CMV MATRICES: FIVE YEARS AFTER

BARRY SIMON*

Abstract. CMV matrices are the unitary analog of Jacobi matrices; we review their general theory.

1. Introduction

The Arnold Principle: If a notion bears a personal name, then this name is not the name of the inventor. The Berry Principle: The Arnold Principle is applicable to itself.

— V. I. Arnold

On Teaching Mathematics, 1997 [8]

(Arnold says that Berry formulated these principles.)

In 1848, Jacobi [45] initiated the study of quadratic forms J(x_{1}, …, x_{n}) = ∑_{k=1}^{n} b_{k}x_{k}^{2} +2 ∑_{k=1}^{n−1} a_{k}x_{k}x_{k+1}, that is, essentially n×n matrices of the form

J = (\begin{matrix}b_{1} & a_{1} & 0 & … & 0 \\ a_{1} & b_{2} & a_{2} & … & 0 \\ 0 & a_{2} & b_{3} & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & … & … & a_{n−1} & b_{n} \end{matrix})  (1.1)

and found that the eigenvalues of J were the zeros of the denominator of the continued fraction

\frac{1}{b_{1} − z − \frac{a_{1}^{2}}{b_{2} − z − \frac{a_{2}^{2}}{⋯}}}  (1.2)

In the era of the birth of the spectral theorem, Toeplitz [79], Hellinger– Toeplitz [44], and especially Stone [75] realized that Jacobi matrices were universal models of selfadjoint operators, A, with a cyclic vector, φ_{0}.

Date: March 10, 2006.

∗ Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125, USA. E-mail: bsimon@caltech.edu. Supported in part by NSF grant DMS-0140592.

To appear in the Proceedings of the W. D. Evans’ 65th Birthday Conference.

To avoid technicalities, consider the case where A is bounded, and suppose initially that H is infinite-dimensional. By cyclicity, {A^{k}φ_{0}}_{k=0}^{∞} are linearly independent, so by applying Gram–Schmidt to φ_{0}, Aφ_{0}, A^{2}φ_{0}, …, we get polynomials p_{j}(A) of degree exactly j with positive leading coefficients so that

φ_{j} = p_{j}(A)φ_{0}  (1.3)

are an orthonormal basis for H. By construction,

φ_{j} ⊥ φ_{0}, Aφ_{0}, …, A^{j−1}φ_{0}

so

⟨φ_{j}, Aφ_{k}⟩ = 0 j ≥ k + 2  (1.4)

Because A is selfadjoint, we see ⟨φ_{j}, Aφ_{k}⟩ = 0 also if j ≤ k − 2. Thus, the matrix ⟨φ_{j}, Aφ_{k}⟩ has exactly the form (1.1) where a_{j} > 0 (since p_{j}(A) has leading positive coefficient).

Put differently, for all A, φ_{0}, there is a unitary U : H → ℓ^{2} (given by Fourier components in the φ_{j} basis), so UAU^{−1} has the form J and φ_{0} = (1, 0, 0, …)^{t}. The Jacobi parameters, {a_{n}, b_{n}}_{n=1}^{∞}, are intrinsic, which shows there is exactly one J (with φ_{0} = (1, 0, 0, …)^{t}) in the unitary equivalence class of (A, φ_{0}).

There is, of course, another way of describing unitary invariants for (A, φ_{0}): the spectral measure dμ defined by

∫ x^{n} dμ(x) = ⟨φ_{0}, A^{n}φ_{0}⟩  (1.5)

There is a direct link from dμ to the Jacobi parameters: the p_{j}(x) are orthonormal polynomials associated to dμ, and the Jacobi parameters are associated to the three-term recursion relation obeyed by the p’s:

xp_{j}(x) = a_{j+1}p_{j+1} + b_{j+1}p_{j}(x) + a_{j}p_{j−1}(x)  (1.6)

(where p_{−1} ≡ 0).

Here we are interested in the analog of these structures for unitary matrices. We begin by remarking that for a general normal operator, N, the right form of cyclicity is that {N^{k}(N^{∗})^{ℓ}φ_{0}}_{k,ℓ=0}^{∞} is total. Since A = A^{∗}, only {A^{k}φ_{0}}_{k=0}^{∞} enters. Since U^{∗} = U^{−1}, for unitaries U^{k}(U^{∗})^{ℓ} = U^{k−ℓ} and the right notion of cyclicity is that {U^{k}φ_{0}}_{k=−∞}^{∞} is total.

Some parts of the above four-fold equivalence:

(1) unitary equivalence classes of (A, φ_{0})

(2) spectral measures, that is, probability measures dμ on ℝ with bounded support and infinite support

(3) Jacobi parameters

(4) Jacobi matrices are immediate for the unitary case. Namely, (1) ⇔ (2) holds since there is a spectral theorem for unitaries, and so, a one-one correspondence between unitary equivalence classes of (U, φ_{0}) on infinite-dimensional spaces and probability measures on ∂𝔻 (𝔻 = {z | |z| < 1}) with infinite support.

More subtle is the analog of Jacobi parameters. Starting from such a probability measure on ∂𝔻, one can form the monic orthogonal polynomials Φ_{n}(z) and find (see Szegő [77]; see also Section 1.5 of [69]) {α_{n}}_{n=0}^{∞} ∈ 𝔻^{∞}, so

zΦ_{n}(z) = Φ_{n+1}(z) + ᾱ_{n}z^{n} \overline{Φ_{n}(1/z̄)}  (1.7)

While Verblunsky [81] defined the α_{n} in a different (albeit equivalent) way, he proved a theorem (called Verblusnky’s theorem in [69]; see also [68]) that says this map dμ → {α_{n}}_{n=0}^{∞} is one-one and onto all of 𝔻^{∞}, so (1)–(3) for the unitary case have been well understood for sixty-five years.

Surprisingly, (4) (i.e., the canonical matrix form for unitaries) is of much more recent vintage. The key paper by Cantero, Moral, and Velázquez [12] was submitted in April 2001 — so we are witnessing five years of study in the area — it is reviewing these developments that is the main scope of this review article. Spectral theory of differential and difference operators has been an enduring theme of Des Evans’ research and I am pleased to dedicate this review to him.

There is an “obvious” matrix to try, namely, G_{kℓ} = ⟨φ_{k}, zφ_{ℓ}⟩ with φ_{k} = Φ_{k}/∥Φ_{k}∥ the orthonormal polynomials. This GGT matrix (as it is named in [69]; see Section 10 below) has two defects. First, {φ_{k}}_{k=0}^{∞} is a basis if and only if ∑_{n=0}^{∞}|α_{n}|^{2} = ∞, and if it is not, G_{kℓ} is not unitary and is not conjugate to multiplication by z (in that case, one can look at the minimal dilation of G, which is discussed in [15, 69]). Second, it obeys (1.4), but in general, ⟨φ_{j}, U^{∗}φ_{k}⟩ ≠ 0 for all j ≥ k + 1, that is, G is not of finite width measured from the diagonal. CMV [12] has the following critical ideas:

(a) The basis χ_{k} obtained by orthonormalizing 1, z, z^{−1}, z^{2}, z^{−2}, … can be written in terms of φ_{ℓ}(z), \overline{φ_{ℓ}(1/z̄)}, and powers of z.

(b) The matrix C_{kℓ} = ⟨χ_{k}, zχ_{ℓ}⟩ is unitary and five-diagonal.

(c) C can be factorized into C = LM where L is a direct sum of 2 × 2 unitary matrices and M the direct sum of a single 1 × 1 and 2 × 2 matrices.

It turns out that these key ideas appeared about ten years earlier in the numeric matrix literature (still, of course, much later than the 1930’s resolution of (1)–(3)). Intimately related to this history is what we will call the AGR factorization in Section 11 — the ability to write G in the case of n × n matrices as a product Θ̃_{0} … Θ̃_{n−1} ˜Θ̃_{n−1} of matrices with a single 2 × 2 block placed in 1 and a finite matrix which is diagonal, differing from 1 in a single place (see Section 11 for details).

In 1986, Ammar, Gragg, and Reichel [5] found the AGR factorization for orthogonal matrices — here the α_{j} are real and the Θ(α_{j}) are reflections, so the AGR factorization can be viewed as an iteration of a Householder algorithm. In this paper, they also had a proof of the LM factorization for this case. This proof (a variant of which appears in Section 10), which works in general to go from the AGR factorization of the GGT matrix to an LM factorization, was only given in the orthogonal case since they did not yet have the AGR factorization for general unitaries.

In 1988 (published 1991), AGR [6] extended the AGR factorization to the general unitary case and realized the connection to Szegő recursion. While they could have proven an LM factorization from this using the method in [5], they did not and the general LM factorization only appeared in [11].

In 1991, Bunse-Gerstner and Elsner [11] found the LM factorization for a general finite unitary and noted it was a five-diagonal representation. Watkins [82] codified and cleaned up those results and emphasized the connection to OPUC and found a proof of Szegő recursion from the LM factorization. Virtually all the main results from [12] are already in Watkins [82].

We will continue to use the name CMV matrices, in part because the analytic revolution we discuss here was ushered in by their work and in part because the name has been used now in many, many publications.

Here is a summary of the rest of this review. Section 2 presents the basics, essentially notation and (a)–(c) above. Section 3 discusses “other” CMV matrices. In particular, we consider two kinds of finite variants. In the selfadjoint case, restricting the matrix by taking the first n rows and columns preserves selfadjointness but the analog for unitaries does not, and we have both the nonunitary cutoff CMV matrices obtained from the first n rows and columns and the unitary finite CMV matrices which are models of finite unitary matrices with a cyclic vector. Section 4 discusses CMV matrices for matrix-valued measures — something that is new here. Section 5 discusses the effect on Verblunsky coefficients of rank one multiplication perturbations, and Section 6 the formula for the resolvent of the CMV matrices, the analog of well-known Green’s function formulae for Jacobi matrices. Sections 7 and 9 discuss perturbation results, and Section 8 a general theorem on the essential spectrum of CMV matrices. Section 10 discusses the AGR factorization discussed above as preparation for the Killip–Nenciu discussion of five-diagonal models for β-distribution of eigenvalues, the subject of Section 11. Section 12 discusses the defocusing AL flows, which bear the same relation to CMV matrices as Toda flows do to Jacobi matrices. Finally, Section 13 discusses a natural reduction of CMV matrices to a direct sum of two Jacobi matrices when all Verblunsky coefficients are real.

