3.31)–(3.34) hold for some n ≥ 0. Then,

φ_{n+1}^{L}ψ_{n+1}^{L,∗} + ψ_{n+1}^{L}φ_{n+1}^{L,∗} =
= (ρ_{n}^{L})^{−1} [(zφ_{n}^{L} − α_{n}^{†}φ_{n}^{R,∗})(ψ_{n}^{L,∗} + zψ_{n}^{R}α_{n})
+ (zψ_{n}^{L} + α_{n}^{†}ψ_{n}^{R,∗})(φ_{n}^{L,∗} − zφ_{n}^{R}α_{n})] (ρ_{n}^{L})^{−1}

= (ρ_{n}^{L})^{−1} [z(φ_{n}^{L}ψ_{n}^{L,∗} + ψ_{n}^{L}φ_{n}^{L,∗}) − zα_{n}^{†}(ψ_{n}^{R,∗}φ_{n}^{R} + φ_{n}^{R,∗}ψ_{n}^{R})α_{n}

+ α_{n}^{†}(ψ_{n}^{R,∗}φ_{n}^{L,∗} − φ_{n}^{R,∗}ψ_{n}^{L,∗}) + z^{2}(φ_{n}^{L}ψ_{n}^{R} − ψ_{n}^{L}φ_{n}^{R})α_{n}] (ρ_{n}^{L})^{−1}

= (ρ_{n}^{L})^{−1} [2z^{n+1}(1 − α_{n}^{†}α_{n})] (ρ_{n}^{L})^{−1} = 2z^{n+1}1,

where we used (3.31)–(3.34) for n in the third step. Thus, (3.31) holds for n + 1.

For (3.32), we note that

ψ_{n+1}^{R,∗}φ_{n+1}^{R} + φ_{n+1}^{R,∗}ψ_{n+1}^{R} =
= (ρ_{n}^{R})^{−1}[(zα_{n}ψ_{n}^{L} + ψ_{n}^{R,∗})(zφ_{n}^{R} − φ_{n}^{L,∗}α_{n}^{†})
+ (φ_{n}^{R,∗} − zα_{n}φ_{n}^{L})(zψ_{n}^{R} + ψ_{n}^{L,∗}α_{n}^{†})] (ρ_{n}^{R})^{−1}

= (ρ_{n}^{R})^{−1}[z(ψ_{n}^{R,∗}φ_{n}^{R} + φ_{n}^{R,∗}ψ_{n}^{R}) − zα_{n}(ψ_{n}^{L}φ_{n}^{L,∗} + φ_{n}^{L}ψ_{n}^{L,∗})α_{n}^{†}

+ z^{2}α_{n}(ψ_{n}^{L}φ_{n}^{R} − φ_{n}^{L}ψ_{n}^{R}) − (ψ_{n}^{R,∗}φ_{n}^{L,∗} − φ_{n}^{R,∗}ψ_{n}^{L,∗})α_{n}^{†}] (ρ_{n}^{R})^{−1}

= (ρ_{n}^{R})^{−1}2z^{n+1}(1 − α_{n}α_{n}^{†}) (ρ_{n}^{R})^{−1} = 2z^{n+1}1,

again using (3.31)–(3.34) for n in the third step.

Next,

φ_{n+1}^{L}ψ_{n+1}^{R} − ψ_{n+1}^{L}φ_{n+1}^{R} =
= (ρ_{n}^{L})^{−1}[(zφ_{n}^{L} − α_{n}^{†}φ_{n}^{R,∗})(zψ_{n}^{R} + ψ_{n}^{L,∗}α_{n}^{†})
− (zψ_{n}^{L} + α_{n}^{†}ψ_{n}^{R,∗})(zφ_{n}^{R} − φ_{n}^{L,∗}α_{n}^{†})](ρ_{n}^{R})^{−1}
= (ρ_{n}^{L})^{−1}[z^{2}(φ_{n}^{L}ψ_{n}^{R} − ψ_{n}^{L}φ_{n}^{R}) − α_{n}^{†}(φ_{n}^{R,∗}ψ_{n}^{L,∗} − ψ_{n}^{R,∗}φ_{n}^{L,∗})α_{n}^{†}

− zα_{n}^{†}(φ_{n}^{R,∗}ψ_{n}^{R} + ψ_{n}^{R,∗}φ_{n}^{R}) + z(φ_{n}^{L}ψ_{n}^{L,∗} + ψ_{n}^{L}φ_{n}^{L,∗})α_{n}^{†}](ρ_{n}^{R})^{−1} = 0

which implies first (3.33) for n+1 and then, by applying the ∗-operation of order 2n + 2, also (3.34) for n + 1. This concludes the proof of the proposition. □

3.4. Christoffel–Darboux Formulas.

Proposition 3.6. (a) (CD)-left orthogonal

(1 − ξ̄z) ∑_{k=0}^{n} φ_{k}^{L}(ξ)^{†}φ_{k}^{L}(z) = φ_{n}^{R,∗}(ξ)^{†}φ_{n}^{R,∗}(z) − ξ̄zφ_{n}^{L}(ξ)^{†}φ_{n}^{L}(z)
= φ_{n+1}^{R,∗}(ξ)^{†}φ_{n+1}^{R,∗}(z) − φ_{n+1}^{L}(ξ)^{†}φ_{n+1}^{L}(z)

(b) (CD)-right orthogonal

(1 − ξ̄z) ∑_{k=0}^{n} φ_{k}^{R}(z)φ_{k}^{R}(ξ)^{†} = φ_{n}^{L,∗}(z)φ_{n}^{L,∗}(ξ)^{†} − ξ̄zφ_{n}^{R}(z)φ_{n}^{R}(ξ)^{†}
= φ_{n+1}^{L,∗}(z)φ_{n+1}^{L,∗}(ξ)^{†} − φ_{n+1}^{R}(z)φ_{n+1}^{R}(ξ)^{†}

(c) (Mixed CD)-left orthogonal

(1 − ξ̄z) ∑_{k=0}^{n} ψ_{k}^{L}(ξ)^{†}φ_{k}^{L}(z) = 2 · 1 − ψ_{n}^{R,∗}(ξ)^{†}φ_{n}^{R,∗}(z) − ξ̄zψ_{n}^{L}(ξ)^{†}φ_{n}^{L}(z)

= 2 · 1 − ψ_{n+1}^{R,∗}(ξ)^{†}φ_{n+1}^{R,∗}(z) − ψ_{n+1}^{L}(ξ)^{†}φ_{n+1}^{L}(z)

(d) (Mixed CD)-right orthogonal

(1 − ξ̄z) ∑_{k=0}^{n} φ_{k}^{R}(z)ψ_{k}^{R}(ξ)^{†} = 2 · 1 − φ_{n}^{L,∗}(z)ψ_{n}^{L,∗}(ξ)^{†} − ξ̄zφ_{n}^{R}(z)ψ_{n}^{R}(ξ)^{†}

= 2 · 1 − φ_{n+1}^{L,∗}(z)ψ_{n+1}^{L,∗}(ξ)^{†} − φ_{n+1}^{R}(z)ψ_{n+1}^{R}(ξ)^{†}

Remark. Since the ψ’s are themselves MOPUCs, the analogue of (a) and (b), with all φ’s replaced by ψ’s, holds.

Proof. (a) Write

F_{n}^{L}(z) = (\begin{matrix}φ_{n}^{L}(z) \\ φ_{n}^{R,∗}(z) \end{matrix}) , J = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , J̃ = ( \begin{matrix}ξ̄z1 & 0 \\ 0 & −1 \end{matrix} ) .

Then,

F_{n+1}^{L}(z) = A^{L}(α_{n}, z)F_{n}^{L}(z)

and

A^{L}(α, ξ)^{†}JA^{L}(α, z) =

= (\begin{matrix}ξ̄(ρ^{L})^{−1} & −ξ̄α^{†}(ρ^{R})^{−1} \\ −α(ρ^{L})^{−1} & (ρ^{R})^{−1} \end{matrix}) (\begin{matrix}z(ρ^{L})^{−1} & −(ρ^{L})^{−1}α^{†} \\ z(ρ^{R})^{−1}α & −(ρ^{R})^{−1} \end{matrix})

= (\begin{matrix}ξ̄z(ρ^{L})^{−2} − ξ̄zα^{†}(ρ^{R})^{−2}α & −ξ̄(ρ^{L})^{−2}α^{†} + ξ̄α^{†}(ρ^{R})^{−2} \\ −zα(ρ^{L})^{−2} + z(ρ^{R})^{−2}α & α(ρ^{L})^{−2}α^{†} − (ρ^{R})^{−2} \end{matrix})

= (\begin{matrix}ξ̄z1 & 0 \\ 0 & −1 \end{matrix}) = J̃.

Thus,

F_{n+1}^{L}(ξ)^{†}JF_{n+1}^{L}(z) = F_{n}^{L}(ξ)^{†}A^{L}(α_{n}, ξ)^{†}JA^{L}(α_{n}, z)F_{n}^{L}(z)
= F_{n}^{L}(ξ)^{†}J̃F_{n}^{L}(z)

and hence φ_{n+1}^{L}(ξ)^{†}φ_{n+1}^{L}(z) − φ_{n+1}^{R,∗}(ξ)^{†}φ_{n+1}^{R,∗}(z) = ξ̄zφ_{n}^{L}(ξ)^{†}φ_{n}^{L}(z) − φ_{n}^{R,∗}(ξ)^{†}φ_{n}^{R,∗}(z) which shows that the last two expressions in (a) are equal. Denote their common value by Q_{n}^{L}(z, ξ). Then,

Q_{n}^{L}(z, ξ) − Q_{n−1}^{L}(z, ξ) = φ_{n}^{R,∗}(ξ)^{†}φ_{n}^{R,∗}(z) − ξ̄zφ_{n}^{L}(ξ)^{†}φ_{n}^{L}(z)
− φ_{n}^{R,∗}(ξ)^{†}φ_{n}^{R,∗}(z) + φ_{n}^{L}(ξ)^{†}φ_{n}^{L}(z)
= (1 − ξ̄z)φ_{n}^{L}(ξ)^{†}φ_{n}^{L}(z).

Summing over n completes the proof since Q_{−1}^{L}(z, ξ) = 0.

(b) The proof is analogous to (a): Write F_{n}^{R}(z) = (φ_{n}^{R}(z) φ_{n}^{L,∗}(z)). Then, F_{n+1}^{R}(z) = F_{n}^{R}(z)A^{R}(α_{n}, z) and A^{R}(α, z)JA^{R}(α, ξ)^{†} = J̃. Thus,

F_{n+1}^{R}(z)JF_{n+1}^{R}(ξ)^{†} = F_{n}^{R}(z)A^{R}(α_{n}, z)JA^{R}(α_{n}, ξ)^{†}F_{n}^{R}(ξ)^{†}
= F_{n}^{R}(z)J̃F_{n}^{R}(ξ)^{†}

and hence φ_{n+1}^{R}(z)φ_{n+1}^{R}(ξ)^{†} − φ_{n+1}^{L,∗}(z)φ_{n+1}^{L,∗}(ξ)^{†} = ξ̄zφ_{n}^{R}(z)φ_{n}^{R}(ξ)^{†} − φ_{n}^{L,∗}(z)φ_{n}^{L,∗}(ξ)^{†} which shows that the last two expressions in (b) are equal. Denote their common value by Q_{n}^{R}(z, ξ). Then,

Q_{n}^{R}(z, ξ) − Q_{n−1}^{R}(z, ξ) = (1 − ξ̄z)φ_{n}^{R}(z)φ_{n}^{R}(ξ)^{†}

and the assertion follows as before.