We do not discuss the use of CMV matrices to compute the zeros of OPUC. These zeros are the eigenvalues of the cutoff CMV matrix. We note that this method of computing zeros was used in the recent paper of Martínez-Finkelshtein, McLaughlin, and Saff [57]. Numerical aspects of CMV matrices deserve further study.

While this is primarily a review article, there are numerous new results, including:

(1) an analysis of what matrices occur as cutoff CMV matrices (Section 3)

(2) an analysis following Watkins [82] of the LM factorization without recourse to Szegő recursion (Section 3)

(3) the basics of CMV matrices for matrix-valued measures (Section 4)

(4) a new proof of AGR factorization using intermediate bases (Section 10)

(5) a new trace class estimate for GGT matrices that relies on AGR factorization (Section 10)

(6) a reworked proof of the Killip–Nenciu [50] theorem on the measure that Haar measure on 𝕌(n) induces on Verblunsky coefficients (Section 11)

(7) an argument of AGR is made explicit and streamlined to go from AGR to LM factorization (Section 10)

It is a pleasure to thank M. Cantero, P. Deift, L. Golinskii, F. Marcellán, A. Martínez-Finkelshtein, L. Moral, I. Nenciu, P. Nevai, L. Velázquez, and D. Watkins for useful input.

2. CMV Matrices: The Basics

In this section, we define the CMV basis, the CMV matrix, and the LM factorization.

CMV matrices can be thought of in terms of unitary matrices or OPUC. We start with the OPUC point of view. A measure dμ in ∂𝔻 is called nontrivial if it is not supported on a finite set; equivalently, if every polynomial, which is not identically zero, is nonzero in L^{2}(∂𝔻, dμ). Then one can define orthonormal polynomials, φ_{n}(z) (or φ_{n}(z, dμ)), by

(i)  φ_{n}(z) = κ_{n}z^{n} + lower order; κ_{n} > 0  (2.1)
(ii) φ_{n} ⊥ {1, z, z^{2}, …, z^{n−1}}  (2.2)

We define the monic polynomials Φ_{n}(z) by Φ_{n}(z) = φ_{n}(z)/κ_{n}. The Szegő dual is defined by

P_{n}^{∗}(z) = z^{n} \overline{P_{n}(1/z̄)}  (2.3)

that is,

P_{n}(z) = ∑_{j=0}^{n} c_{j}z^{j} ⇒ P_{n}^{∗}(z) = ∑_{j=0}^{n} c̄_{n−j}z^{j}  (2.4)

The symbol ^{∗} is n-dependent and is sometimes applied to polynomials of degree at most n, making the notation ambiguous!

Then there are constants {α_{n}}_{n=0}^{∞} in 𝔻, called Verblunsky coefficients, (sometimes we will write α_{n}(dμ)) so that

ρ_{n} φ_{n+1}(z) = zφ_{n}(z) − ᾱ_{n}φ_{n}^{∗}(z)  (2.5)

where

ρ_{n} = (1 − |α_{n}|^{2})^{1/2}  (2.6)

Moreover, μ → {α_{n}}_{n=0}^{∞} sets up a one-one correspondence between nontrivial measures on ∂𝔻 and points of 𝔻^{∞} (as we will show below). (2.5) (called Szegő recursion after [77]) and this one-one correspondence are discussed in [69, 70]; see also [68]. Applying ^{∗} for P_{n+1} to (2.5), we get

ρ_{n} φ_{n+1}^{∗}(z) = φ_{n}^{∗}(z) − α_{n}zφ_{n}(z)  (2.7)

If one defines (of course, ρ = (1 − |α|^{2})^{1/2})

A(α) = \frac{1}{ρ} (\begin{matrix}z & −ᾱ \\ −αz & 1 \end{matrix})  (2.8)

then (2.5)/(2.7) can be written

\binom{φ_{n+1}}{φ_{n+1}^{∗}} = A(α_{n})\binom{φ_{n}}{φ_{n}^{∗}}  (2.9)

Since det(A) = z, we have

A(α)^{−1} = \frac{1}{ρz} (\begin{matrix}1 & ᾱ \\ αz & z \end{matrix})  (2.10)

and thus

ρ_{n} φ_{n}(z) = \frac{φ_{n+1}(z) + ᾱ_{n}φ_{n+1}^{∗}(z)}{z}  (2.11)
ρ_{n} φ_{n}^{∗}(z) = (φ_{n+1}^{∗}(z) + α_{n}φ_{n+1}(z))  (2.12)

Introduce the notation [y_{1}, …, y_{k}] for the span of the vectors y_{1}, …, y_{k} and P_{[y_{1},...,y_{k}]} for the projection onto the space [y_{1}, …, y_{k}]. For x ∉ [y_{1}, …, y_{k}], define

[x; y_{1}, …, y_{k}] = \frac{(1 − P_{[y_{1},...,y_{k}]})x}{∥(1 − P_{[y_{1},...,y_{k}]})x∥}  (2.13)

the normalized projection of x onto [y_{1}, …, y_{k}]^{⊥}, that is, the result of adding x to a Gram–Schmidt procedure.

We define π_{n} to be P_{[1,...,z^{n−1}]}.

By the definition of φ_{n} and the fact that ^{∗} is anti-unitary on Ran π_{n} and takes z^{j} to z^{n−j}, we have

φ_{n} = [z^{n}; 1, …, z^{n−1}] φ_{n}^{∗} = [1; z, …, z^{n}]  (2.14)

With this notation out of the way, we can define the CMV basis {χ_{n}}_{n=0}^{∞} and alternate CMV basis {x_{n}}_{n=0}^{∞} as the Laurent polynomials (i.e., polynomials in z and z^{−1}) obtained by applying Gram–Schmidt to 1, z, z^{−1}, z^{2}, z^{−2}, … and 1, z^{−1}, z, z^{−2}, z^{2}, …, that is, for k = 0, 1, …,

χ_{2k} = [z^{−k}; 1, z, …, z^{−k+1}, z^{k}]  χ_{2k−1} = [z^{k}; 1, z, …, z^{k−1}, z^{−k+1}] (2.15)
x_{2k} = [z^{k}; 1, z^{−1}, …, z^{k−1}, z^{−k}]  x_{2k−1} = [z^{−k}; 1, z^{−1}, …, z^{−k+1}, z^{k−1}] (2.16)

So, in particular, as functions in L^{2}(∂𝔻, dμ),

x_{n} = χ̄_{n}  (2.17)

and as Laurent polynomials,

x_{n}(z) = \overline{χ_{n}(1/z̄)}  (2.18)

As realized by CMV [12], the {χ_{n}}_{n=0}^{∞} and {x_{n}}_{n=0}^{∞} are always a basis of L^{2}(∂𝔻, dμ) since the Laurent polynomials are dense on C(∂𝔻), while {φ_{n}}_{n=0}^{∞} may or may not be a basis (it is known that this is a basis if and only if ∑_{n}|α_{n}|^{2} = ∞; see Theorem 1.5.7 of [69]). On the other hand, the χ and x bases can be expressed in terms of φ and φ^{∗} by (2.14) and the fact that multiplication by z^{ℓ} is unitary. For example,

x_{2k} = z^{−k}[z^{2k}; z^{k}, z^{k−1}, …, z^{2k−1}, 1]
= z^{−k}[z^{2k}; 1, …, z^{2k−1}]
= z^{−k}φ_{2k}(z)

The full set is

χ_{2k}(z) = z^{−k}φ_{2k}^{∗}(z) χ_{2k−1}(z) = z^{−k+1}φ_{2k−1}(z)  (2.19)
x_{2k}(z) = z^{−k}φ_{2k}(z)  x_{2k−1}(z) = z^{−k}φ_{2k−1}^{∗}(z)  (2.20)

Since χ and x are bases, the matrices of multiplication by z in these bases are unitary. So we have the unitary matrices

C_{mℓ} = ⟨χ_{m}, zχ_{ℓ}⟩  (2.21)
C̃_{mℓ} = ⟨x_{m}, zx_{ℓ}⟩  (2.22)

called the CMV matrix and the alternate CMV matrix, respectively. By (2.18), the unitarity of C and z̄ = z^{−1}, we see

C̃_{mk} = C_{km}  (2.23)

that is, C and C̃ are transposes of each other. We will see shortly that C is five-diagonal, but this follows now by noting that both z and z^{−1} map [χ_{0}, …, χ_{k}] into [χ_{0}, …, χ_{k+2}].

CMV [12], Ammar–Gragg–Reichel [5], Bunse-Gerstner and Elsner [11], and Watkins [82] also discussed the important factorization C = LM as follows:

L_{mk} = ⟨χ_{m}, zx_{k}⟩ M_{mk} = ⟨x_{m}, χ_{k}⟩  (2.24)

Since {x_{k}}_{k=1}^{∞} is a basis, ⟨f, g⟩ = ∑_{k=0}^{∞}⟨f, x_{k}⟩⟨x_{k}, g⟩, and thus

C = LM C̃ = ML  (2.25)

The point of this factorization is that L and M have a simpler structure than C. Indeed, L is a direct sum of 2 × 2 blocks and M of a single 1 × 1 block and then 2 × 2 blocks.

One can (and we will) see this based on calculations, but it is worth seeing why it is true in terms of the structure of the CMV and alternate CMV basis. Notice that χ_{2n−1} and χ_{2n} span the two-dimensional space [1, z, z^{−1}, …, z^{n}, z^{−n}]∩[1, z, z^{−1}, …, z^{n−1}, z^{−n+1}]^{⊥} and so do x_{2n−1} and x_{2n}. This shows that M is a direct sum of 1_{1×1} and 2 × 2 matrices. Similarly, χ_{2n} and χ_{2n+1} span [1, …, z^{−n}, z^{n+1}] ∩ [1, …, z^{−n+1}, z^{n}]^{⊥}, as do zx_{2n} and zx_{2n+1} (even for n = 0). Thus L has a 2×2 block structure.