(c) Write

F̃_{n}^{L}(z) = (\begin{matrix}ψ_{n}^{L}(z) \\ −ψ_{n}^{R,∗}(z) \end{matrix})

with the second kind polynomials ψ_{n}^{L,R}. As in (a), we see that

F̃_{n+1}^{L}(ξ)^{†}JF_{n+1}^{L}(z) = F̃_{n}^{L}(ξ)^{†}A^{L}(α_{n}, ξ)^{†}JA^{L}(α_{n}, z)F_{n}^{L}(z)

= F̃_{n}^{L}(ξ)^{†}J̃F_{n}^{L}(z)

and hence

ψ_{n+1}^{L}(ξ)^{†}φ_{n+1}^{L}(z)+ψ_{n+1}^{R,∗}(ξ)^{†}φ_{n+1}^{R,∗}(z) = ξ̄zψ_{n}^{L}(ξ)^{†}φ_{n}^{L}(z)+ψ_{n}^{R,∗}(ξ)^{†}φ_{n}^{R,∗}(z).

Denote

Q̃_{n}^{L}(z, ξ) = 2 · 1 − ψ_{n+1}^{R,∗}(ξ)^{†}φ_{n+1}^{R,∗}(z) − ψ_{n+1}^{L}(ξ)^{†}φ_{n+1}^{L}(z).

Then,

Q̃_{n}^{L}(z, ξ) − Q̃_{n−1}^{L}(z, ξ) = −ψ_{n}^{R,∗}(ξ)^{†}φ_{n}^{R,∗}(z) − ξ̄zψ_{n}^{L}(ξ)^{†}φ_{n}^{L}(z)
+ ψ_{n}^{R,∗}(ξ)^{†}φ_{n}^{R,∗}(z) + ψ_{n}^{L}(ξ)^{†}φ_{n}^{L}(z)
= (1 − ξ̄z)ψ_{n}^{L}(ξ)^{†}φ_{n}^{L}(z)

and the assertion follows as before.

(d) Write F̃_{n}^{R}(z) = (ψ_{n}^{R}(z) − ψ_{n}^{L,∗}(z)). As in (b), we see that

F_{n+1}^{R}(z)JF̃_{n+1}^{R}(ξ)^{†} = F_{n}^{R}(z)A^{R}(α_{n}, z)JA^{R}(α_{n}, ξ)^{†}F̃_{n}^{R}(ξ)^{†}
= F_{n}^{R}(z)J̃F̃_{n}^{R}(ξ)^{†}

and hence

φ_{n+1}^{R}(z)ψ_{n+1}^{R}(ξ)^{†} +φ_{n+1}^{L,∗}(z)ψ_{n+1}^{L,∗}(ξ)^{†} = ξ̄zφ_{n}^{R}(z)ψ_{n}^{R}(ξ)^{†} +φ_{n}^{L,∗}(z)ψ_{n}^{L,∗}(ξ)^{†}.

With Q̃_{n}^{R}(z, ξ) = 2 · 1 − φ_{n+1}^{L,∗}(z)ψ_{n+1}^{L,∗}(ξ)^{†} − φ_{n+1}^{R}(z)ψ_{n+1}^{R}(ξ)^{†}, we have

Q̃_{n}^{R}(z, ξ) − Q̃_{n−1}^{R}(z, ξ) = (1 − ξ̄z)φ_{n}^{R}(z)ψ_{n}^{R}(ξ)^{†}

and we conclude as in (c). □

3.5. Zeros of MOPUC. Our main result in this section is:

Theorem 3.7. All the zeros of det(φ_{n}^{R}(z)) lie in 𝔻 = {z : |z| < 1}.

We will also prove:

Theorem 3.8. For each n,

det(φ_{n}^{R}(z)) = det(φ_{n}^{L}(z)).  (3.35)

The scalar analogue of Theorem 3.7 has seven proofs in [167]! The simplest is due to Landau [135] and its MOPUC analogue is Theorem 2.13.7 of [167]. There is also a proof in Delsarte et al. [37] who attribute the theorem to Whittle [194]. We provide two more proofs here, not only for their intrinsic interest: our first proof we need because it depends only on the recursion relation (it is related to the proof of Delsarte et al. [37]). The second proof is here since it relates zeros to eigenvalues of a cutoff CMV matrix.

Theorem 3.9. We have (i) For z ∈ ∂𝔻, all of φ_{n}^{R,∗}(z), φ_{n}^{L,∗}(z), φ_{n}^{R}(z), φ_{n}^{L}(z) are invertible. (ii) For z ∈ ∂𝔻, φ_{n}^{L}(z)(φ_{n}^{R,∗}(z))^{−1} and (φ_{n}^{∗,L}(z))^{−1}φ_{n}^{R}(z) are unitary. (iii) For z ∈ 𝔻, φ_{n}^{R,∗}(z) and φ_{n}^{L,∗}(z) are invertible. (iv) For z ∈ 𝔻, φ_{n}^{L}(z)(φ_{n}^{R,∗}(z))^{−1} and (φ_{n}^{∗,L}(z))^{−1}φ_{n}^{R}(z) are of norm at

most 1 and, for n ≥ 1, strictly less than 1.

(v) All zeros of det(φ_{n}^{R,∗}(z)) and det(φ_{n}^{L,∗}(z)) lie in ℂ \ \overline{𝔻}. (vi) All zeros of det(φ_{n}^{R}(z)) and det(φ_{n}^{L}(z)) lie in 𝔻.

Remark. (vi) is our first proof of Theorem 3.7.

Proof. All these results are trivial for n = 0, so we can hope to use an inductive argument. So suppose we have the result for n − 1.

By (3.13),

φ_{n}^{R,∗} = (ρ_{n−1}^{R})^{−1}(1 − zα_{n−1}φ_{n−1}^{L}(φ_{n−1}^{R,∗})^{−1})φ_{n−1}^{R,∗}.  (3.36)

Since |α_{n−1}| < 1, if |z| ≤ 1, each factor on the right of (3.36) is invertible. This proves (i) and (iii) for φ_{n}^{R,∗} and a similar argument works for φ_{n}^{L,∗}. If z = e^{iθ}, φ_{n}^{R}(e^{iθ}) = e^{inθ}φ_{n}^{R,∗}(e^{iθ})^{†} is also invertible, so we have (i) and (iii) for n.

Next, we claim that if z ∈ ∂𝔻, then

φ_{n}^{R,∗}(z)^{†}φ_{n}^{R,∗}(z) = φ_{n}^{L}(z)^{†}φ_{n}^{L}(z).  (3.37)

This follows from taking z = ξ ∈ ∂𝔻 in Proposition 3.6(a). Given that φ_{n}^{R,∗}(z) is invertible, this implies

1 = (φ_{n}^{L}(z)φ_{n}^{R,∗}(z)^{−1})^{†}(φ_{n}^{L}(z)φ_{n}^{R,∗}(z)^{−1})  (3.38)

proving the first part of (ii) for n. The second part of (ii) is proven similarly by using Proposition 3.6(b).

For z ∈ \overline{𝔻}, let

F (z) = φ_{n}^{L}(z)φ_{n}^{R,∗}(z)^{−1}.

Then F is analytic in 𝔻, continuous in \overline{𝔻}, and ∥F (z)∥ = 1 on ∂𝔻, so (iv) follows from the maximum principle.

Since φ_{n}^{R,∗}(z) is invertible on \overline{𝔻}, its det is non-zero there, proving (v). (vi) then follows from

det(φ_{n}^{R}(z)) = z^{nl} \overline{det(φ_{n}^{R,∗}(1/z̄))} .  (3.39)
□

Let V be the ℂ^{l}-valued functions on ∂𝔻 and V_{n} the span of the ℂ^{l}-valued polynomials of degree at most n, so

dim(V_{n}) = ℂ^{l(n+1)}.

Let V_{∞} be the set ∪_{n}V_{n} of all ℂ^{l}-valued polynomials. Let π_{n} be the projection onto V_{n} in the V inner product (1.35).

It is easy to see that

V_{n} ∩ V_{n−1}^{⊥} = {Φ_{n}^{R}(z)v : v ∈ ℂ^{l}}  (3.40)

since ⟨z^{l}, Φ_{n}^{R}(z)v⟩ = 0 for l = 0, …, n − 1 and the dimensions on the left and right of (3.40) coincide. V_{n} ∩ V_{n−1}^{⊥} can also be described as the set of (v^{†}Φ_{n}^{L}(z))^{†} for v ∈ ℂ^{l}.

We define M_{z}: V_{n−1} → V_{n} or V_{∞} → V_{∞} as the operator of multiplication by z.

Theorem 3.10. For all n, we have

det_{ℂ^{l}}(Φ_{n}^{R}(z)) = det_{V_{n−1}}(z1 − π_{n−1}M_{z}π_{n−1}).  (3.41)

Remarks. 1. Since ∥M_{z}∥ ≤ 1, (3.41) immediately implies zeros of det(φ_{n}^{R}(z)) lie in \overline{𝔻}, and a small additional argument proves they lie in 𝔻. As we will see, this also implies Theorem 3.8.

2. Of course, π_{n−1}M_{z}π_{n−1} is a cutoff CMV matrix if written in a CMV basis.

Proof. If Q ∈ V_{n−k}, then by (3.40),

π_{n−1}[(z − z_{0})^{k}Q] = 0 ⇔ (z − z_{0})^{k}Q = Φ_{n}^{R}(z)v  (3.42)

for some v ∈ ℂ^{l}. Thus writing det(Φ_{n}^{R}(z)) = Φ_{n}^{R}(z)v_{1}∧⋯∧Φ_{n}^{R}(z)v_{l} in a Jordan basis for Φ_{n}^{R}(z), we see that the order of the zeros of det(Φ_{n}^{R}(z)) at z_{0} is exactly the order of z_{0} as an algebraic eigenvalue of π_{n−1}M_{z}π_{n−1}, that is, the order of z_{0} as a zero of the right side of (3.41).

Since both sides of (3.41) are monic polynomials of degree nl and their zeros including multiplicity are the same, we have proven (3.41).

□

Proof of Theorem 3.8. On the right side of (3.42), we can put (Φ_{n}^{L}(z)v^{†})^{†} and so conclude (3.41) holds with Φ_{n}^{L}(z) on the left.

This proves (3.35) if φ is replaced by Φ. Since α_{j}^{∗}α_{j} and α_{j}α_{j}^{∗} are unitarily equivalent, det(ρ_{j}^{L}) = det(ρ_{j}^{R}). Thus, det(κ_{n}^{L}) = det(κ_{n}^{R}), and we obtain (3.35) for φ. □

It is a basic fact (Theorem 1.7.5 of [167]) that for the scalar case, any set of n zeros in 𝔻 are the zeros of a unique OPUC Φ_{n} and any monic polynomial with all its zeros in 𝔻 is a monic OPUC. It is easy to see that any set of nl zeros in 𝔻 is the set of zeros of an OPUC Φ_{n}, but certainly not unique. It is an interesting open question to clarify what matrix monic OPs are monic MOPUCs.

3.6. Bernstein–Szegő Approximation. Given {α_{j}}_{j=0}^{n−1} ∈ 𝔻^{n}, we use Szegő recursion to define polynomials φ_{j}^{R}, φ_{j}^{L} for j = 0, 1, …, n. We define a measure dμ_{n} on ∂𝔻 by

dμ_{n}(θ) = [φ_{n}^{R}(e^{iθ})φ_{n}^{R}(e^{iθ})^{†}]^{−1} \frac{dθ}{2π}  (3.43)

Notice that (3.37) can be rewritten

φ_{n}^{R}(e^{iθ})φ_{n}^{R}(e^{iθ})^{†} = φ_{n}^{L}(e^{iθ})^{†}φ_{n}^{L}(e^{iθ}).  (3.44)

We use here and below the fact that the proof of Theorem 3.9 only depends on Szegő recursion and not on the a priori existence of a measure. That theorem also shows the inverse in (3.43) exists. Thus,

dμ_{n}(θ) = [φ_{n}^{L}(e^{iθ})^{†}φ_{n}^{L}(e^{iθ})]^{−1} \frac{dθ}{2π} .  (3.45)

Theorem 3.11. The measure dμ_{n} is normalized (i.e., μ_{n}(∂𝔻) = 1) and its right MOPUC for j = 0, …, n are {φ_{j}^{R}}_{j=0}^{n}, and for j > n,

φ_{j}^{R}(z) = z^{j−n}φ_{n}^{R}(z).  (3.46)

The Verblunsky coefficients for dμ_{n} are

α_{j}(dμ_{n}) = \begin{cases}α_{j}, & j ≤ n, \\ 0, & j ≥ n + 1. \end{cases}  (3.47)

Remarks. 1. In the scalar case, one can multiply by a constant and renormalize, and then prove the constant is 1. Because of commutativity issues, we need a different argument here.

2. Of course, using (3.45), φ_{n}^{L} are left MOPUC for dμ_{n}.

3. Our proof owes something to the scalar proof in [80].

Proof. Let ⟨⟨·, ·⟩⟩_{R} be the inner product associated with μ_{n}. By a direct computation, ⟨⟨φ_{n}^{R}, φ_{n}^{R}⟩⟩_{R} = 1, and for j = 0, 1, …, n − 1,

⟨⟨z^{j}, φ_{n}^{R}⟩⟩_{R} = \frac{1}{2π} ∫_{0}^{2π} e^{−ijθ}(φ_{n}^{R}(e^{iθ})^{†})^{−1} dθ

= \frac{1}{2πi} ∮ z^{n−j−1}(φ_{n}^{R,∗}(z))^{−1} dz = 0

by analyticity of φ_{n}^{R,∗}(z)^{−1} in 𝔻 (continuity in \overline{𝔻}).

This proves φ_{n}^{R} is a MOPUC for dμ_{n} (and a similar calculation works for the right side of (3.46) if j ≥ n). By the inverse Szegő recursion and induction downwards, {φ_{j}^{R}}_{j=0}^{n−1} are also OPs, and by the Szegő recursion, they are normalized. In particular, since φ_{0}^{R} ≡ 1 is normalized, dμ_{n} is normalized. □

3.7. Verblunsky’s Theorem. We can now prove the analogue of Favard’s theorem for MOPUC; the scalar version is called Verblunsky’s theorem in [167] after [192]. A history and other proofs can be found in [167]. The proof below is essentially the matrix analogue of that of Geronimus [90] (rediscovered in [37, 80]). Delsarte et al. [37] presented their proof in the MOPUC case and they seem to have been the first with a matrix Verblunsky theorem. One can extend the CMV and the Geronimus theorem proofs from the scalar case to get alternate proofs of the theorem below.

Theorem 3.12 (Verblunsky’s Theorem for MOPUC). Any sequence {α_{j}}_{j=0}^{∞} ∈ 𝔻^{∞} is the set of Verblunsky coefficients of a unique measure.

Proof. Uniqueness is easy, since the α’s determine the φ_{j}^{R}’s and so the Φ_{j}^{R}’s which determine the moments.

Given a set of {α_{j}}_{j=0}^{∞}, let dμ_{n} be the measures of the last section. By compactness of l×l matrix-valued probability measures on ∂𝔻, they have a weak limit. By using limits, {φ_{j}^{R}}_{j=0}^{∞} are the right MOPUC for dμ and they determine the proper Verblunsky coefficients. □

3.8. Matrix POPUC. Analogously to the scalar case (see [15, 100, 119, 171, 197]), given any unitary β in M_{l}, we define

φ_{n}^{R}(z; β) = zφ_{n−1}^{R}(z) − φ_{n−1}^{L,∗}(z)β^{†}.  (3.48)

As in the scalar case, this is related to the secular determinant of unitary extensions of the cutoff CMV matrix. Moreover,

Theorem 3.13. Fix β. All the zeros of (φ_{n}(z; β)) lie on ∂𝔻.

Proof. If |z| < 1, φ_{n−1}^{L,∗}(z) is invertible and

φ_{n}^{R}(z; β) = −φ_{n−1}^{L,∗}(z)β^{†}(1 − zβ φ_{n−1}^{L,∗}(z)^{−1}φ_{n−1}^{R}(z))

is invertible since the last factor differs from 1 by a strict contraction. A similar argument shows invertibility if |z| > 1. Thus, the only zeros of det( ·) lie in ∂𝔻. □

3.9. Matrix-Valued Carathéodory and Schur Functions. An analytic matrix-valued function F defined on 𝔻 is called a (matrix-valued) Carathéodory function if F (0) = 1 and Re F (z) ≡ \frac{1}{2}(F (z)+F (z)^{†}) ≥ 0 for every z ∈ 𝔻. The following result can be found in [44, Thm. 2.2.2].

Theorem 3.14 (Riesz–Herglotz). If F is a matrix-valued Carathéodory function, then there exists a unique positive semi-definite matrix measure dμ such that

F (z) = ∫ \frac{e^{iθ} + z}{e^{iθ} − z} dμ(θ).  (3.49)

The measure dμ is given by the unique weak limit of the measures dμ_{r}(θ) = Re F (re^{iθ})\frac{dθ}{2π} as r ↑ 1. Moreover,

F (z) = c_{0} + 2 ∑_{n=1}^{∞} c_{n}z^{n}

where

c_{n} = ∫ e^{−inθ} dμ(θ).

Conversely, if dμ is a positive semi-definite matrix measure, then (3.49) defines a matrix-valued Carathéodory function.

An analytic matrix-valued function f defined on 𝔻 is called a (matrix-valued) Schur function if f(z)^{†}f(z) ≤ 1 for every z ∈ 𝔻. This condition is equivalent to f(z)f(z)^{†} ≤ 1 for every z ∈ 𝔻 and to ∥f(z)∥ ≤ 1 for every z ∈ 𝔻. By the maximum principle, if f is not constant, the inequalities are strict. The following can be found in [167, Prop. 4.5.3]:

Proposition 3.15. The association

f(z) = z^{−1}(F (z) − 1)(F (z) + 1)^{−1},  (3.50)
F (z) = (1 + zf(z))(1 − zf(z))^{−1}  (3.51)

sets up a one-one correspondence between matrix-valued Carathéodory functions and matrix-valued Schur functions.

Proposition 3.16. For z ∈ 𝔻, we have

Re F (z) = (1 − z̄f(z)^{†})^{−1}(1 − |z|^{2}f(z)^{†}f(z))(1 − zf(z))^{−1} (3.52)

and the non-tangential boundary values Re F (e^{iθ}) and f(e^{iθ}) exist for Lebesgue almost every θ.

Write dμ(θ) = w(θ)\frac{dθ}{2π} + dμ_{s}. Then, for almost every θ,

w(θ) = Re F (e^{iθ})  (3.53)

and for a.e. θ, det(w(θ)) ≠ 0 if and only if f(e^{iθ})^{†}f(e^{iθ}) < 1.

Proof. The identity (3.52) follows from (3.51). The existence of the boundary values of f follows by application of the scalar result to the individual entries of f. Then (3.51) gives the boundary values of F. We also used the following fact: Away from a set of zero Lebesgue measure, det(1 − zf(z)) has non-zero boundary values by general H^{∞} theory.

(3.53) holds for ⟨η, F (z)η⟩_{ℂ^{l}} and ⟨η, dμη⟩_{ℂ^{l}} for any η ∈ ℂ^{l} by the scalar result. We get (3.53) by polarization. From

w(θ) = (1 − e^{−iθ}f(e^{iθ})^{†})^{−1}(1 − f(e^{iθ})^{†}f(e^{iθ}))(1 − e^{iθ}f(e^{iθ}))^{−1}

it follows immediately that f(e^{iθ})^{†}f(e^{iθ}) < 1 implies det(w(θ)) > 0. Conversely, if f(e^{iθ})^{†}f(e^{iθ}) ≤ 1 but not f(e^{iθ})^{†}f(e^{iθ}) < 1, then det(1 − f(e^{iθ})^{†}f(e^{iθ})) = 0 and by our earlier arguments det(1 − e^{−iθ}f(e^{iθ})^{†})^{−1} and det(1 − e^{iθ}f(e^{iθ}))^{−1} exist and are finite; hence det(w(θ)) = 0. All previous statements are true away from suitable sets of zero Lebesgue measure. □

3.10. Coefficient Stripping, the Schur Algorithm, and Geronimus’ Theorem. The matrix version of Geronimus’ theorem goes back at least to the book of Bakonyi–Constantinescu [6]. Let F (z) be the matrix-valued Carathéodory function (3.49) (with the same measure μ as the one used in the definition of ⟨⟨·, ·⟩⟩_{R}). Let us denote

u_{n}^{L}(z) = ψ_{n}^{L}(z) + φ_{n}^{L}(z)F (z),
u_{n}^{R}(z) = ψ_{n}^{R}(z) + F (z)φ_{n}^{R}(z).

We also define

u_{n}^{L,∗}(z) = ψ_{n}^{L,∗}(z) − F (z)φ_{n}^{L,∗}(z),
u_{n}^{R,∗}(z) = ψ_{n}^{R,∗}(z) − φ_{n}^{R,∗}(z)F (z).

Proposition 3.17. For any |z| < 1, the sequences u_{n}^{L}(z), u_{n}^{R}(z), u_{n}^{L,∗}(z), u_{n}^{R,∗}(z) are square summable.

Proof. Denote

f(θ) = \frac{e^{−iθ} + z̄}{e^{−iθ} − z̄} , g(θ) = \frac{e^{iθ} + z}{e^{iθ} − z} .

By the definitions (3.19)–(3.24), we have

u_{n}^{L}(z) = ⟨⟨f, φ_{n}^{L}⟩⟩_{L},
−u_{n}^{R,∗}(z) = z^{n}⟨⟨φ_{n}^{R}, g⟩⟩_{R},
−u_{n}^{L,∗}(z) = z^{n}⟨⟨φ_{n}^{L}, g⟩⟩_{L},
u_{n}^{R}(z) = ⟨⟨f, φ_{n}^{R}⟩⟩_{R}.

Using the Bessel inequality and the fact that |z| < 1, we obtain the required statements. □

Next we will consider sequences defined by

(\begin{matrix}s_{n} \\ t_{n} \end{matrix}) = A^{L}(α_{n−1}, z) … A^{L}(α_{0}, z) (\begin{matrix}s_{0} \\ t_{0} \end{matrix})  (3.54)

where s_{n}, t_{n} ∈ M_{l}. Similarly, we will consider the sequences

(s_{n}, t_{n}) = (s_{0}, t_{0})A^{R}(α_{0}, z) … A^{R}(α_{n−1}, z)  (3.55)

Theorem 3.18. Let z ∈ 𝔻 and let f be the Schur function associated with dμ via (3.49) and (3.50). Then:

(i) A solution of (3.54) is square summable if and only if the initial condition is of the form

(\begin{matrix}s_{0} \\ t_{0} \end{matrix}) = (\begin{matrix}c \\ zf(z)c \end{matrix})

for some matrix c in M_{l}. (ii) A solution of (3.55) is square summable if and only if the initial

condition is of the form

(s_{0}, t_{0}) = (c, czf(z))

for some matrix c.

Proof. We shall prove (i); the proof of (ii) is similar.

1. By Proposition 3.17 and (3.14), (3.30), we have the square summable solution

(\begin{matrix}s_{n} \\ t_{n} \end{matrix}) = (u_{n}^{L}(z) − u_{n}^{R,∗}(z)) , (\begin{matrix}s_{0} \\ t_{0} \end{matrix}) = (\begin{matrix}d \\ zf(z)d \end{matrix}) , d = (F (z) + 1).

(3.56)

The matrix d = F (z) + 1 is invertible. Thus, multiplying the above solution on the right by d^{−1}c for any given matrix c, we get the “if” part of the theorem.

2. Let us check that φ_{n}^{R,∗}c is not square summable for any matrix c ≠ 0. By the CD formula with ξ = z, we have

(1 − |z|^{2}) ∑_{k=1}^{n−1} φ_{k}^{L}(z)^{†}φ_{k}^{L}(z) + φ_{n}^{L}(z)^{†}φ_{n}^{L}(z) = φ_{n}^{R,∗}(z)^{†}φ_{n}^{R,∗}(z)

and so

φ_{n}^{R,∗}(z)^{†}φ_{n}^{R,∗}(z) ≥ (1 − |z|^{2})φ_{0}^{L}(z)^{†}φ_{0}^{L}(z) = (1 − |z|^{2})1.

Thus, we get

∥φ_{n}^{R,∗}c∥_{R}^{2} ≥ (1 − |z|^{2}) Tr c^{†}c > 0.