In fact, we can use Szegő recursion in the form (2.5), (2.7), (2.11), (2.12) to find the 2 × 2 matrices explicitly. For example, taking (2.12) for n = 2m − 1, we get

φ_{2m}^{∗} = ρ_{2n−1}φ_{2m−1}^{∗} − α_{2m−1}φ_{2m}

and multiplying by z^{−m} yields (by (2.20)/(2.20)),

χ_{2m} = −α_{2m−1}x_{2m} + ρ_{2m−1}x_{2m−1}

This plus similar calculations imply

Theorem 2.1 ([5, 11, 12, 82]). Let

Θ(α) = (\begin{matrix}ᾱ & ρ \\ ρ & −α \end{matrix})  (2.26)

Then C = LM and

L = Θ(α_{0}) ⊕ Θ(α_{2}) ⊕ Θ(α_{4}) ⊕ ⋯ ⊕ Θ(α_{2m}) ⊕ ⋯  (2.27)

and

M = 1_{1×1} ⊕ Θ(α_{1}) ⊕ Θ(α_{3}) ⊕ ⋯ ⊕ Θ(α_{2m+1}) ⊕ ⋯  (2.28)

Doing the multiplication yields

C = ( \begin{matrix}ᾱ_{0} & ᾱ_{1}ρ_{0} & ρ_{1}ρ_{0} & 0 & 0 & … \\ ρ_{0} & −ᾱ_{1}α_{0} & −ρ_{1}α_{0} & 0 & 0 & … \\ 0 & ᾱ_{2}ρ_{1} & −ᾱ_{2}α_{1} & ᾱ_{3}ρ_{2} & ρ_{3}ρ_{2} & … \\ 0 & ρ_{2}ρ_{1} & −ρ_{2}α_{1} & −ᾱ_{3}α_{2} & −ρ_{3}α_{2} & … \\ 0 & 0 & 0 & ᾱ_{4}ρ_{3} & −ᾱ_{4}α_{3} & … \\ … & … & … & … & … & … \end{matrix} )  (2.29)

C is five-diagonal, that is, only nonzero in those diagonals C_{k k+j} with j = 0, ±1, ±2. Notice that half of the elements with j = ±2 are zero, so it is only “barely” five-diagonal — and it cannot be tridiagonal or even four-diagonal since

Proposition 2.2 ([13]). If {A_{jk}}_{1≤j, k<∞} is a semi-infinite unitary matrix and

k − j ∉ {−1, …, n} ⇒ A_{jk} = 0

then A is a direct sum of finite blocks of size at most n + 1.

This was proven for n = 1 in [10] and conjectured for n = 2 in a draft of [69] before motivating [13].

While our construction has been for α_{n}’s which come from a dμ and, in particular, which obey

|α_{j}(dμ)| < 1  (2.30)

Θ defines a unitary so long as |α_{n}| ≤ 1. We thus define a CMV matrix to be a matrix of the form (2.25)–(2.29) for any {α_{n}}_{n=0}^{∞} with |α_{n}| ≤ 1. If |α_{n}| < 1 for all n, we call C a proper CMV matrix, and if |α_{n}| = 1 for some n, we call it an improper CMV matrix.

To state the analog of Stone’s selfadjoint cyclic model theorem, we need another definition. A cyclic unitary model is a unitary operator, U, on a (separable) Hilbert space, H, with a distinguished unit vector, v_{0}, which is cyclic, that is, finite linear combinations of {U^{n}v_{0}}_{n=−∞}^{∞} are dense in H. We call the model proper if dim(H) = ∞ and improper if dim(H) < ∞. It is easy to see that the model is improper if and only if P (U) = 0 for some polynomial, P, which can be taken to have degree dim(H) − 1. Two cyclic unitary models, (H, U, v_{0}) and (H̃, Ũ, ṽ_{0}), are called equivalent if and only if there is a unitary W from H onto H̃ so that

W v_{0} = ṽ_{0} W UW^{−1} = Ũ  (2.31)

Theorem 2.3. There is a one-one correspondence between proper cyclic unitary models and proper CMV matrices, C, in that δ_{0} = (1, 0, 0, …)^{t} is cyclic for any such C and every equivalence class contains exactly one proper CMV model: (ℓ^{2}, C, δ_{0}).

Remarks. 1. There is behind this a four-fold equivalence:

(i) equivalence classes of proper cyclic unitary models (ii) nontrivial probability measures on ∂𝔻 (iii) Verblunsky coefficients {α_{n}(dμ)}_{n=0}^{∞} in 𝔻^{∞} (iv) proper CMV matrices. The spectral theorem sets up a one-one correspondence between (i) and (ii), while the definition of CMV matrices between (iii) and (iv). Szegő recursion sets up a map from dμ to {α_{n}(dμ)}_{n=1}^{∞}. As we will show, each (ℓ^{2}, C, δ_{0}) is a cyclic model, so the key remaining fact is the uniqueness.

2. A corollary of this is Verblunsky’s theorem (also called “Favard’s theorem for the unit circle”) that each {α_{n}}_{n=0}^{∞} ∈ 𝔻 is the Verblunsky coefficient for some dμ. See [69, 68] for further discussion and other proofs.

Proof. As explained in Remark 1, we need only prove that any proper CMV matrix has δ_{0} as a cyclic vector, and that if {α_{n}^{(0)}}_{n=0}^{∞} are the Verblunsky coefficients for C and dμ the spectral measure for δ_{0}, then

α_{n}(dμ) = α_{n}^{(0)}  (2.32)

Let δ_{n} be the unit vector in ℓ^{2} with coefficient 1 in place n and 0 elsewhere; index labelling for our vectors starts at 0. By direct calculations using the LM representation,

C^{n+1}δ_{0} − ρ_{0}^{(0)}ρ_{1}^{(0)} … ρ_{2n}^{(0)}δ_{2n+1} ∈ [δ_{0}, …, δ_{2n}]  (2.33)
(C^{∗})^{n}δ_{0} − ρ_{0}^{(0)} … ρ_{2n−1}^{(0)}δ_{2n} ∈ [δ_{0}, …, δ_{2n−1}]  (2.34)

It follows that δ_{0} is cyclic and

χ_{n}(dμ) = W δ_{n}  (2.35)

where W is the spectral representation from ℓ^{2} to L^{2}(∂𝔻, dμ). (2.35) follows from (2.33)–(2.34), induction, and the Gram–Schmidt definition of χ.

By (2.35) and

⟨δ_{0}, Cδ_{0}⟩ = ᾱ_{0}^{(0)}  ⟨χ_{0}, zχ_{0}⟩ = α_{0}(dμ)

⟨δ_{2n−2}, Cδ_{2n−1}⟩ = ᾱ_{2n−1}^{(0)}ρ_{2n−2}^{(0)} ⟨χ_{2n−2}, zχ_{2n−1}⟩ = ᾱ_{2n−1}ρ_{2n−2}
⟨δ_{2n}, Cδ_{2n−1}⟩ = ᾱ_{2n}^{(0)}ρ_{2n−1}^{(0)}  ⟨χ_{2n}, zχ_{2n−1}⟩ = ᾱ_{2n}(dμ)ρ_{2n−1}(dμ)

we obtain (2.32) by induction. □

3. Cutoff, Finite, Two-Sided, Periodic, and Floquet CMV Matrices

In this section, we will discuss various matrices constructed from or related to CMV matrices. Some are finite, and in that case, we will also discuss the associated characteristic polynomial which turns out to be equal or related to the basic ordinary or Laurent polynomials of OPUC: the monic orthogonal and paraorthogonal polynomials and the discriminant. The basic objects we will discuss are:

(i) Cutoff CMV matrices, that is, π̃_{n}Cπ̃_{n} where π̃_{n} is projection onto the span of the first n of 1, z, z^{−1}, . . ..

(ii) Finite CMV matrices, the upper n × n block of an improper CMV

matrix with α_{n−1} ∈ ∂𝔻. (iii) Two-sided CMV matrices defined for {α_{n}}_{n=−∞}^{∞} via extending L

and M in the obvious way to a two-sided form. (iv) Periodic CMV matrices. The special case of two-sided CMV ma-

trices when α_{n+p} = α_{n} for some p.

(v) Floquet CMV matrices. Periodic CMV matrices have a direct integral decomposition whose fibers are p×p matrices that are finite CMV matrices with a few changed matrix elements.

Cutoff CMV matrices. A cutoff CMV matrix is the restriction of a proper CMV matrix to the upper n × n block, that is, top n rows and leftmost n columns. We use C^{(n)} to denote the cutoff matrix associated to C. A glance at (2.29) shows that C^{(n)} depends on {α_{j}}_{j=0}^{n−1}. Here is a key fact:

Proposition 3.1. Let Φ_{n}(z) be the monic orthogonal polynomial associated to C (i.e., Φ_{n} = κ_{n}^{−1}φ_{n}). Then

Φ_{n}(z) = det(z1 − C^{(n)})  (3.1)

Proof. If π_{n} is the projection onto [1, …, z^{n−1}] and π̃_{n} on the span of the first n of 1, z, z^{−1}, …, then π_{n} and π̃_{n} are unitarily equivalent under a power of z. So if M_{z}f = zf, then π_{n}M_{z}π_{n} and

π̃_{n}M_{z}π̃_{n} ≡ C^{(n)}  (3.2)

are unitarily equivalent, and thus, (3.1) is equivalent to

Φ_{n}(z) = det(z1 − π_{n}M_{z}π_{n})  (3.3)

Let z_{j} be a zero of Φ_{n} of multiplicity k_{j} and let P_{j}(z) = Φ_{n}(z)/(z − z_{j})^{k_{j}}. Then with A = π_{n}M_{z}π_{n}, we have

(A − z_{j})^{k_{j}}P_{j} = 0 (A − z_{j})^{k_{j}−1}P_{j} ≠ 0

Thus, as z_{j} runs through the distinct zeros, {(A − z_{j})^{ℓ}P_{j} | ℓ = 0, 1, …, k_{j} − 1} gives us a Jordan basis in which A has a k_{j} × k_{j} block for each z_{j} of the form

(\begin{matrix}z_{j} & & & 1 & & … & & & … & & & 0 \\ 0 & & & z_{j} & & 1 & & & … & & & 0 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 0 & & & 0 & & … & & & … & & & 1 \\ 0 & & & 0 & & … & & & … & & & z_{j} \end{matrix})

and thus

det(z − A) = ∏(z − z_{j})^{k_{j}} = Φ_{n}(z)  □

Corollary 3.2. The zeros of Φ_{n}(z) lie in 𝔻.

Remark. See Section 1.7 of [69] for six other proofs of this theorem.

Proof. Let A = π_{n}M_{z}π_{n}. Then ∥A∥ ≤ 1, so obviously, eigenvalues lie in \overline{𝔻}. If Aη = z_{0}η with η ∈ Ran π_{n} and z_{0} ∈ ∂𝔻, then ∥Aη∥ = ∥η∥, so π_{n}zη = zη and thus as polynomials (z−z_{0})η = 0. Since the polynomials are a division ring, η = 0, that is, there are no eigenvalues on ∂𝔻. □

To classify cutoff CMV matrices, we need to understand how Θ(α) arises from a 2 × 2 change of basis.

Lemma 3.3. Let f, g be two independent unit vectors with

⟨f, g⟩ = α  (3.4)

Let φ_{1}, φ_{2} be the result of applying Gram–Schmidt to f, g and ψ_{1}, ψ_{2} to g, f. Let M be the matrix of change of basis

φ_{1} = m_{11}ψ_{1} + m_{12}ψ_{2}  (3.5)
φ_{2} = m_{21}ψ_{1} + m_{22}ψ_{2}  (3.6)

Then α ∈ 𝔻 and M = Θ(α).

Proof. |α| < 1 by the Schwarz inequality and the independence of f and g. Note that

∥g − ⟨f, g⟩f∥^{2} = ∥g∥^{2} + |⟨f, g⟩|^{2}∥f∥^{2} − 2 Re[⟨f, g⟩⟨g, f⟩]
= 1 − |α|^{2} ≡ ρ^{2}

so

φ_{2} = ρ^{−1}(g − αf)  (3.7)

ψ_{2} = ρ^{−1}(f − ᾱg)  (3.8)

From this, a direct calculation shows that

m_{11} = ⟨ψ_{1}, φ_{1}⟩ = ⟨g, f⟩ = ᾱ
m_{12} = ⟨ψ_{2}, φ_{1}⟩ = ρ^{−1}(1 − |α|^{2}) = ρ
m_{21} = ⟨ψ_{1}, φ_{2}⟩ = ρ^{−1}(1 − |α|^{2}) = ρ
m_{22} = ⟨ψ_{2}, φ_{2}⟩ = ρ^{−2}(α + α|α|^{2} − 2α) = −α  □

Remark. One can use this lemma to deduce the form of L and M in the LM factorization without recourse to Szegő recursion, and then use their form to deduce the Szegő recursion. This is precisely Watkins’ approach [82] to the factorization and Szegő recursion.

Given any matrix A and vector δ_{0}, define

V_{k} = span(δ_{0}, Aδ_{0}, A^{∗}δ_{0}, …, A^{k}δ_{0}, (A^{∗})^{k}δ_{0})  (3.9)

which has dimension 2k + 1 if and only if the vectors are independent.

Here is a complete classification of cutoff CMV matrices analogous to Theorem 2.3:

Theorem 3.4. Let A be an n × n cutoff CMV matrix with δ_{0} = (1, 0, …, 0)^{t}. Then:

(i) If n = 2k + 1, V_{k} has dimension n. If n = 2k, then span[V_{k−1} ∪ {A^{k}δ_{0}}] has dimension n.

(ii) If n = 2k + 1, A^{∗} is an isometry on span(V_{k−1} ∪ {(A)^{k}δ_{0}}) and A is an isometry on span(V_{k−1} ∪ {(A^{∗})^{k}δ_{0}}). If n = 2k, A^{∗} is an isometry on span(V_{k−2} ∪ {A^{k−1}δ_{0}, A^{k}δ_{0}}) and A is an isometry on V_{k−1}.

(iii) ∥A∥ ≤ 1. (iv) A is not unitary.

Conversely, if A_{0}, δ_{0} are a pair of an n × n matrix and vector δ_{0} obeying (i)–(iv), then there is a basis in which δ_{0} = (1, 0, …, 0)^{t} and A is a cutoff CMV matrix.

(A, δ_{0}) determine the Verblunsky coefficients (α_{0}, …, α_{n−1}) uniquely. In particular, two cutoff CMV matrices with distinct {α_{j}}_{j=0}^{n} are not unitarily equivalent by a unitary preserving δ_{0}.

Proof. Suppose first that A is a cutoff CMV matrix, that is, A = π̃_{n}Cπ̃_{n}. By definition of π̃_{n},

π̃_{n}C^{j}δ_{0} = C^{j}δ_{0} π̃_{n}(C^{∗})^{ℓ}δ_{0} = (C^{∗})^{ℓ}δ_{0}

for j = 0, 1, …, k and ℓ = 0, 1, …, k (resp., k − 1) if n = 2k + 1 (resp., 2k). It follows that for those values of j and ℓ,

C^{j}δ_{0} = A^{j}δ_{0} (C^{∗})^{ℓ}δ_{0} = (A^{∗})^{ℓ}δ_{0}  (3.10)

so that (i) holds.

This also shows that A^{∗}A^{j}δ_{0} = A^{j−1}δ_{0} for j = 1, …, k, and from this and (3.10), it follows that A^{∗} is unitary on span{A^{j}δ_{0}}_{j=0}^{k} ∪ {(A^{∗})^{ℓ}δ_{0}}_{ℓ=0}^{k−1 (or k − 2)}. Similarly, we get the unitarity result for A.

(iii) is obvious since ∥π̃_{n}∥ = ∥C∥ = 1 and (iv) follows since there is a vector φ in Ran(π̃_{n}) with π̃_{n}Cφ ≠ Cφ. This completes the proof of the first paragraph of the theorem.

As a preliminary to the converse, we note that π̃_{n} commutes with either L or M, so a finite CMV matrix has the form L_{n}M_{n} where if n = 2k + 1 is odd (1 = 1 × 1 identity matrix),

L_{n} = Θ(α_{0}) ⊕ ⋯ ⊕ Θ(α_{2k−2}) ⊕ α_{2k}1  (3.11)
M_{n} = 1 ⊕ Θ(α_{1}) ⊕ ⋯ ⊕ Θ(α_{2k−1})  (3.12)

and if n = 2k is even,

L_{n} = Θ(α_{0}) ⊕ ⋯ ⊕ Θ(α_{2k−2})  (3.13)
M_{n} = 1 ⊕ Θ(α_{1}) ⊕ ⋯ ⊕ Θ(α_{2k−3}) ⊕ α_{2k−1}1  (3.14)

We will prove that when A obeys (i)–(iv), then A has an L_{n}M_{n} factorization with parameter α_{j} given intrinsically by A. This will not only prove the converse but the uniqueness of the map from {α_{j}}_{j=0}^{N−1} to cutoff CMV matrices, and so it will complete the proof of the theorem.

We first consider the case n = 2k + 1 odd. Define χ_{ℓ} to be the basis obtained by Gram–Schmidt on δ_{0}, Aδ_{0}, A^{∗}δ_{0}, …, A^{k}δ_{0}, (A^{∗})^{k}δ_{0} (this is possible because (i) implies these vectors are linearly independent) and define x_{ℓ} to be the result of Gram–Schmidt on δ_{0}, A^{∗}δ_{0}, Aδ_{0}, …, (A^{∗})^{k}δ_{0}, A^{k}δ_{0}. Then if A is written in χ_{ℓ} basis,

A = LM  (3.15)

where

M_{kℓ} = ⟨x_{k}, χ_{ℓ}⟩ L_{kℓ} = ⟨χ_{k}, Ax_{ℓ}⟩  (3.16)

We need to show that L, M have the form (3.11)/(3.12).

If P_{m} is the projection to the orthogonal complement of V_{m−1} and f = P_{m−1}(A^{∗})^{m}δ_{0}/∥P_{m−1}(A^{∗})^{m}δ_{0}∥ and g = P_{m−1}A^{m}δ_{0}/∥P_{m−1}A^{m}δ_{0}∥, then {χ_{ℓ}, x_{ℓ}}_{ℓ=m,m+1} are given by Lemma 3.3. So M has the form 1 ⊕ Θ(α_{1}) ⊕ ⋯ ⊕ Θ(α_{2k−1}) as required.

Let W_{ℓ} be the projection onto the span of the first 2ℓ of δ_{0}, Aδ_{0}, A^{∗}δ_{0}, A^{2}δ_{0}, … and W̃_{ℓ} the span of the first 2ℓ of δ_{0}, A^{∗}δ_{0}, Aδ_{0}, (A^{∗})^{2}δ_{0}, . . .. By hypothesis (ii), A is an isometry on W̃_{1}, W̃_{2}, …, W̃_{k}, and by the same hypothesis, AA^{∗}φ = φ for φ = δ_{0}, A^{∗}δ_{0}, …, (A^{∗})^{k}δ_{0}. So it follows that A maps W̃_{ℓ} to W_{ℓ} for ℓ = 1, …, k. Thus, by Lemma 3.3, the 2k × 2k upper block of L is Θ(α_{0}) ⊕ Θ(α_{2}) ⊕ ⋯ ⊕ Θ(α_{2k−2}). Since A and A^{∗} are contractions, L must have 0’s in the bottom and rightmost column, except for the lower corner. That corner value, call it α_{2k}, must have |α_{2k}| ≤ 1 by (iii) and |α_{2k}| < 1 by (iv). Thus, we have the required LM factorization if n = 2k + 1.