3. Let ( \begin{matrix}s_{n} \\ t_{n} \end{matrix}) be any square summable solution to (3.54). Let us write this solution as

(\begin{matrix}s_{n} \\ t_{n} \end{matrix}) = (\begin{matrix}φ_{n}^{L}a \\ φ_{n}^{R,∗}a \end{matrix}) + (\begin{matrix}ψ_{n}^{L}b \\ −ψ_{n}^{R,∗}b \end{matrix}) , a = \frac{s_{0} + t_{0}}{2} , b = \frac{s_{0} − t_{0}}{2} . (3.57)

Multiplying the solution (3.56) by b and subtracting from (3.57), we get a square summable solution

(\begin{matrix}φ_{n}^{L}(z)(a − F (z)b) \\ φ_{n}^{R,∗}(a − F (z)b) \end{matrix})

It follows that a = F (z)b, which proves the “only if” part. □

The Schur function f is naturally associated with dμ and hence with the Verblunsky coefficients α_{0}, α_{1}, α_{2}, . . .. The Schur functions obtained by coefficient stripping will be denoted by f_{1}, f_{2}, f_{3}, …, that is, f_{n} corresponds to Verblunsky coefficients α_{n}, α_{n+1}, α_{n+2}, . . .. We also write f_{0} ≡ f.

Theorem 3.19 (Schur Algorithm and Geronimus’ Theorem). For the Schur functions f_{0}, f_{1}, f_{2}, … associated with Verblunsky coefficients α_{0}, α_{1}, α_{2}, …, the following relations hold:

f_{n+1}(z) = z^{−1}(ρ_{n}^{R})^{−1}[f_{n}(z) − α_{n}] [1 − α_{n}^{†}f_{n}(z)]^{−1}ρ_{n}^{L},  (3.58)
f_{n}(z) = (ρ_{n}^{R})^{−1}[zf_{n+1}(z) + α_{n}] [1 + zα_{n}^{†}f_{n+1}(z)]^{−1}ρ_{n}^{L}.  (3.59)

Remarks. 1. See (1.81) and Theorem 1.4 to understand this result. 2. (1.83) provides an alternate way to write (3.58) and (3.59).

Proof. It clearly suffices to consider the case n = 0. Consider the solution of (3.54) with initial condition

(\begin{matrix}1 \\ zf_{0}(z) \end{matrix})

By Theorem 3.18, there exists a matrix c such that

(\begin{matrix}c \\ zf_{1}(z)c \end{matrix}) = A^{L}(α_{0}, z) (\begin{matrix}1 \\ zf_{0}(z) \end{matrix})

= (\begin{matrix}z(ρ_{0}^{L})^{−1} − z(ρ_{0}^{L})^{−1}α_{0}^{†}f_{0}(z) \\ −z(ρ_{0}^{R})^{−1}α_{0} + z(ρ_{0}^{R})^{−1}f_{0}(z) \end{matrix})

From this we can compute zf_{1}(z):

zf_{1}(z) = [−z(ρ_{0}^{R})^{−1}α_{0} + z(ρ_{0}^{R})^{−1}f_{0}(z)] [z(ρ_{0}^{L})^{−1} − z(ρ_{0}^{L})^{−1}α_{0}^{†}f_{0}(z)]^{−1}
= (ρ_{0}^{R})^{−1}[f_{0}(z) − α_{0}] [1 − α_{0}^{†}f_{0}(z)]^{−1}ρ_{0}^{L}

which is (3.58).

Similarly, we can express f_{0} in terms of f_{1}. From

(\begin{matrix}1 \\ zf_{0}(z) \end{matrix}) = A^{L}(α_{0}, z)^{−1} (\begin{matrix}c \\ zf_{1}(z)c \end{matrix})

= (\begin{matrix}z^{−1}(ρ_{0}^{L})^{−1} & z^{−1}(ρ_{0}^{L})^{−1}α_{0}^{†} \\ (ρ_{0}^{R})^{−1}α_{0} & (ρ_{0}^{R})^{−1} \end{matrix}) (\begin{matrix}c \\ zf_{1}(z)c \end{matrix})

= (\begin{matrix}z^{−1}(ρ_{0}^{L})^{−1}c + (ρ_{0}^{L})^{−1}α_{0}^{†}f_{1}(z)c \\ (ρ_{0}^{R})^{−1}α_{0}c + (ρ_{0}^{R})^{−1}zf_{1}(z)c \end{matrix})

we find that

zf_{0}(z) = [(ρ_{0}^{R})^{−1}α_{0}c + (ρ_{0}^{R})^{−1}zf_{1}(z)c] [z^{−1}(ρ_{0}^{L})^{−1}c + (ρ_{0}^{L})^{−1}α_{0}^{†}f_{1}(z)c]^{−1}
= [(ρ_{0}^{R})^{−1}α_{0} + (ρ_{0}^{R})^{−1}zf_{1}(z)] [z^{−1}(ρ_{0}^{L})^{−1} + (ρ_{0}^{L})^{−1}α_{0}^{†}f_{1}(z)]^{−1}
= (ρ_{0}^{R})^{−1}[α_{0} + zf_{1}(z)] [z^{−1}1 + α_{0}^{†}f_{1}(z)]^{−1}ρ_{0}^{L}

which gives (3.59). □

3.11. The CMV Matrix. In this section and the next, we discuss CMV matrices for MOPUC. This was discussed first by Simon in [170], which also has the involved history in the scalar case. Most of the results in this section appear already in [170]; the results of the next section are new here—they parallel the discussion in [167, Sect. 4.4] where these results first appeared in the scalar case.

3.11.1. The CMV basis. Consider the two sequences χ_{n}, x_{n} ∈ H, defined by

χ_{2k}(z) = z^{−k}φ_{2k}^{L,∗}(z), χ_{2k−1}(z) = z^{−k+1}φ_{2k−1}^{R}(z),
x_{2k}(z) = z^{−k}φ_{2k}^{R}(z),  x_{2k−1}(z) = z^{−k}φ_{2k−1}^{L,∗}(z).

For an integer k ≥ 0, let us introduce the following notation: i_{k} is the (k + 1)th term of the sequence 0, 1, −1, 2, −2, 3, −3, …, and j_{k} is the (k + 1)th term of the sequence 0, −1, 1, −2, 2, −3, 3, . . .. Thus, for example, i_{1} = 1, j_{1} = −1.

We use the right module structure of H. For a set of functions {f_{k}(z)}_{k=0}^{n} ⊂ H, its module span is the set of all sums ∑ f_{k}(z)a_{k} with a_{k} ∈ M_{l}.

Proposition 3.20. (i) For any n ≥ 1, the module span of {χ_{k}}_{k=0}^{n} coincides with the module span of {z^{i_{k}}}_{k=0}^{n} and the module span of {x_{k}}_{k=0}^{n} coincides with the module span of {z^{j_{k}}}_{k=0}^{n}.

(ii) The sequences {χ_{k}}_{k=0}^{∞} and {x_{k}}_{k=0}^{∞} are orthonormal:

⟨⟨χ_{k}, χ_{m}⟩⟩_{R} = ⟨⟨x_{k}, x_{m}⟩⟩_{R} = δ_{km}.  (3.60)

Proof. (i) Recall that

φ_{n}^{R}(z) = κ_{n}^{R}z^{n} + linear combination of{1, …, z^{n−1}},
φ_{n}^{L,∗}(z) = (κ_{n}^{L})^{†} + linear combination of{z, …, z^{n}},

where both κ_{n}^{R} and (κ_{n}^{L})^{†} are invertible matrices. It follows that

χ_{n}(z) = γ_{n}z^{i_{n}} + linear combination of{z^{i_{0}}, …, z^{i_{n−1}}},
x_{n}(z) = δ_{n}z^{j_{n}} + linear combination of{z^{j_{0}}, …, z^{j_{n−1}}},

where γ_{n}, δ_{n} are invertible matrices. This proves (i).

(ii) By the definition of φ_{n}^{L} and φ_{n}^{R}, we have

⟨⟨φ_{n}^{R}, φ_{m}^{R}⟩⟩_{R} = ⟨⟨φ_{n}^{L,∗}, φ_{m}^{L,∗}⟩⟩_{R} = δ_{nm},  (3.61)

⟨⟨φ_{n}^{R}, z^{m}⟩⟩_{R} = 0, m = 0, …, n − 1; ⟨⟨φ_{n}^{L,∗}, z^{m}⟩⟩_{R} = 0, m = 1, …, n. (3.62)

From (3.61) with n = m, we get

⟨⟨χ_{n}, χ_{n}⟩⟩_{R} = ⟨⟨x_{n}, x_{n}⟩⟩_{R} = 1.

Considering separately the cases of even and odd n, it is easy to prove that

⟨⟨χ_{n}, z^{m}⟩⟩_{R} = 0, m = i_{0}, i_{1}, …, i_{n−1},  (3.63)
⟨⟨x_{n}, z^{m}⟩⟩_{R} = 0, m = j_{0}, j_{1}, …, j_{n−1}.  (3.64)

For example, for n = 2k, m ∈ {i_{0}, …, i_{2k−1}} we have m + k ∈ {1, 2, …, 2k} and so, by (3.62),

⟨⟨χ_{n}, z^{m}⟩⟩_{R} = ⟨⟨z^{−k}φ_{2k}^{L,∗}, z^{m}⟩⟩_{R} = ⟨⟨φ_{2k}^{L,∗}, z^{m+k}⟩⟩_{R} = 0.

The other three cases are considered similarly. From (3.63), (3.64), and (i), we get (3.60) for k ≠ m. □

From the above proposition and the ∥·∥_{∞}-density of Laurent polynomials in C(∂𝔻), it follows that {χ_{k}}_{k=0}^{∞} and {x_{k}}_{k=0}^{∞} are right orthonormal modula bases in H, that is, any element f ∈ H can be represented as a

f = ∑_{k=0}^{∞} χ_{k}⟨⟨χ_{k}, f⟩⟩_{R} = ∑_{k=0}^{∞} x_{k}⟨⟨x_{k}, f⟩⟩_{R}.  (3.65)

3.11.2. The CMV matrix. Consider the matrix of the right homomorphism f(z) ↦ zf(z) with respect to the basis {χ_{k}}. Denote C_{nm} = ⟨⟨χ_{n}, zχ_{m}⟩⟩_{R}. The matrix C is unitary in the following sense:

∑_{k=0}^{∞} C_{kn}^{†}C_{km} = ∑_{k=0}^{∞} C_{nk}C_{mk}^{†} = δ_{nm}1.

The proof follows from (3.65):

δ_{nm}1 = ⟨⟨zχ_{n}, zχ_{m}⟩⟩_{R} = ⟨⟨∑_{k=0}^{∞} χ_{k}⟨⟨χ_{k}, zχ_{n}⟩⟩_{R}, zχ_{m}⟩⟩_{R} = ∑_{k=0}^{∞} C_{kn}^{†}C_{km},

δ_{nm}1 = ⟨⟨z̄χ_{n}, z̄χ_{m}⟩⟩_{R} = ⟨⟨∑_{k=0}^{∞} χ_{k}⟨⟨χ_{k}, z̄χ_{n}⟩⟩_{R}, z̄χ_{m}⟩⟩_{R} = ∑_{k=0}^{∞} C_{nk}C_{mk}^{†}.

We note an immediate consequence:

Lemma 3.21. Let |z| ≤ 1. Then, for every m ≥ 0,

∑_{n=0}^{∞} χ_{n}(z)C_{nm} = zχ_{m}(z),  ∑_{n=0}^{∞} C_{mn}χ_{n}(1/z̄)^{†} = zχ_{m}(1/z̄)^{†}.

Proof. First note that the above series contains only finitely many nonzero terms. Expanding f(z) = zχ_{n} according to (3.65), we see that

zχ_{n}(z) = ∑_{k=0}^{∞} χ_{k}(z)⟨⟨χ_{k}, zχ_{n}⟩⟩_{R} = ∑_{k=0}^{∞} χ_{k}(z)C_{kn}

which is the first identity. Next, taking adjoints, we get

z̄χ_{n}(z)^{†} = ∑_{k=0}^{∞} C_{kn}^{†}χ_{k}(z)^{†}

which yields

z̄∑_{n=0}^{∞} C_{mn}χ_{n}(z)^{†} = ∑_{n=0}^{∞} C_{mn} ∑_{k=0}^{∞} C_{kn}^{†}χ_{k}(z)^{†}

= ∑_{k=0}^{∞}( ∑_{n=0}^{∞} C_{mn} C_{kn}^{†})χ_{k}(z)^{†}

= ∑_{k=0}^{∞} δ_{mk}χ_{k}(z)^{†} = χ_{m}(z)^{†}.

Replacing z by 1/z̄, we get the required statement. □

3.11.3. The LM-representation. Using (3.65) for f = χ_{m}, we obtain:

C_{nm} = ⟨⟨χ_{n}, zχ_{m}⟩⟩_{R} = ∑_{k=0}^{∞}⟨⟨χ_{n}, zx_{k}⟩⟩_{R}⟨⟨x_{k}, χ_{m}⟩⟩_{R} = ∑_{k=0}^{∞} L_{nk}M_{km}.

(3.66)

Denote by Θ(α) the 2l × 2l unitary matrix

Θ(α) = (\begin{matrix}α^{†} & ρ^{L} \\ ρ^{R} & −α \end{matrix}) .

Using the Szegő recursion formulas (3.15) and (3.18), we get

zφ_{n}^{R} = φ_{n+1}^{R}ρ_{n}^{R} + φ_{n}^{L,∗}α_{n}^{†},  (3.67)
φ_{n+1}^{L,∗} = φ_{n}^{L,∗}ρ_{n}^{L} − φ_{n+1}^{R}α_{n}.  (3.68)

Taking n = 2k and multiplying by z^{−k}, we get

zx_{2k} = χ_{2k}α_{2k}^{†} + χ_{2k+1}ρ_{2k}^{R},
zx_{2k+1} = χ_{2k}ρ_{2k}^{L} − χ_{2k+1}α_{2k}.

It follows that the matrix L has the structure

L = Θ(α_{0}) ⊕ Θ(α_{2}) ⊕ Θ(α_{4}) ⊕ · · · .

Taking n = 2k − 1 in (3.67), (3.68) and multiplying by z^{−k}, we get

χ_{2k−1} = x_{2k−1}α_{2k−1}^{†} + x_{2k}ρ_{2k−1}^{R},
χ_{2k} = x_{2k−1}ρ_{2k−1}^{L} − x_{2k}α_{2k−1}.

It follows that the matrix M has the structure

M = 1 ⊕ Θ(α_{1}) ⊕ Θ(α_{3}) ⊕ · · · .  (3.69)

Substituting this into (3.66), we obtain:

C = ( \begin{matrix}α_{0}^{†} & ρ_{0}^{L}α_{1}^{†} & ρ_{0}^{L}ρ_{1}^{L} & 0 & 0 & ⋯ \\ ρ_{0}^{R} & −α_{0}α_{1}^{†} & −α_{0}ρ_{1}^{L} & 0 & 0 & ⋯ \\ 0 & α_{2}^{†}ρ_{1}^{R} & −α_{2}^{†}α_{1} & ρ_{2}^{L}α_{3}^{†} & ρ_{2}^{L}ρ_{3}^{L} & ⋯ \\ 0 & ρ_{2}^{R}ρ_{1}^{R} & −ρ_{2}^{R}α_{1} & −α_{2}α_{3}^{†} & −α_{2}ρ_{3}^{L} & ⋯ \\ 0 & 0 & 0 & α_{4}^{†}ρ_{3}^{R} & −α_{4}^{†}α_{3} & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ) . (3.70)

We note that the analogous formula to this in [170], namely, (4.30), is incorrect! The order of the factors below the diagonal is wrong there.

3.12. The Resolvent of the CMV Matrix. We begin by studying solutions to the equations

∑_{k=0}^{∞} C_{mk}w_{k} = zw_{m}, m ≥ 2,  (3.71)

∑_{k=0}^{∞} w̃_{k}C_{km} = zw̃_{m}, m ≥ 1.  (3.72)

Let us introduce the following functions:

x̃_{n}(z) = χ_{n}(1/z̄)^{†},
Υ_{2n}(z) = −z^{−n}ψ_{2n}^{L,∗}(z),
Υ_{2n−1}(z) = z^{−n+1}ψ_{2n−1}^{R}(z),
y_{2n}(z) = −Υ_{2n}(1/z̄)^{†} = z^{−n}ψ_{2n}^{L}(z),
y_{2n−1}(z) = −Υ_{2n−1}(1/z̄)^{†} = −z^{−n}ψ_{2n−1}^{R,∗}(z),
p_{n}(z) = y_{n}(z) + x̃_{n}(z)F (z),
π_{n}(z) = Υ_{n}(z) + F (z)χ_{n}(z).

Proposition 3.22. Let z ∈ 𝔻 \ {0}.

(i) For each n ≥ 0, a pair of values (w̃_{2n}, w̃_{2n+1}) uniquely determines a solution w̃_{n} to (3.72). Also, for any pair of values (w̃_{2n}, w̃_{2n+1}) in M_{l}, there exists a solution w̃_{n} to (3.72) with these values at (2n, 2n + 1).

(ii) The set of solutions w̃_{n} to (3.72) coincides with the set of sequences

w̃_{n}(z) = aχ_{n}(z) + bπ_{n}(z)  (3.73)

where a, b range over M_{l}. (iii) A solution (3.73) is in ℓ^{2} if and only if a = 0. (iv) A solution (3.73) obeys (3.72) for all m ≥ 0 if and only if b = 0. Proposition 3.23. Let z ∈ 𝔻 \ {0}.

(i) For each n ≥ 1, a pair of values (w_{2n−1}, w_{2n}) uniquely determines a solution w_{n} to (3.71). Also, for any pair of values (w_{2n−1}, w_{2n}) in M_{l}, there exists a solution w_{n} to (3.71) with these values at (2n − 1, 2n).

(ii) The set of solutions w_{n} to (3.71) coincides with the set of sequences

w_{n}(z) = x̃_{n}(z)a + p_{n}(z)b  (3.74)

where a, b range over M_{l}. (iii) A solution (3.74) is in ℓ^{2} if and only if a = 0. (iv) A solution (3.74) obeys (3.71) for all m ≥ 0 if and only if b = 0. Proof of Proposition 3.22. (i) The matrix C − z can be written in the form

C − z = (\begin{matrix}A_{0} & B_{0} & 0 & 0 & ⋯ \\ 0 & A_{1} & B_{1} & 0 & ⋯ \\ 0 & 0 & A_{2} & B_{2} & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix})  (3.75)

where

A_{0} = (\begin{matrix}α_{0}^{†} − z \\ ρ_{0}^{R} \end{matrix}) A_{n} = (\begin{matrix}α_{2n}^{†}ρ_{2n−1}^{R} & −α_{2n}^{†}α_{2n−1} − z \\ ρ_{2n}^{R}ρ_{2n−1}^{R} & −ρ_{2n}^{R}α_{2n−1} \end{matrix})

B_{n} = (\begin{matrix}ρ_{2n}^{L}α_{2n+1}^{†} & ρ_{2n}^{L}ρ_{2n+1}^{L} \\ −α_{2n}α_{2n+1}^{†} − z & −α_{2n}ρ_{2n+1}^{L} \end{matrix})

Define W̃_{n} = (w̃_{2n}, w̃_{2n+1}) for n = 0, 1, 2, . . .. Then (3.72) for m = 2n + 1, 2n + 2 is equivalent to

W̃_{n}B_{n} + W̃_{n+1}A_{n+1} = 0.

It remains to prove that the 2l × 2l matrices A_{j}, B_{j} are invertible. Suppose that for some x, y ∈ ℂ^{l}, A_{n}\binom{x}{y} = \binom{0}{0}. This is equivalent to the system

α_{2n}^{†}ρ_{2n−1}^{R}x − α_{2n}^{†}α_{2n−1}y − zy = 0,

ρ_{2n}^{R}ρ_{2n−1}^{R}x − ρ_{2n}^{R}α_{2n−1}y = 0.

The second equation of this system yields ρ_{2n−1}^{R}x = α_{2n−1}y (since ρ_{2n}^{R} is invertible), and upon substitution into the first equation, we get y = x = 0. Thus, ker(A_{n}) = {0}. In a similar way, one proves that ker(B_{n}) = {0}.

(ii) First note that w̃_{n} = χ_{n} is a solution to (3.72) by Lemma 3.21. Let us check that w̃_{n} = Υ_{n} is also a solution. If U_{km} = (−1)^{k}δ_{km}, then (UCU)_{km} for m ≥ 1 coincides with the CMV matrix corresponding to the coefficients {−α_{n}}. Recall that ψ_{n}^{L,R} are the orthogonal polynomials φ_{n}^{L,R}, corresponding to the coefficients {−α_{n}}. Taking into account the minus signs in the definition of Υ_{n}, we see that w̃_{n} = Υ_{n} solves (3.72) for m ≥ 1. It follows that any w̃_{n} of the form (3.73) is a solution to (3.72).

Let us check that any solution to (3.72) can be represented as (3.73). By (i), it suffices to show that for any w̃_{0}, w̃_{1}, there exist a, b ∈ M_{l} such that

aχ_{0}(z) + bπ_{0}(z) = w̃_{0},
aχ_{1}(z) + bπ_{1}(z) = w̃_{1}.

Recalling that χ_{0} = 1, Υ_{0} = −1, Υ_{1}(z) = (z + α_{0}^{†})(ρ_{0}^{R})^{−1}, χ_{1}(z) = (z − α_{0}^{†})(ρ_{0}^{R})^{−1}, we see that the above system can be easily solved for a, b if z ≠ 0.

(iii) Let us prove that the solution π_{n} is square integrable. We will consider separately the sequences π_{2n} and π_{2n−1} and prove that they both belong to ℓ^{2}. By (3.20) and (3.23), we have

ψ_{n}^{R}(z) + F (z)φ_{n}^{R}(z) = ∫ \frac{e^{iθ} + z}{e^{iθ} − z} dμ(θ)φ_{n}^{R}(e^{iθ}),  (3.76)
ψ_{n}^{L,∗}(z) − F (z)φ_{n}^{L,∗}(z) = −z^{n} ∫ \frac{e^{iθ} + z}{e^{iθ} − z} dμ(θ)φ_{n}^{L}(e^{iθ})^{†}.  (3.77)

Taking n = 2k in (3.77) and n = 2k − 1 in (3.76), we get

π_{2k}(z) = z^{k} ∫ \frac{e^{iθ} + z}{e^{iθ} − z} dμ(θ)φ_{2k}^{L}(e^{iθ})^{†},  (3.78)
π_{2k−1}(z) = z^{−k+1} ∫ \frac{e^{iθ} + z}{e^{iθ} − z} dμ(θ)φ_{2k−1}^{R}(e^{iθ}).  (3.79)

As φ_{2k}^{L} is an orthonormal sequence, using the Bessel inequality, from (3.78) we immediately get that π_{2k} is in ℓ^{2}.

Consider the odd terms π_{2k−1}. We claim that

z^{−k+1} ∫ \frac{e^{iθ} + z}{e^{iθ} − z} dμ(θ)φ_{2k−1}^{R}(e^{iθ}) = ∫ \frac{e^{iθ} + z}{e^{iθ} − z} dμ(θ)e^{i(−k+1)θ}φ_{2k−1}^{R}(e^{iθ}).