Now let ℓ = 2k be even. Define χ_{ℓ} as before, but define x̃_{ℓ} by Gram– Schmidt on Aδ_{0}, δ_{0}, A^{2}δ_{0}, A^{∗}δ_{0}, …, A^{k}δ_{0}, (A^{∗})^{k−1}δ_{0}. Then A written in χ_{ℓ} basis has the form (3.15) where

M_{kℓ} = ⟨x̃_{k}, Aχ_{ℓ}⟩ L_{kℓ} = ⟨χ_{k}, x̃_{ℓ}⟩  (3.17)

We need to show that L, M have the form (3.13)/(3.14).

Define W_{ℓ} to be the span of δ_{0}, Aδ_{0}, A^{∗}δ_{0}, …, A^{ℓ}δ_{0}, (A^{ℓ})^{∗}δ_{0} and W̃_{ℓ} the span of Aδ_{0}, δ_{0}, …, (A^{∗})^{ℓ−1}δ_{0}, A^{ℓ+1}δ_{0}. As above, A is an isometry of W_{ℓ} to W̃_{ℓ}, so M has the form 1⊕Θ(α_{1})⊕⋯⊕Θ(α_{2k−3})⊕α_{2k−1}1 where |α_{2k−1}| < 1 by condition (iv). Similarly, L has a Θ(α_{0})⊕⋯⊕Θ(α_{2k−2}) block structure. This proves (3.13)/(3.14) and completes the case n = 2k. □

Remark. This theorem sets up a one-one correspondence between {α_{j}}_{j=0}^{n−1} ∈ 𝔻^{n} and cutoff CMV matrices.

Finite CMV matrices. As discussed in Section 2, C, originally defined for |α_{j}| < 1, has an extension to |α_{j}| ≤ 1 via the Θ formula for L, M. Since |α_{j_{0}}| = 1 implies ρ_{j_{0}} = 0 and Θ(α_{j_{0}}) is diagonal, if |α_{j_{0}}| = 1, C({α_{j}}) leaves ℂ^{j_{0}+1} (i.e., vectors φ with φ_{k} = 0 if k ≥ j_{0}+1) invariant and C ↾ ℂ^{j_{0}+1} is a (j_{0} + 1) × (j_{0} + 1) unitary matrix. If |α_{0}|, …, |α_{n−2}| < 1 and |α_{n−1}| = 1, the corresponding n × n matrix is called a finite n × n CMV matrix, C_{n}({α_{0}, …, α_{n−2}, α_{n−1}}). C_{n} has the form L_{n}M_{n} where (3.11)/(3.12) or (3.13)/(3.14) hold, and now α_{n−1} ∈ ∂𝔻.

If U is an n × n matrix and δ_{0} a cyclic vector in the sense that {U^{m}δ_{0}}_{m=−∞}^{∞} is total, δ_{0} cannot be orthogonal to any eigenvector. So U has to have n distinct eigenvalues {λ_{j}}_{j=1}^{n} and the eigenvectors {ψ_{j}}_{j=1}^{n} obey |⟨δ_{0}, ψ_{j}⟩|^{2} = a_{j} ≠ 0. The unitary invariants of the pair (U, δ_{0}) are the spectral measures ∑_{j=1}^{n} a_{j}δ_{λ_{j}} where {λ_{j}}_{j=1}^{n} are arbitrary distinct points and the a_{j} > 0 have the single restriction ∑_{j=1}^{n} a_{j} = 1. Thus, the number of real parameters is n + (n − 1) = 2n − 1. The number of free parameters in an n × n finite CMV matrix is n − 1 complex numbers in 𝔻 and one in ∂𝔻, that is, 2(n − 1) + 1 = 2n − 1. This suggests that

Theorem 3.5. There is a one-one correspondence between unitary equivalence classes of n × n unitary matrices with a cyclic vector and finite CMV matrices in that each equivalence class contains one CMV matrix (fixed by δ_{0} = (1, 0, …, 0)^{t}) and two CMV matrices with distinct parameters are not unitarily equivalent by a unitary fixing (1, 0, …, 0)^{t}.

The proof is identical to the proof of Theorem 3.4 except that A nonunitary is replaced by A unitary so |α_{n−1}| = 1. As noted in the discussion after Lemma 3.3, this approach is close to that in Watkins [82]. This theorem is related to results in Ammar–Gragg–Reichel [5] and Killip–Nenciu [51]. The latter talk about matrices with CMV shape having the CMV form.

Instead of the cutoff CMV matrix, π_{n}M_{z}π_{n}, one can look at π̂_{n}M_{z}π̂_{n} where π̂_{n} is a not necessary selfadjoint projection. CMV [14] have shown that finite CMV matrices have this form and that they are the only normal operators of this form.

Two-sided CMV matrices. In a sense, CMV matrices are twosided. For example, if α_{n} ≡ 0, C is unitarily equivalent to a two-sided shift since C^{k}δ_{0} = δ_{2k−1} and C^{−k}δ_{0} = δ_{2k}. However, as structures, the matrix is semi-infinite and there is a cyclic vector which is often not true for two-sided matrices. Thus, there is an extension to “two-sided” examples.

Let \{α_{n}\}_{n=−∞}^{∞} be a two-sided sequence of numbers in \overline{𝔻}. Let H = ℓ^{2}(ℤ), that is, two-sided sequences {u_{n}}_{n=−∞}^{∞} with ∑_{n=−∞}^{∞}|u_{n}|^{2} < ∞. Let Θ_{j}(β) be Θ(β) acting on the two indices j and j + 1. Define

E({α_{j}}_{j=−∞}^{∞}) = L̃({α_{j}}_{j=−∞}^{∞})M̃({α_{j}}_{j=−∞}^{∞})  (3.18)

where

M̃ = ⨁_{j=−∞}^{∞} Θ_{2j−1}(α_{2j−1})

L̃ = ⨁_{j=−∞}^{∞} Θ_{2j}(α_{2j})

E is called the extended CMV matrix.

The extended CMV matrix was introduced in [69]. Earlier, Bourget, Howland, and Joye [10] had considered some doubly infinite fivediagonal matrices which factor into a product of two direct sums of 2 × 2 matrices, but the 2 × 2 blocks were general unitaries rather than Θ’s.

While E is natural and important for the periodic case, we will also see that it arises in the theory of essential spectrum of C (see Section 8).

One reason for the name “extended CMV matrix” is:

Proposition 3.6. If α_{−1} = −1, then E is a direct sum on ℓ^{2}(−∞, −1]⊕ ℓ^{2}[0, ∞) and E ↾ ℓ^{2}[0, ∞) is the CMV matrix C({α_{j}}_{j=0}^{∞}). Moreover, E ↾ ℓ^{2}(−∞, −1] is unitarily equivalent to C({ᾱ_{−j−2}}_{j=0}^{∞}).

Remark. ℓ^{2}[0, ∞) means those u ∈ ℓ^{2}(ℤ) with u_{n} = 0 if n < 0 and ℓ^{2}(−∞, −1] those with u_{n} = 0 if n > −1.

Proof. Θ(−1) = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ), so both L̃ and M̃ leave ℓ^{2}[0, ∞) and ℓ^{2}(−∞, −1] invariant. Thus, E does.

M̃ ↾ ℓ^{2}[0, ∞) = M and L̃ ↾ ℓ^{2}[0, ∞) = L, so E ↾ ℓ^{2}[0, ∞) is C({α_{j}}_{j=0}^{∞}).

For the restriction to ℓ^{2}(−∞, −1], note first that ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) Θ(α) ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) = Θ(−ᾱ). Thus, by labeling the basis backwards, E is unitarily equivalent to something that looks very much like C({−ᾱ_{−j−2}}_{j=0}^{∞}) except M starts with −1, not 1. By the discussion in Section 5, there is a unitary that flips the sign of this −1 and all the α_{j}’s. □

Changing α_{−1} from its value to α_{−1} = −1 is a perturbation of rank at most two, so by the Kato–Rosenblum theorem [65], the a.c. spectrum of E is that of a direct sum of two C’s. Since these a.c. spectra are only restricted by simplicity, we see that the a.c. spectrum of E has multiplicity at most 2, but is otherwise arbitrary: it can be partially multiplicity 0, partially 1, and partially 2. In particular, E may not have a cyclic vector.

It is a theorem of Simon [67] that the singular spectrum of E is simple. This is an analog of a theorem of Kac [47, 48] and Gilbert [37, 38] for Schrödinger operators.

Periodic CMV matrices. If {α_{j}}_{j=0}^{∞} is a sequence of Verblunsky coefficients with

α_{j+p} = α_{j}  (3.19)

for j ≥ 0, p fixed, and ≥ 1, then α_{j} has a unique extension to j ∈ ℤ obeying (3.19). The corresponding E is called a periodic CMV matrix. The theory is simpler if p is even, which we henceforth assume. As explained in [70], there are several ways to analyze odd p once one has understood even p.

Associated to {α_{j}}_{j=0}^{p−1} is a natural Laurent polynomial, called the discriminant, Δ(z; {α_{j}}_{j=0}^{p−1}),

Δ(z) = z^{−p/2}Tr(A(α_{p−1}, z)A(α_{p−2}, z) … A(α_{0}, z))  (3.20)

where

A(α, z) = ρ^{−1} (\begin{matrix}z & −ᾱ \\ −zα & 1 \end{matrix})  (3.21)

This is analyzed in Section 11.1 of [70]. Δ(z) is real on ∂𝔻 and has positive leading coefficient. This means Δ(z) has p real parameters. This suggests the map from {α_{j}}_{j=0}^{p} (of real dimension 2p) to Δ is many to 1, with inverse images generically of dimension p (= 2p − p). This is in fact true: the inverse images are tori of dimension d ≤ p (we will see what d is in a moment). They are called isospectral tori.

For fixed {α_{j}}_{j=0}^{p−1}, Δ^{−1}([−2, 2]) (which is the spectrum of E) lies in ∂𝔻 and is naturally p closed intervals whose endpoints are Δ^{−1}({−2, 2}). Generically (in α), the intervals are disjoint, that is, their complement (called the gaps) is p nonempty open intervals. In general, the number of open intervals in the gaps is d, the dimension of the isospectral torus.

Floquet CMV matrices. If T: ℓ^{2}(ℤ) → ℓ^{2}(ℤ) by (T u)_{n} = u_{n+p} and p is even, and if α_{n} = α_{n+p}, then T M̃T^{−1} = M̃ and T L̃T^{−1} = L̃, so

T ET^{−1} = E  (3.22)

(We will not consider odd p in detail, but we note in that case, T M̃T^{−1} = L̃ and T L̃T^{−1} = M̃ so, since M̃^{t} = M̃ and L̃^{t} = L̃ (on account of Θ^{t} = Θ where ^{t} is transpose), we have that T ET^{−1} = E^{t}.)

Since T and E commute, they can be “simultaneously” diagonalized, in this case represented on a direct integral representation. One way of making this explicit is to define, for each β ∈ ∂𝔻, the space ℓ_{β}^{∞}, the sequences {u_{n}}_{n=−∞}^{∞} obeying u_{n+p} = βu_{n}. This is clearly a space of dimension p since {u_{n}}_{n=−∞}^{∞} mapping to {u_{n}}_{n=0}^{p−1} (i.e., restriction) maps ℓ_{β}^{∞} to ℂ^{p}.

By (3.22), E, which maps bounded sequences to bounded sequences, maps ℓ_{β}^{∞} to ℓ_{β}^{∞}, and so defines a finite-dimensional operator E_{p}(β) under the explicit relation of ℓ_{β}^{∞} mentioned above. One sees

E_{p}(β) = L_{p}M_{p}(β)  (3.23)

where

L_{p} = Θ_{0}(α_{0}) ⊕ ⋯ ⊕ Θ_{p−2}(α_{p−2})  (3.24)
M_{p}(θ) = Θ_{1}(α) ⊕ ⋯ ⊕ Θ_{p−3}(α_{p−3}) ⊕ Θ_{p−1}^{(β)}(α_{p−1})  (3.25)

where Θ_{p−1}^{(β)}(α) acts on δ_{p−1} and δ_{0}, and in that (ordered) basis has the form

(\begin{matrix}ᾱ & ρβ \\ ρβ̄ & −α \end{matrix})  (3.26)

E_{p}(β) is called the Floquet CMV matrix. To make precise the connection to E, we define the unitary Fourier transform F: ℓ^{2}(ℤ) → L^{2}(∂𝔻, \frac{dθ}{2π}; ℂ^{p}), the set of L^{2} functions on ∂𝔻 with values in ℂ^{p} by

(Fu)_{k}(β) = ∑_{n=−∞}^{∞} β^{−n}u_{k+np}  (3.27)

Then

(FEF^{−1}g)(β) = E_{p}(β)g(β)  (3.28)

(For details, see Section 11.2 of [70].)

Finally, we note a general relation of the eigenvalues of E_{p}(β) and the discriminant, Δ(z), of (3.19). For z_{0} ∈ ∂𝔻 is an eigenvalue of E_{p}(β) if and only if there is (u_{1}, u_{0})^{t} so that after a p-step transfer, we get β(u_{1}, u_{0})^{t}, that is, if and only if z_{0}^{p/2}β is an eigenvalue of T_{p}(z_{0}). This is true if and only if z_{0}^{−p/2}T_{p}(z_{0}) has eigenvalues β and β^{−1} if and only if Δ(z_{0}) = β + β^{−1}. It follows that

det(z − E_{p}(β)) = ( ∏_{j=0}^{p−1} ρ_{j})[z^{p/2}[Δ(z) − β − β^{−1}]]  (3.29)

for both sides are monic polynomials of degree p and they have the same zeros.

4. CMV Matrices for Matrix-Valued Measures

Because of applications to perturbations of periodic Jacobi and CMV matrices [16], interest in matrix-valued measures (say, k × k matrices) has increased. Here we will provide the CMV basis and CMV matrices in this matrix-valued situation; these results are new here. Since adjoints of finite-dimensional matrices enter but we want to use ^{∗} for Szegő reversed polynomials, in this section we use ^{†} for matrix adjoint.

Measures which are nontrivial in a suitable sense are described by a sequence {α_{j}}_{j=0}^{∞} of Verblunsky coefficients that are k × k matrices with ∥α_{j}∥ < 1.

To jump to the punch line, we will see that C still has an LM factorization, where Θ(α) is the 2k × 2k matrix

Θ(α) = (\begin{matrix}α^{†} & ρ^{L} \\ ρ^{R} & −α \end{matrix})  (4.1)

where

ρ^{L} = (1 − α^{†}α)^{1/2} ρ^{R} = (1 − αα^{†})^{1/2}  (4.2)

It is an interesting calculation to check that Θ is unitary, that is,

(\begin{matrix}α & ρ^{R} \\ ρ^{L} & −α^{†} \end{matrix}) (\begin{matrix}α^{†} & ρ^{L} \\ ρ^{R} & −α \end{matrix}) = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix})  (4.3)

That αα^{†} + (ρ^{R})^{2} = 1 = (ρ^{L})^{2} + α^{†}α follows from (4.2). That αρ^{L} − ρ^{R}α = ρ^{L}α^{†} − α^{†}ρ^{R} = 0 follows by expanding the square roots in (4.2) in a Taylor series and using α(α^{†}α)^{m} = (αα^{†})^{m}α.

To describe the model specifically, we have a k×k matrix-valued (normalized, positive) measure which can be described as follows: dμ_{t}(θ) is a positive scalar measure on ∂𝔻 and for a.e. (dμ_{t}(θ)) a matrix A(θ) obeying

A(θ) ≥ 0 Tr(A(θ)) = 1  (4.4)

We write dμ(θ) = A(θ) dμ_{t}(θ). We assume dμ is normalized in the sense that ∫ A(θ) dμ_{t} = 1. We will consider H_{R} to be the k×k matrix-valued functions, f, on ∂𝔻 with

∫ Tr(f(θ)^{†}A(θ)f(θ)) dμ_{t}(θ) < ∞  (4.5)

The measure dμ is called nontrivial if

dim[span{B_{ℓ}z^{ℓ}}_{ℓ=0}^{n−1}] = nk^{2}  (4.6)

for each n. Equivalently, for each n and {B_{ℓ}}_{ℓ=0}^{n−1} in L(ℂ^{k}), we have ∑_{ℓ=0}^{n−1} B_{ℓ}z^{ℓ} = 0 in H_{R} implies B_{0} = B_{1} = ⋯ = B_{n−1} = 0. Also equivalent is that ⟨φ, A(θ)φ⟩ dμ_{t}(θ) is nontrivial for each φ ∈ ℂ^{k}\{0}.

Similarly, we define H_{L} to be f’s with

∫ Tr(f(θ)A(θ)f^{†}(θ)) dμ_{t}(θ) < ∞  (4.7)

It is easy to see that nontriviality implies (4.6) holds also in H_{L}.

We define two “inner products,” sesquilinear forms from H_{R} and H_{L} to L(ℂ^{k}), the k × k matrices:

⟨⟨f, g⟩⟩_{R} = ∫ f^{†}(θ) dμ(θ)g(θ)  (4.8)
⟨⟨f, g⟩⟩_{L} = ∫ g(θ) dμ(θ)f^{†}(θ)  (4.9)

The right side of (4.8) is shorthand for

∫ f^{†}(θ)A(θ)g(θ) dμ_{t}(θ)

so the LHS of (4.5) is Tr(⟨⟨f, f⟩⟩_{R}). The symbols L, R (for left and right) come from

⟨⟨f, gB⟩⟩_{R} = ⟨⟨f, g⟩⟩_{R}B  (4.10)
⟨⟨f, Bg⟩⟩_{L} = B⟨⟨f, g⟩⟩_{L}  (4.11)

for constant k × k matrices, B.

The normalized matrix OPUC, φ_{n}^{R}, φ_{n}^{L}, are polynomials in z of degree n with matrix coefficients with

⟨⟨φ_{n}^{R}, φ_{m}^{R}⟩⟩_{R} = δ_{nm}1 ⟨⟨φ_{n}^{L}, φ_{m}^{L}⟩⟩_{R} = δ_{nm}1  (4.12)

This determines φ uniquely up to a unitary right (resp., left) prefactor. We will pick this prefactor by demanding

φ_{n}^{R,L}(z) = κ_{n}^{R,L}z^{n} + lower order  (4.13)
κ_{n+1}^{L}(κ_{n}^{L})^{−1} > 0 (κ_{n}^{R})^{−1}κ_{n+1}^{R} > 0  (4.14)

With this choice of normalization, one has a sequence of k×k matrices, {α_{n}}_{n=0}^{∞}, and the recursion relations

zφ_{n}^{L} = ρ_{n}^{L}φ_{n+1}^{L} + α_{n}^{†}(φ_{n}^{R})^{∗}  (4.15)
zφ_{n}^{R} = φ_{n+1}^{R}ρ_{n}^{R} + (φ_{n}^{L})^{∗}α_{n}^{†}  (4.16)
(φ_{n}^{L})^{∗} = (φ_{n+1}^{L})^{∗}ρ_{n}^{L} + zφ_{n}^{R}α_{n}  (4.17)
(φ_{n}^{R})^{∗} = ρ_{n}^{R}φ_{n+1}^{∗} + α_{n}zφ_{n}^{L}  (4.18)

where ρ_{n}^{R}, ρ_{n}^{L} are given by (4.2) and P_{n}^{∗}(z) = z^{n}P_{n}(1/z̄)^{†}. For construction of φ_{n}^{L,R} and proof of (4.15)–(4.18), see [7] or [69, Section 2.13] following Delsarte et al. [17] and Geronimo [29].

It will help to also have the following, which can be derived from (4.15)–(4.18):

φ_{n+1}^{L} = ρ_{n}^{L}zφ_{n}^{L} − α_{n}^{†}(φ_{n+1}^{R})^{∗}  (4.19)
(φ_{n+1}^{R})^{∗} = ρ_{n}^{R}(φ_{n}^{R})^{∗} − α_{n}φ_{n+1}^{L}  (4.20)

Following (2.19) and (2.20), we define the CMV and alternate CMV basis by

χ_{2k}(z) = z^{−k}(φ_{2k}^{R}(z))^{∗} χ_{2k−1}(z) = z^{−k+1}φ_{2k−1}^{L}(z)  (4.21)
x_{2k}(z) = z^{−k}φ_{2k}^{L}(z)  x_{2k−1}(z) = z^{−k}(φ_{2k−1}^{R}(z))^{∗}  (4.22)

Proposition 4.1. {χ_{ℓ}(z)}_{ℓ=0}^{∞} and {x_{ℓ}(z)}_{ℓ=0}^{∞} are ⟨⟨· , ·⟩⟩_{L} orthonormal, that is,

⟨⟨χ_{ℓ}, χ_{m}⟩⟩_{L} = δ_{ℓm}  ⟨⟨x_{ℓ}, x_{m}⟩⟩_{L} = δ_{ℓm}  (4.23)

Moreover, χ_{ℓ} is in the module span of the first ℓ of 1, z, z^{−1}, … and x_{ℓ} of 1, z^{−1}, z, . . ..