(3.80)

Indeed, using the right orthogonality of φ_{2k−1}^{R} to e^{imθ}, m = 0, 1, …, 2k − 2, we get

∫ \frac{e^{iθ} + z}{e^{iθ} − z} dμ(θ)φ_{2k−1}^{R}(e^{iθ}) = ⟨⟨1 + 2 ∑_{m=1}^{∞} z̄^{m}e^{imθ}, φ_{2k−1}^{R}⟩⟩_{R}

= ⟨⟨2 ∑_{m=2k−1}^{∞} z̄^{m}e^{imθ}, φ_{2k−1}^{R}⟩⟩_{R}

and

∫ \frac{e^{iθ} + z}{e^{iθ} − z} z^{k−1} e^{i(−k+1)θ} dμ(θ)φ_{2k−1}^{R}(e^{iθ}) =

= ⟨⟨z̄^{k−1}e^{i(k−1)θ}(1 + 2 ∑_{m=1}^{∞} z̄^{m}e^{imθ}), φ_{2k−1}^{R}⟩⟩_{R}

= ⟨⟨2 ∑_{m=2k−1}^{∞} z̄^{m}e^{imθ}, φ_{2k−1}^{R}⟩⟩_{R}

which proves (3.80). The identities (3.80) and (3.79) yield

π_{2k−1}(z) = ⟨⟨\frac{e^{−iθ} + z̄}{e^{−iθ} − z̄}, χ_{2k−1}⟩⟩_{R}

and, since χ_{2k−1} is a right orthogonal sequence, the Bessel inequality ensures that π_{2k−1}(z) is in ℓ^{2}. Thus, π_{k}(z) is in ℓ^{2}.

Next, as in the proof of Theorem 3.18, using the CD formula, we check that the sequence ∥φ_{n}^{L,∗}(z)∥_{R} is bounded below and therefore the sequence χ_{2n}(z) is not in ℓ^{2}. This proves the statement (iii).

(iv) By Lemma 3.21, the solution χ_{n}(z) obeys (3.72) for all m ≥ 0. It is easy to check directly that the solution π_{n}(z) does not obey (3.72) for m = 0 if z ≠ 0. This proves the required statement. □

Proof of Proposition 3.23. (i) For j = 1, 2, …, define W_{j} = (w_{2j−1}, w_{2j}). Then, using the block structure (3.75), we can rewrite (3.71) for m = 2j, 2j + 1 as A_{j}W_{j} + B_{j}W_{j+1} = 0. By the proof of Proposition 3.22, the matrices A_{j} and B_{j} are invertible, which proves

(ii) Lemma 3.21 ensures that x̃_{n}(z) is a solution of (3.71). As in the proof of Proposition 3.22, by considering the matrix (UCU)_{km}, one checks that y_{n}(z) is also a solution to (3.71).

Let us prove that any solution to (3.71) can be represented in the form (3.74). By (i), it suffices to show that for any w_{1}, w_{2}, there exist a, b ∈ M_{l} such that

x̃_{1}(z)a + p_{1}(z)b = w_{1},
x̃_{2}(z)a + p_{2}(z)b = w_{2}.

We claim that this system of equations can be solved for a, b. Here are the main steps. Substituting the definitions of x̃_{n}(z) and p_{n}(z), we rewrite this system as

φ_{1}^{R,∗}(a + F (z)b) − ψ_{1}^{R,∗}(z)b = zw_{1},
φ_{2}^{L}(a + F (z)b) + ψ_{2}^{L}(z)b = zw_{2}.

Using Szegő recurrence, we can substitute the expressions for φ_{2}^{L}, ψ_{2}^{L}, which helps rewrite our system as

φ_{1}^{R,∗}(a + F (z)b) − ψ_{1}^{R,∗}(z)b = zw_{1},
φ_{1}^{L}(a + F (z)b) + ψ_{1}^{L}(z)b = ρ_{1}^{L}w_{2} + α_{1}^{†}w_{1}.

Substituting explicit formulas for φ_{1}^{R,∗}, φ_{1}^{L}, ψ_{1}^{R,∗}, ψ_{1}^{L}, and expressing F (z) in terms of f(z), we can rewrite this as

(ρ_{0}^{R})^{−1}(1 − α_{0}z)a + 2z(ρ_{0}^{R})^{−1}(f(z) − α_{0})(1 − zf(z))^{−1}b = zw_{1},
(ρ_{0}^{L})^{−1}(z − α_{0}^{†})a + 2z(ρ_{0}^{L})^{−1}(1 − α_{0}^{†}f(z))(1 − zf(z))^{−1}b = ρ_{1}^{L}w_{2} + α_{1}^{†}w_{1}.

Denote

a_{1} = (ρ_{0}^{R})^{−1}(1 − α_{0}z)a,
b_{1} = 2z(ρ_{0}^{L})^{−1}(1 − α_{0}^{†}f(z))(1 − zf(z))^{−1}b.

Then in terms of a_{1}, b_{1}, our system can be rewritten as

a_{1} + X_{1}b_{1} = zw_{1},
X_{2}a_{1} + b_{1} = ρ_{1}^{L}w_{2} + α_{1}^{†}w_{1},

where

X_{1} = (ρ_{0}^{R})^{−1}(f(z) − α_{0})(1 − α_{0}f(z))^{−1}ρ_{0}^{L},
X_{2} = (ρ_{0}^{L})^{−1}(z − α_{0}^{†})(1 − α_{0}z)^{−1}ρ_{0}^{R}.

Since ∥f(z)∥ < 1 and |z| < 1, we can apply Corollary 1.5, which yields ∥X_{1}∥ < 1 and ∥X_{2}∥ < 1. It follows that our system can be solved for a_{1}, b_{1}.

(iii) As p_{n}(z) = −π_{n}(1/z̄)^{†}, by Proposition 3.22, we get that p_{n}(z) is in ℓ^{2}. In the same way, as x̃_{n}(z) = χ_{n}(1/z̄)^{†}, we get that x̃_{n}(z) is not in ℓ^{2}.

(iv) By Lemma 3.21, the solution x̃_{n}(z) obeys (3.71) for all m ≥ 0. Using the explicit formula for y_{n}(z), one easily checks that the solution y_{n}(z) does not obey (3.71) for m = 0, 1. □

Theorem 3.24. We have for z ∈ 𝔻,

[(C − z)^{−1}]_{k,l} = \begin{cases}(2z)^{−1}x̃_{k}(z)π_{l}(z), & l > k or k = l even, \\ (2z)^{−1}p_{k}(z)χ_{l}(z), & k > l or k = l odd. \end{cases}

Proof. Fix z ∈ 𝔻. Write G_{k,l}(z) = [(C − z)^{−1}]_{k,l}. Then G_{·,l}(z) is equal to (C − z)^{−1}δ_{l}, which means that G_{k,l}(z) solves (3.71) for m ≠ l. Since G_{·,l}(z) is ℓ^{2} at infinity and obeys the equation at m = 0, we see that it is a right-multiple of p for large k and a right-multiple of x̃ for small k. Thus,

G_{k,l}(z) = \begin{cases}x̃_{k}(z)a_{l}(z), & k < l or k = l even, \\ p_{k}(z)b_{l}(z), & k > l or k = l odd. \end{cases}

Similarly,

G_{k,l}(z) = \begin{cases}b̃_{k}(z)π_{l}(z), & k < l or k = l even, \\ ã_{k}(z)χ_{l}(z), & k > l or k = l odd. \end{cases}

Equating the two expressions, we find

x̃_{k}(z)a_{l}(z) = b̃_{k}(z)π_{l}(z) k < l or k = l even,  (3.81)
p_{k}(z)b_{l}(z) = ã_{k}(z)χ_{l}(z) k > l or k = l odd.  (3.82)

Putting k = 0 in (3.81) and setting b̃_{0}(z) = c_{1}(z), we find a_{l}(z) = c_{1}(z)π_{l}(z). Putting l = 0 in (3.82) and setting b_{0}(z) = c_{2}(z), we find p_{k}(z)c_{2}(z) = ã_{k}(z). Thus,

G_{k,l}(z) = \begin{cases}x̃_{k}(z)c_{1}(z)π_{l}(z), & k < l or k = l even, \\ p_{k}(z)c_{2}(z)χ_{l}(z), & k > l or k = l odd. \end{cases}

We claim that c_{1}(z) = c_{2}(z) = (2z)^{−1}1. Consider the case k = l = 0. Then, on the one hand, by the definition,

G_{0,0}(z) = ∫ \frac{1}{e^{iθ} − z} dμ(e^{iθ})

= ∫ (2z)^{−1}[\frac{e^{iθ} + z}{e^{iθ} − z} − 1]dμ(e^{iθ})

= (2z)^{−1}(F (z) − 1)  (3.83)

and on the other hand,

G_{0,0}(z) = x̃_{0}(z)c_{1}(z)π_{0}(z) = c_{1}(z)(F (z) − 1).

This shows c_{1}(z) = (2z)^{−1}1. Next, consider the case k = 1, l = 0. Then, on the one hand, by the definition,

G_{1,0}(z) = ⟨⟨χ_{1}, (e^{iθ} − z)^{−1}χ_{0}⟩⟩_{R}

and on the other hand,

G_{1,0}(z) = p_{1}(z)c_{2}(z)χ_{0}(z).

Let us calculate the expressions on the right-hand side. We have

p_{1}(z)c_{2}(z)χ_{0}(z) = (ρ_{0}^{R})^{−1}(−z^{−1} − α_{0} + (z^{−1} − α_{0})F (z))c_{2}(z) (3.84)

and

⟨⟨χ_{1},(e^{iθ} − z)^{−1}χ_{0}⟩⟩_{R} =

= (ρ_{0}^{R})^{−1} ∫ (e^{−iθ} − α_{0})dμ(θ)(e^{iθ} − z)^{−1}

= (ρ_{0}^{R})^{−1} ∫ [z^{−1}(e^{iθ} − z)^{−1} − z^{−1}e^{−iθ} − α_{0}(e^{iθ} − z)^{−1}] dμ(θ)

= (ρ_{0}^{R})^{−1}[\frac{1}{2z^{2}} (F (z) − 1) − \frac{1}{2z} α_{0}(F (z) − 1) − \frac{1}{z} ∫ e^{−iθ} dμ(θ)].

Taking into account the identity

∫ e^{−iθ}dμ(θ) = α_{0}

(which can be obtained, e.g., by expanding ⟨⟨φ_{1}^{R}, φ_{0}^{R}⟩⟩_{R} = 0), we get

⟨⟨χ_{1}, (e^{iθ} − z)^{−1}χ_{0}⟩⟩_{R} = \frac{1}{2z} (ρ_{0}^{R})^{−1}(−z^{−1} − α_{0} + (z^{−1} − α_{0})F (z)).

Comparing this with (3.84), we get c_{2}(z) = (2z)^{−1}1. □

As an immediate corollary, evaluating the kernel on the diagonal for even and odd indices, we obtain the formulas

∫ φ_{2n}^{L}(e^{iθ}) \frac{dμ(θ)}{e^{iθ} − z} φ_{2n}^{L}(e^{iθ})^{†} = −\frac{1}{2z^{2n+1}} φ_{2n}^{L}(z)u_{2n}^{L,∗}(z), (3.85)
∫ φ_{2n−1}^{R}(e^{iθ})^{†} \frac{dμ(θ)}{e^{iθ} − z} φ_{2n−1}^{R}(e^{iθ}) = −\frac{1}{2z^{2n}} u_{2n−1}^{R,∗}(z)φ_{2n−1}^{R}(z). (3.86)

Combining this with (3.31) and (3.32), we find

u_{n}^{L}(z)φ_{n}^{L,∗}(z) + φ_{n}^{L}(z)u_{n}^{L,∗}(z) = 2z^{n},  (3.87)
φ_{n}^{R,∗}(z)u_{n}^{R}(z) + u_{n}^{R,∗}(z)φ_{n}^{R}(z) = 2z^{n}.  (3.88)

3.13. Khrushchev Theory. Among the deepest and most elegant methods in OPUC are those of Khrushchev [125, 126, 101]. We have not been able to extend them to MOPUC! We regard their extension as an important open question; we present the first very partial steps here.

Let

Ω = {θ : det w(θ) > 0}.

Theorem 3.25. For every n ≥ 0,

{θ : f_{n}(e^{iθ})^{†}f_{n}(e^{iθ}) < 1} = Ω

up to a set of zero Lebesgue measure.