Remark. By module span of {f_{j}(z)}_{j=1}^{m} of scalar functions, f, we mean elements in H_{L} of the form ∑_{j=1}^{m} B_{j}f_{j}(z) where B_{1}, B_{2}, … are fixed k × k matrices.

Proof. (4.23) for ℓ = m holds by (4.12) and (4.21), (4.22) if we note that

⟨⟨P_{n}^{∗}, Q_{n}^{∗}⟩⟩_{L} = ⟨⟨Q_{n}, P_{n}⟩⟩_{R}  (4.24)

It is obvious from the definition of χ_{ℓ} and x_{ℓ} that they lie in the proper span. To get (4.23) for ℓ < m, we need to know that χ_{ℓ} is orthogonal to the first ℓ − 1 of 1, z, z^{−1}, … and x_{ℓ} to the first ℓ − 1 of 1, z, z^{−1}, . . .. For cases where χ_{ℓ} or x_{ℓ} is given by φ^{L}, this follows from ⟨⟨z^{k}, φ_{ℓ}^{L}⟩⟩ = 0 for 0 ≤ k < ℓ and when it is a (φ_{ℓ}^{R})^{∗} from (4.24) which says

⟨⟨z^{k}, (φ_{ℓ}^{R})^{∗}⟩⟩_{L} = ⟨⟨φ_{ℓ}^{R}, z^{ℓ−k}⟩⟩ = 0

for 0 ≤ ℓ − k < ℓ. □

By a (left-) module basis for H_{L}, we mean a sequence {f_{j}}_{j=0}^{∞} orthonormal in ⟨⟨· , ·⟩⟩_{L}, that is, ⟨⟨f_{j}, f_{ℓ}⟩⟩_{L} = δ_{jℓ} so that as {B_{j}}_{j=0}^{N} runs through all N-tuples of k × k matrices, ∑_{j=0}^{N} B_{j}f_{j} is a sequence of subspaces whose union is dense in H_{L}. For any such basis, any η ∈ H_{L} has a unique convergent expansion,

η = ∑_{j=0}^{∞}⟨⟨f_{j}, η⟩⟩f_{j}  (4.25)

{χ_{j}}_{j=0}^{∞} and {x_{j}}_{j=0}^{∞} are both module bases. That means, if C_{jℓ} is defined by

C_{jℓ} = ⟨χ_{j}, zχ_{ℓ}⟩  (4.26)

then the matrix, obtained by using the k × j blocks, C_{jℓ}, is unitary. Moreover,

C_{jℓ} = ∑_{m} L_{jm}M_{mℓ}  (4.27)

where

L_{jℓ} = ⟨χ_{j}, zx_{ℓ}⟩ M_{jℓ} = ⟨x_{j}, χ_{ℓ}⟩  (4.28)

In (4.19), set n = 2k − 1 and multiply by z^{−k} to get

x_{2k} = −α_{n}^{†}χ_{2k} + ρ_{n}^{L}χ_{2k−1}  (4.29)

where a bottom row of Θ is clear. In this way, using (4.15), (4.18), (4.19), and (4.20), one obtains:

Theorem 4.2. With Θ_{j}(α) given by (4.1) acting on ℂ^{2k} corresponding to δ_{j}, δ_{j+1}, we have

M = 1_{1×1} ⊕ Θ_{1}(α_{1}) ⊕ Θ_{3}(α_{3}) ⊕ ⋯
L = Θ_{0}(α_{0}) ⊕ Θ_{2}(α_{2}) ⊕ Θ_{4}(α_{4}) ⊕ ⋯

The analog of (2.29) is

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 & ρ_{1}^{R}α_{2}^{†} & −α_{1}α_{2}^{†} & ρ_{2}^{L}α_{3}^{†} & ρ_{2}^{L}ρ_{3}^{L} & … \\ 0 & ρ_{1}^{R}ρ_{2}^{R} & −α_{1}ρ_{2}^{R} & −α_{2}α_{3}^{†} & −α_{2}ρ_{3}^{L} & … \\ 0 & 0 & 0 & ρ_{3}^{R}α_{4}^{†} & −α_{3}α_{4}^{†} \end{matrix} )  (4.30)

We note for later purposes that for this matrix case, the GGT matrix, which we will discuss in Section 10, has the form

G_{kℓ} = \begin{cases}−α_{k−1}ρ_{k}^{L}ρ_{k+1}^{L} … ρ_{ℓ−1}^{L}α_{ℓ}^{†} & 0 ≤ k ≤ ℓ \\ ρ_{ℓ}^{R} & k = ℓ + 1 \\ 0 & k ≥ ℓ + 2 \end{cases}  (4.31)

that is,

G = ( \begin{matrix}α_{0}^{†} & ρ_{0}^{L}α_{1}^{†} & ρ_{0}^{L}ρ_{1}^{L}α_{2}^{†} & ρ_{0}^{L}ρ_{1}^{L}ρ_{2}^{L}α_{3}^{†} \\ ρ_{0}^{R} & −α_{0}α_{1}^{†} & −α_{0}ρ_{1}^{L}α_{2}^{†} & −α_{0}ρ_{1}^{L}ρ_{2}^{L}α_{3}^{†} \\ 0 & ρ_{1}^{R} & −α_{1}α_{2}^{†} & −α_{1}ρ_{2}^{L}α_{3}^{†} \\ … & … & … & … \end{matrix} )  (4.32)

5. Rank One Covariances

For selfadjoint matrices, the most elementary rank one perturbations are diagonal, that is, J ↦ J + λ(δ_{n}, ·)δ_{n}, where δ_{n} is the vector with 1 in position n and 0 elsewhere. The impact of such a change on Jacobi parameters is trivial: a_{m} → a_{m}, b_{m} → b_{m} + λδ_{nm} (if we label vectors in the selfadjoint case starting at n = 1). One of our goals is to find the analog for CMV matrices, where we will see that the impact on Verblunsky coefficients is more subtle.

We will also address a related issue: In the spectral theory of OPUC, the family of measures, dμ_{λ} with α_{n}(dμ_{λ}) = λα_{n} for a fixed {α_{n}}_{n=0}^{∞}, called an Aleksandrov family, plays an important role analogous to a change of boundary condition in ODE’s. If φ_{n} are the normalized OPUC, the GGT matrix,

G_{kℓ}({α_{n}}_{n=0}^{∞}) = ⟨φ_{k}, zφ_{ℓ}⟩  (5.1)

has the property that G({λα_{n}}_{n=0}^{∞}) − G({α_{n}}_{n=0}^{∞}) is rank one (see [69, page 259]). But for the CMV basis, C({λα_{n}}_{n=0}^{∞})−C({α_{n}}_{n=0}^{∞}) is easily seen to be infinite rank (if the α’s are not mainly 0). However, we will see here that for a suitable U_{λ} (depending on λ but not on α!), U_{λ}C({λα_{n}}_{n=0}^{∞})U_{λ}^{−1} − C({α_{n}}_{n=0}^{∞}) is rank one.

We need to begin by figuring out what are natural rank one perturbations. The key realization is that the proper format is multiplicative:

Let P be a rank one projection and W_{θ} = e^{iθP} = (1 − P) + e^{iθ}P. Then W_{θ} −1 = (e^{iθ} −1)P is rank one, and for any U, UW_{θ} is a rank one perturbation of U. It will be convenient to parametrize by λ = e^{iθ} ∈ ∂𝔻. Thus, we define

W^{(m)}(λ) = 1 + (e^{iθ} − 1)⟨δ_{m}, · ⟩δ_{m}  (5.2)

and given any CMV matrix C, we let

C^{m}(λ) = CW^{(m)}(λ)  (5.3)

We will use C^{m}(λ; {α_{k}}) where we want to make the α-dependence explicit. Notice that

C_{ℓk}^{m}(λ) = \begin{cases}C_{ℓk} & if k ≠ m \\ λC_{ℓk} & if k = m \end{cases}  (5.4)

that is, we multiply column m by λ.

Part of the result we are heading towards is that

α_{ℓ}(C^{m}(λ)) = \begin{cases}α_{ℓ}(C) & ℓ < m \\ λ^{−1}α_{ℓ}(C) & ℓ ≥ m \end{cases}  (5.5)

In particular, C^{0}(λ̄) realizes the fact that C({λα_{k}}_{k=0}^{∞}) is unitarily equivalent to a rank one perturbation of C({α_{k}}_{k=0}^{∞}). (5.5) for the important case m = 0 is due to Simon [69, Theorem 4.2.9] and for the general case to Simon [72]. We will sketch the various proofs.

While we will eventually provide explicit unitaries that show C^{m}(λ; {α_{j}}_{j=0}^{∞}) is unitarily equivalent to C (right side of (5.5)), we begin with a direct proof of (5.5) in case m = 0.

Theorem 5.1. C^{m=0}(λ; {α_{j}}_{j=0}^{∞}) has Verblunsky coefficients {λ^{−1}α_{j}}_{j=0}^{∞}.

Remark. If M^{(λ)} = λ1 ⊕ Θ(α_{1}) ⊕ Θ(α_{3}) ⊕ ⋯, that is, the 1 in the upper left corner is replaced by λ, then LM^{(λ)} = CW^{(0)}(λ).