Consequently,

∫ ∥f_{n}(e^{iθ})∥ \frac{dθ}{2π} ≥ 1 − \frac{|Ω|}{2π} .  (3.89)

Proof. Recall that, by Proposition 3.16, up to a set of zero Lebesgue measure,

{θ : f_{0}(e^{iθ})^{†}f_{0}(e^{iθ}) < 1} = {θ : det w(θ) > 0}

so, by induction, it suffices to show that, up to a set of zero Lebesgue measure,

{θ : f_{0}(e^{iθ})^{†}f_{0}(e^{iθ}) < 1} = {θ : f_{1}(e^{iθ})^{†}f_{1}(e^{iθ}) < 1}.

This in turn follows from the fact that the Schur algorithm, which relates the two functions, preserves the property g^{†}g < 1.

Notice that away from Ω, f_{n}(e^{iθ}) has norm one and therefore,

∫ ∥f_{n}(e^{iθ})∥ \frac{dθ}{2π} ≥ ∫_{Ω^{c}} ∥f_{n}(e^{iθ})∥ \frac{dθ}{2π} = ∫_{Ω^{c}} 1 \frac{dθ}{2π}

which yields (3.89). □

Define

b_{n}(z; dμ) = φ_{n}^{L}(z; dμ)φ_{n}^{R,∗}(z; dμ)^{−1}.

Proposition 3.26. (a) b_{n+1} = (ρ_{n}^{L})^{−1}(zb_{n} − α_{n}^{†})(1 − zα_{n}b_{n})^{−1}ρ_{n}^{R} (b) The Verblunsky coefficients of b_{n} are (−α_{n−1}^{†}, −α_{n−2}^{†}, …, −α_{0}^{†}, 1).

Proof. (a) By the Szegő recursion, we have that

b_{n+1} = φ_{n+1}^{L}(φ_{n+1}^{R,∗})^{−1}

= ((ρ_{n}^{L})^{−1}zφ_{n}^{L} − (ρ_{n}^{L})^{−1}α_{n}^{†}φ_{n}^{R,∗})((ρ_{n}^{R})^{−1}φ_{n}^{R,∗} − z(ρ_{n}^{R})^{−1}α_{n}φ_{n}^{L})^{−1}

= (ρ_{n}^{L})^{−1}(zφ_{n}^{L} − α_{n}^{†}φ_{n}^{R,∗})(φ_{n}^{R,∗} − zα_{n}φ_{n}^{L})^{−1}ρ_{n}^{R}
= (ρ_{n}^{L})^{−1}(zφ_{n}^{L}(φ_{n}^{R,∗})^{−1} − α_{n}^{†}φ_{n}^{R,∗}(φ_{n}^{R,∗})^{−1})

(φ_{n}^{R,∗}(φ_{n}^{R,∗})^{−1} − zα_{n}φ_{n}^{L}(φ_{n}^{R,∗})^{−1})^{−1}ρ_{n}^{R}

= (ρ_{n}^{L})^{−1}(zb_{n} − α_{n}^{†})(1 − zα_{n}b_{n})^{−1}ρ_{n}^{R}

(b) It follows from part (a) that the first Verblunsky coefficient of b_{n} is −α_{n−1}^{†} and that its first Schur iterate is b_{n−1}; compare Theorem 3.19. This gives the claim by induction and the fact that b_{0} = 1. □

4. The Szegő Mapping and the Geronimus Relations

In this chapter, we present the matrix analogue of the Szegő mapping and the resulting Geronimus relations. This establishes a correspondence between certain matrix-valued measures on the unit circle and matrix-valued measures on the interval [−2, 2] and, consequently, a correspondence between Verblunsky coefficients and Jacobi parameters. Throughout this chapter, we will denote measures on the circle by dμ_{C} and measures on the interval by dμ_{I}.

The scalar versions of these objects are due to Szegő [182] and Geronimus [90]. There are four proofs that we know of: the original argument of Geronimus [90] based on Szegő’s formula in [182], a proof of Damanik–Killip [32] using Schur functions, a proof of Killip–Nenciu [127] using CMV matrices, and a proof of Faybusovich–Gekhtman [81] using canonical moments.

The matrix version of these objects was studied by Yakhlef– Marcellán [199] who proved Theorem 4.2 below using the Geronimus– Szegő approach. Our proof uses the Killip–Nenciu–CMV approach. In comparing our formula with [199], one needs the following dictionary (their objects on the left of the equal sign and ours on the right):

H_{n} = −α_{n+1}^{†},
D_{n} = A_{n},
E_{n} = B_{n+1}.

Dette–Studden [43] have extended the theory of canonical moments from OPRL to MOPRL. It would be illuminating to use this to extend the proof that Faybusovich–Gekhtman [81] gave of Geronimus relations for scalar OPUC to MOPUC.

Suppose dμ_{C} is a non-trivial positive semi-definite Hermitian matrix measure on the unit circle that is invariant under θ ↦ −θ (i.e., z ↦ z̄ = z^{−1}). Then we define the measure dμ_{I} on the interval [−2, 2] by

∫ f(x) dμ_{I}(x) = ∫ f(2 cos θ) dμ_{C}(θ)

for f measurable on [−2, 2]. The map

Sz : dμ_{C} ↦ dμ_{I}

is called the Szegő mapping.

The Szegő mapping can be inverted as follows. Suppose dμ_{I} is a non-degenerate positive semi-definite matrix measure on [−2, 2]. Then we define the measure dμ_{C} on the unit circle which is invariant under θ ↦ −θ by

∫ g(θ) dμ_{C}(θ) = ∫ g (arccos(x/2)) dμ_{I}(x)

for g measurable on ∂𝔻 with g(θ) = g(−θ).

We first show that for the measures on the circle of interest in this section, the Verblunsky coefficients are always Hermitian.

Lemma 4.1. Suppose dμ_{C} is a non-trivial positive semi-definite Hermitian matrix measure on the unit circle. Denote the associated Verblunsky coefficients by {α_{n}}. Then, dμ_{C} is invariant under θ ↦ −θ if and only if α_{n}^{†} = α_{n} for every n.

Proof. For a polynomial P, denote P̃(z) = P (z̄)^{†}.

1. Suppose that dμ_{C} is invariant under θ ↦ −θ. Then we have

⟨⟨f, g⟩⟩_{L} = ⟨⟨g̃, f̃⟩⟩_{R}

for all f, g. Inspecting the orthogonality conditions which define Φ_{n}^{L} and Φ_{n}^{R}, we see that

Φ̃_{n}^{L} = Φ_{n}^{R} and ⟨⟨Φ_{n}^{L}, Φ_{n}^{L}⟩⟩_{L} = ⟨⟨Φ_{n}^{R}, Φ_{n}^{R}⟩⟩_{R}.  (4.1)

Next, we claim that

κ_{n}^{L} = κ_{n}^{R,†}.  (4.2)

Indeed, recall the definition of κ_{n}^{L}, κ_{n}^{R}:

κ_{n}^{L} = u_{n}⟨⟨Φ_{n}^{L}, Φ_{n}^{L}⟩⟩_{L}^{−1/2}, u_{n} is unitary, κ_{n+1}^{L}(κ_{n}^{L})^{−1} > 0, κ_{0}^{L} = 1,
κ_{n}^{R} = ⟨⟨Φ_{n}^{R}, Φ_{n}^{R}⟩⟩_{R}^{−1/2}v_{n}, v_{n} is unitary, (κ_{n}^{R})^{−1}κ_{n+1}^{R} > 0, κ_{0}^{R} = 1.

Using this definition and (4.1), one can easily prove by induction that v_{n} = u_{n}^{†} and therefore (4.2) holds true.

Next, taking z = 0 in (3.11), we get

α_{n} = −(κ_{n}^{R})^{−1}Φ_{n+1}^{L}(0)^{†}(κ_{n}^{L})^{†},
α_{n} = −(κ_{n}^{R})^{†}Φ_{n+1}^{R}(0)^{†}(κ_{n}^{L})^{−1}.

From here and (4.1), (4.2), we get α_{n} = α_{n}^{†}.

2. Assume α_{n}^{†} = α_{n} for all n. Then, by Theorem 3.3(c), we have ρ_{n}^{L} = ρ_{n}^{R}. It follows that in this case the Szegő recurrence relation is invariant with respect to the change φ_{n}^{L} ↦ φ̃_{n}^{R}, φ_{n}^{R} ↦ φ̃_{n}^{L}. It follows that φ_{n}^{L} = φ̃_{n}^{R}, φ_{n}^{R} = φ̃_{n}^{L}. In particular, we get

⟨⟨φ_{n}^{L}, φ_{m}^{L}⟩⟩_{L} = ⟨⟨φ̃_{m}^{R}, φ̃_{n}^{R}⟩⟩_{R}.  (4.3)

Now let f and g be any polynomials; we have

f(z) = ∑_{n} f_{n}φ_{n}^{L}(z), f̃(z) = ∑_{n} φ̃_{n}^{L}(z)f_{n}^{†},

and a similar expansion for g. Using these expansions and (4.3), we get

⟨⟨f, g⟩⟩_{L} = ⟨⟨g̃, f̃⟩⟩_{R} for all polynomials f, g.

From here it follows that the measure dμ_{C} is invariant under θ ↦ −θ. □

Now consider two measures dμ_{C} and dμ_{I} = Sz(dμ_{C}) and the associated CMV and Jacobi matrices. What are the relations between the parameters of these matrices?

Theorem 4.2. Given dμ_{C} and dμ_{I} = Sz(dμ_{C}) as above, the coefficients of the associated CMV and Jacobi matrices satisfy the Geronimus relations:

B_{k+1} = \sqrt{1 − α_{2k−1}} α_{2k}\sqrt{1 − α_{2k−1}} − \sqrt{1 + α_{2k−1}} α_{2k−2}\sqrt{1 + α_{2k−1}} , (4.4)

A_{k+1} = \sqrt{1 − α_{2k−1}}\sqrt{1 − α_{2k}^{2}}\sqrt{1 + α_{2k+1}} .  (4.5)

Remarks. 1. For these formulas to hold for k = 0, we set α_{−1} = −1.

2. There are several proofs of the Geronimus relations in the scalar case. We follow the proof given by Killip and Nenciu in [127].

3. These A’s are, in general, not type 1 or 2 or 3.

Proof. For a Hermitian l × l matrix α with ∥α∥ < 1, define the unitary 2l × 2l matrix S(α) by

S(α) = \frac{1}{\sqrt{2}} (\begin{matrix}\sqrt{1 − α} & −\sqrt{1 + α} \\ \sqrt{1 + α} & \sqrt{1 − α} \end{matrix})

Since α^{†} = α, the associated ρ^{L} and ρ^{R} coincide and will be denoted by ρ. We therefore have

Θ(α) = (\begin{matrix}α & ρ \\ ρ & −α \end{matrix})

and hence, by a straightforward calculation,

S(α)Θ(α)S(α)^{−1} = \frac{1}{2} (\begin{matrix}\sqrt{1 − α} & −\sqrt{1 + α} \\ \sqrt{1 + α} & \sqrt{1 − α} \end{matrix}) (\begin{matrix}α & ρ \\ ρ & −α \end{matrix}) (\begin{matrix}\sqrt{1 − α} & \sqrt{1 + α} \\ −\sqrt{1 + α} & \sqrt{1 − α} \end{matrix})

= (\begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix})

Thus, if we define

S = 1 ⊕ S(α_{1}) ⊕ S(α_{3}) ⊕ …

and

R = 1 ⊕ (−1) ⊕ 1 ⊕ (−1) ⊕ …

it follows that (see (3.69))

SMS^{†} = R.

The matrix LM + ML is unitarily equivalent to

A = S(LM + ML)S^{†} = SLS^{†}R + RSLS^{†}.

Observe that A is the direct sum of two block Jacobi matrices. Indeed, it follows quickly from the explicit form of R that the even-odd and odd-even block entries of A vanish. Consequently, A is the direct sum of its odd-odd and even-even block entries. We will call these two block Jacobi matrices J and J̃, respectively.

Consider C as an operator on H_{v}. Then dμ_{C} is the spectral measure of C in the following sense:

[C^{m}]_{0,0} = ∫ e^{imθ}dμ(θ);

see (3.83). Then, by the spectral theorem, dμ_{C}(−θ) is the spectral measure of C^{−1} and so dμ_{I} is the spectral measure of C + C^{−1} = LM + (LM)^{−1} = LM + ML. Since S leaves (1 0 0 0 ⋯)^{t} invariant, we see that dμ_{I} is the spectral measure of J.

To determine the block entries of J, we only need to compute the odd-odd block entries of A. For k ≥ 0, we have

A_{2k+1,2k+1} = (\sqrt{1 + α_{2k−1}} \sqrt{1 − α_{2k−1}}) (\begin{matrix}−α_{2k−2} & 0 \\ 0 & α_{2k} \end{matrix}) (\begin{matrix}√ & \overline{1 + α_{2k−1}} \\ √ & \overline{1 − α_{2k−1}} \end{matrix})
= \sqrt{1 − α_{2k−1}} α_{2k}\sqrt{1 − α_{2k−1}} − \sqrt{1 + α_{2k−1}} α_{2k−2}\sqrt{1 + α_{2k−1}}

and

A_{2k+1,2k+3} = (\sqrt{1 + α_{2k−1}} \sqrt{1 − α_{2k−1}}) (\begin{matrix}0 & 0 \\ ρ_{2k} & 0 \end{matrix}) (\begin{matrix}√ & \overline{1 + α_{2k+1}} \\ √ & \overline{1 − α_{2k+1}} \end{matrix})
= \sqrt{1 − α_{2k−1}}\sqrt{1 − α_{2k}^{2}}\sqrt{1 + α_{2k+1}} .

The result follows. □

As in [127, 168], one can also use this to get Geronimus relations for the second Szegő map.

5. Regular MOPRL

The theory of regular (scalar) OPs was developed by Stahl–Totik [180] generalizing a definition of Ullman [190] for [−1, 1]. (See Simon [172] for a review and more about the history.) Here we develop the basics for MOPRL; it is not hard to do the same for MOPUC.

5.1. Upper Bound and Definition.

Theorem 5.1. Let dμ be a non-trivial l×l matrix-valued measure on ℝ with E = supp(dμ) compact. Then (with C(E) = logarithmic capacity of E)

lim sup_{n→∞} |det(A_{1} … A_{n})|^{1/n} ≤ C(E)^{l}.  (5.1)

Remarks. 1. |det(A_{1} … A_{n})| is constant over equivalent Jacobi parameters.

2. For the scalar case, this is a result of Widom [195] (it might be older) whose proof extends to the matrix case.

Proof. Let T_{n} be the Chebyshev polynomials for E (see [172, Appendix B] for a definition) and let T_{n}^{(l)} be T_{n} ⊗ 1, that is, the l × l matrix polynomial obtained by multiplying T_{n}(x) by 1. T_{n}^{(l)} is monic so, by (2.12) and (2.34),

|det(A_{1} … A_{n})|^{1/n} ≤ |det(∫ |T_{n}^{(l)}(x)|^{2} dμ(x))|^{1/2n}

≤ sup_{n} |T_{n}(x)|^{l/n}.

By a theorem of Szegő [183], sup_{n}|T_{n}(x)|^{1/n} → C(E), so (5.1) follows. □

Definition. Let dμ be a non-trivial l × l matrix- valued measure with E = supp(dμ) compact. We say μ is regular if equality holds in (5.1).

5.2. Density of Zeros. The following is a simple extension of the scalar results (see [172, Sect. 2]):

Theorem 5.2. Let dμ be a regular measure with E = supp(dμ). Let dν_{n} be the zero counting measure for det(p_{n}^{L}(x)), that is, if {x_{j}^{(n)}}_{j=1}^{nl} are the zeros of this determinant (counting degenerate zeros multiply), then

dν_{n} = \frac{1}{nl} ∑_{j=1}^{nl} δ_{x_{j}^{(n)}}.  (5.2)

Then dν_{n} converges weakly to dρ_{E}, the equilibrium measure for E.

Remark. For a discussion of capacity, equilibrium measure, quasi-every (q.e.), etc., see [172, Appendix A] and [115, 136, 158].

Proof. By (2.80) and (2.93),

|det(p_{n}^{R}(x))| ≥ (\frac{d}{D})^{l}(1 + (\frac{d}{D})^{2})^{(n−1)l/2}  (5.3)

so, in particular,

lim inf_{n} |det(p_{n}^{R}(x))|^{1/nl} ≥ (1 + (\frac{d}{D})^{2})^{1/2} ≥ 1.  (5.4)

But

|det(p_{n}^{R}(x))| = |det(A_{1} … A_{n})|^{−1} exp(−nlΦ_{ν_{n}}(x))  (5.5)

where Φ_{ν} is the potential of the measure ν. Let ν_{∞} be a limit point of ν_{n} and use equality in (5.1) and (5.4) to conclude, for x ∉ cvh(E),

exp(−Φ_{ν_{∞}}(x)) ≥ C(E)

which, as in the proof of Theorem 2.4 of [172], implies that ν_{∞} = ρ_{e}. □

The analogue of the almost converse of this last theorem has an extra subtlety relative to the scalar case:

Theorem 5.3. Let dμ be a non-trivial l × l matrix-valued measure on ℝ with E = supp(dμ) compact. If dν_{n} → dρ_{E}, then either μ is regular or else, with dμ = M(x) dμ_{tr}(x), there is a set S of capacity zero, so det(M(x)) = 0 for dμ_{tr}-a.e. x ∉ S.

Remark. By taking direct sums of regular point measures, it is easy to find regular measures where det(M(x)) = 0 for dμ_{tr}-a.e. x.

Proof. For a.e. x with det(M(x)) ≠ 0, we have (see Lemma 5.7 below)

p_{n}^{R}(x) ≤ C(n + 1)1.

The theorem then follows from the proof of Theorem 2.5 of [172]. □

5.3. General Asymptotics. The following generalizes Theorem 1.10 of [172] from OPRL to MOPRL—it is the matrix analogue of a basic result of Stahl–Totik [180]. Its proof is essentially the same as in [172]. By σ_{ess}(μ), we mean the essential spectrum of the block Jacobi matrix associated to μ.

Theorem 5.4. Let E ⊂ ℝ be compact and let μ be an l×l matrix-valued measure of compact support with σ_{ess}(μ) = E. Then the following are equivalent:

(i) μ is regular, that is, lim_{n→∞}|det(A_{1} … A_{n})|^{1/n} = C(E)^{l}. (ii) For all z in ℂ, uniformly on compacts,

lim sup|det(p_{n}^{R}(z))|^{1/n} ≤ e^{G_{E}(z)}.  (5.6)

(iii) For q.e. z in E, we have

lim sup|det(p_{n}^{R}(z))|^{1/n} ≤ 1.  (5.7)

Moreover, if (i)–(iii) hold, then (iv) For every z ∈ ℂ \ cvh(supp(dμ)), we have

lim_{n→∞} |det(p_{n}^{R}(z))|^{1/n} = e^{G_{E}(z)}.  (5.8)

(v) For q.e. z ∈ ∂Ω,

lim sup_{n→∞} |det(p_{n}^{R}(z))|^{1/n} = 1.  (5.9)

Remarks. 1. G_{E}, the potential theorists’ Green’s function for E, is defined by G_{E}(z) = − log(C(E)) − Φ_{ρ_{E}}(z).

2. There is missing here one condition from Theorem 1.10 of [172] involving general polynomials. Since the determinant of a sum can be much larger than the sum of the determinants, it is not obvious how to extend this result.

5.4. Weak Convergence of the CD Kernel and Consequences. The results of Simon in [174] extend to the matrix case. The basic result is:

Theorem 5.5. The measures dν_{n} and \frac{1}{(n+1)l} Tr(K_{n}(x, x)) dμ(x) have the same weak limits. In particular, if dμ is regular,

\frac{1}{(n + 1)l} Tr(K_{n}(x, x)dμ(x)) ⟶^{w} dρ_{E}.  (5.10)

As in [174], (π_{n}M_{x}π_{n})^{j} and (π_{n}M_{x}^{j}π_{n}) have a difference of traces which is bounded as n → ∞, and this implies the result. Once one has this, combining it with Theorem 2.20 leads to:

Theorem 5.6. Let I = [a, b] ⊂ E ⊂ ℝ with E compact. Let σ_{ess}(dμ) = E for an l × l matrix-valued measure, and suppose μ is regular for E and

dμ = W (x) dx + dμ_{s}  (5.11)

where dμ_{s} is singular and det(W (x)) > 0 for a.e. x ∈ I. Then,

(1)  lim_{n→∞} ∫_{I} p_{n}^{R}(x)^{†} dμ_{s}(x)p_{n}^{R}(x) → 0  (5.12)
(2)  ∫_{I} ∥\frac{1}{n + 1} ∑_{j=0}^{n} p_{j}^{R}(x)^{†}W (x)p_{j}(x) − ρ_{E}(x)1∥ dx → 0 (5.13)

5.5. Widom’s Theorem.

Lemma 5.7. Let dμ be an l × l matrix-valued measure supported on a compact E ⊂ ℝ and let dη be a scalar measure on ℝ so

dμ(x) = W (x) dη(x) + dμ_{s}(x)  (5.14)

where dμ_{s} is dη singular. Suppose for η-a.e. x,

det(W (x)) > 0.  (5.15)

Then for η-a.e. x, there is a positive real function C(x) so that

∥p_{n}^{R}(x)∥ ≤ C(x)(n + 1)1.  (5.16)

In particular,

|det(p_{n}^{R}(x))| ≤ C(x)^{l}(n + 1)^{l}.  (5.17)

Proof. Since ∥p_{n}^{R}∥_{R}^{2} = 1, we have

∑_{n=0}^{∞}(n + 1)^{−2}∥p_{n}^{R}∥_{R}^{2} < ∞  (5.18)

so

∑_{n=0}^{∞}(n + 1)^{−2} Tr(p_{n}^{R}(x)^{†}W (x)p_{n}^{R}(x)) < ∞

for η-a.e. x. Since (5.15) holds, for a.e. x,

W (x) ≥ b(x)1

for some scalar function b(x). Thus, for a.e. x,

∑_{n=0}^{∞}(n + 1)^{−2} Tr(p_{n}^{R}(x)^{†}p_{n}^{R}(x)) ≤ C(x)^{2}.

Since ∥A∥^{2} ≤ Tr(A^{†}A), we find (5.16), which in turn implies (5.17). □

This lemma replaces Lemma 4.1 of [172] and then the proof there of Theorem 1.12 extends to give (a matrix version of the theorem of Widom [195]):

Theorem 5.8. Let dμ be an l×l matrix-valued measure with σ_{ess}(dμ) = E ⊂ ℝ compact. Suppose

dμ(x) = W (x) dρ_{E}(x) + dμ_{s}(x)  (5.19)

with dμ_{s} singular with respect to dρ_{E}. Suppose for dρ_{E}-a.e. x, det(W (x)) > 0. Then μ is regular.

5.6. A Conjecture. We end our discussion of regular MOPRL with a conjecture—an analog of a theorem of Stahl–Totik [180]; see also Theorem 1.13 of [172] for a proof and references. We expect the key will be some kind of matrix Remez inequality. For direct sums, this conjecture follows from Theorem 1.13 of [172].

Conjecture 5.9. Let E be a finite union of disjoint closed intervals in ℝ. Suppose μ is an l×l matrix-valued measure on ℝ with σ_{ess}(dμ) = E. For each η > 0 and m = 1, 2, …, define

S_{m,η} = \{x : μ([x − \frac{1}{m}, x + \frac{1}{m}]) ≥ e^{−ηm}1\}.  (5.20)

Suppose that for each η (with |·| = Lebesgue measure)

lim_{m→∞} |E \ S_{m,η}| = 0

Then μ is regular.

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Department of Mathematics, Rice University, Houston, TX 77005, USA

  E-mail address: damanik@rice.edu

Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK

  E-mail address: alexander.pushnitski@kcl.ac.uk

Mathematics 253–37, California Institute of Technology, Pasadena, CA 91125, USA

  E-mail address: bsimon@caltech.edu