Sketch. (See Theorems 4.2.9 and Subsection 1.4.16 of [69].) By definition,

C_{λ}^{m=0} − C = (λ − 1)CP_{0}  (5.6)

where P_{0} = ⟨δ_{0}, · ⟩δ_{0}. Define, for z ∈ 𝔻, F_{λ} and the f_{λ} by

F_{λ}(z) = ⟨δ_{0}, (C_{λ}^{m=0} − z)(C_{λ}^{m=0} + z)^{−1}δ_{0}⟩  (5.7)
= \frac{1 + zf_{λ}(z)}{1 − zf_{λ}(z)}  (5.8)

Using the second resolvent formula and (5.6) implies (see Subsection 1.4.16 of [69]) that

f_{λ}(z) = λ^{−1}f(z)  (5.9)

The Schur algorithm and Geronimus theorem (see Chapter 3 of [69]) then imply (5.5) for m = 0. □

For discussion of the movement of eigenvalues under the perturbations of Theorem 5.1, see [6, 13, 25] and Theorem 3.2.17 of [69].

The key to an explicit unitary equivalence is the following. Let

ν(λ) = (\begin{matrix}1 & 0 \\ 0 & λ \end{matrix})  ν̃(λ) = (\begin{matrix}λ & 0 \\ 0 & 1 \end{matrix})  (5.10)

Then, by a simple calculation,

ν(λ)Θ(λ^{−1}α)ν(λ) = λΘ(α)  (5.11)
ν̃(λ)^{−1}Θ(λ^{−1}α)ν̃(λ)^{−1} = λ^{−1}Θ(α)  (5.12)

Note that (5.11) does not use ν(λ) and ν(λ)^{−1} but ν(λ) in both places. Similarly, (5.12) has ν̃(λ)^{−1} in both places. In the full calculation, one does not use

ULMU^{−1} = (ULU^{−1})(UMU^{−1})

but rather

ULMU^{−1} = (ULU)(U^{−1}MU^{−1})  (5.13)

We need a notation for diagonal matrices. D(1^{2k}(1λ)^{∞}) indicates the diagonal matrix with entries 1 2k times, then alternating 1’s and λ’s. Thus,

W^{(m)}(λ) = D(1^{m}λ1^{∞})  (5.14)

Using (5.14), (5.13), (5.11), and (5.12), a direct calculation (see Section 5 of [72]) shows:

Theorem 5.2. For n = 0, 1, 2, …, define

U_{2k−1} = D(1^{2k}(1λ)^{∞})  (5.15)
U_{2k} = D(λ^{2k}(1λ)^{∞})  (5.16)
T_{n,λ}({α_{j}}_{j=0}^{∞}) = β_{j}  (5.17)

where

β_{j} = \begin{cases}α_{j} & j < n \\ λα_{j} & j ≥ n \end{cases}  (5.18)

Then

U_{n}C(T_{n,λ^{−1}}({α_{j}}_{j=0}^{∞}))U_{n}^{−1} = C({α_{j}}_{j=0}^{∞})W^{(n)}(λ)  (5.19)

In particular, (5.5) holds.

Remarks. 1. It is important that δ_{0} is an eigenvector of U_{n} since Verblunsky coefficients involve a unitary and a cyclic vector. (5.19) also shows that CW^{(n)}(λ) has δ_{0} as a cyclic vector.

2. One can also ask about Verblunsky coefficients of W^{(n)}(λ)C({α_{j}}_{j=0}^{∞}). Since Verblunsky coefficients are invariant under unitaries that have δ_{0} as an eigenvector and

W^{(n)}C = W^{(n)}CW^{(n)}(W^{(n)})^{−1}

the Verblunsky coefficients of CW^{(n)} and W^{(n)}C are the same.

(5.11), (5.12), and (5.13) imply a result about extended CMV matrices. For λ ∈ ∂𝔻, let W̃(λ) be the two-sided diagonal matrix with d_{2j} = 1, d_{2j+1} = λ. Then

Theorem 5.3. Let λ ∈ ∂𝔻. Then W̃(λ)E({α_{n}}_{n=−∞}^{∞})W̃(λ)^{−1} = E({λα_{n}}_{n=−∞}^{∞}).

Remark. In particular, spectral properties of E({α_{n}}_{n=−∞}^{∞}) and E({λα_{n}}_{n=−∞}^{∞}) are identical and α_{n} → λα_{n} preserves isospectral tori in the periodic case.

6. Resolvents of CMV Matrices

In this section, we will present formulae for the resolvent of C analogous to the Green’s function formula for Jacobi matrices (see Theorem 4.4.3 of [69]). These formulae appeared first in Section 4.4 of [69]. Similar formulae for GGT matrices appeared earlier in Geronimo– Teplyaev [33] (see also [31, 32]).

Clearly, we need an analog of Jost solutions. For OPUC, these were found by Golinskii–Nevai [41] who proved

Theorem 6.1. Fix z ∈ 𝔻. Let φ_{n} be the normalized OPUC for a probability measure dμ on ∂𝔻, and ψ_{n} the normalized OPUC for Verblunsky coefficients −α_{n}(dμ) (so-called second kind polynomials). Then

∑_{n=0}^{∞} |ψ_{n}(z) + F (z)φ_{n}(z)|^{2} + |ψ_{n}^{∗}(z) − F (z)φ_{n}^{∗}(z)|^{2} < ∞  (6.1)

where F is the Carathéodory function:

F (z) ≡ ∫ \frac{e^{iθ} + z}{e^{iθ} − z} dμ(θ)  (6.2)

Remarks. 1. This is an analog of Weyl’s formula; see (1.2.53) of [69]. 2. See [41] or [69, Section 3.2] for a proof.

With this in mind, we define

y_{n} = \begin{cases}z^{−ℓ}ψ_{2ℓ} & n = 2ℓ \\ −z^{−ℓ}ψ_{2ℓ−1}^{∗} & n = 2ℓ − 1 \end{cases}  (6.3)

Υ_{n} = \begin{cases}−z^{−ℓ}ψ_{2ℓ}^{∗} & n = 2ℓ \\ z^{−ℓ+1}ψ_{2ℓ−1} & n = 2ℓ − 1 \end{cases}  (6.4)

p_{n} = y_{n} + F (z)x_{n}  (6.5)
π_{n} = Υ_{n} + F (z)χ_{n}  (6.6)

Then Theorem 4.4.1 of [69] says:

Theorem 6.2. We have that for z ∈ 𝔻,

[(C − z)^{−1}]_{kℓ} = \begin{cases}(2z)^{−1}χ_{ℓ}(z)p_{k}(z) & k > ℓ or k = ℓ = 2n − 1 \\ (2z)^{−1}π_{ℓ}(z)x_{k}(z) & ℓ > k or k = ℓ = 2n \end{cases} (6.7)

As a special case, since δ_{n} = χ_{n}(C)δ_{0} and |χ_{n}(e^{iθ})| = |φ_{n}(e^{iθ})|, we obtain from a spectral representation

∫ \frac{|φ_{n}(e^{iθ})|^{2}}{e^{iθ} − z} dμ(θ) = (2z^{n+1})^{−1}φ_{n}(z)[−ψ_{n}^{∗}(z) + F (z)φ_{n}^{∗}(z)] (6.8)

As shown in remarks to Theorem 9.2.4 of [70], this is equivalent to a formula of Khrushchev [49] for ∫ \frac{e^{iθ}+z}{e^{iθ}−z}|φ_{n}(e^{iθ})|^{2} dμ(θ). For an application of Theorem 6.2, see Stoiciu [74].

7. I^{p} Perturbations

In this section, we give some elementary estimates of Golinskii– Simon [42] on the I^{p} norm of C({α_{n}}_{n=0}^{∞}) − C({β_{n}}_{n=0}^{∞}). (For definition and background on Schatten p-classes, see Gohberg–Krein [39] and Simon [71].)

In this section, we will use these estimates to write the Szegő function in terms of Fredholm determinants of the CMV matrices and to discuss scattering theory. Further applications appear in Section 9.

A diagonal matrix, A, has I_{p} norm (∑_{j}|a_{jj}|^{p})^{1/p}) and ∥A∥_{p} is invariant under multiplication by a unitary. So if A has only a nonvanishing kth principal diagonal, A has I_{p} norm (∑_{j}|a_{j j+k}|^{p})^{1/p}. Since (a^{1/p} + b^{1/p} + c^{1/p}) ≤ (a + b + c)^{1/p}3^{1−1/p} (by Hölder’s inequality), we see for tridiagonal matrices that

∥A − B∥_{p} ≤ 3^{1−1/p}( ∑_{i,j}|a_{ij} − b_{ij}|^{p})^{1/p}  (7.1)

This lets us estimate ∥L({α_{n}}_{n=0}^{∞}) − L({β_{n}}_{n=0}^{∞})∥_{p}, and similarly for M. Using unitarity of L and M, ∥LM − L′M′∥_{p} ≤ ∥(L − L′)M∥_{p} + ∥L′(M − M′)∥_{p} ≤ ∥L − L′∥_{p} + ∥M − M′∥_{p}. So using (a^{1/p} + b^{1/p}) ≤ 2^{1−1/p}(a + b)^{1/p}, we find

Theorem 7.1 (= Theorem 4.3.2 of [69]). Let {α_{n}}_{n=0}^{∞}, {β_{n}}_{n=0}^{∞} be two sequences in \overline{𝔻}^{∞} and let ρ_{n} = (1 − |α_{n}|^{2})^{1/2}, σ_{n} = (1 − |β_{n}|^{2})^{1/2}. Then

∥C({α_{n}}_{n=0}^{∞}) − C({β_{n}}_{n=0}^{∞})∥_{p} ≤ 6^{1−1/p}( ∑_{n=0}^{∞} |α_{n} − β_{n}|^{p} + |ρ_{n} − σ_{n}|^{p})^{1/p}

(7.2)

Remark. [69] has the constants 2 for 1 ≤ p ≤ 2 and 2 · 3^{1−1/2p} for 2 ≤ p ≤ ∞, but the proof there actually shows 2^{1−1/p} and 2^{1−1/p} 3^{1−1/2p}. This improves the constant in (7.2).

To rephrase in terms of |α_{n} − β_{n}| only, we first note that

sup_{|z|≤R} | \frac{d}{dz} (1 − |z|^{2})^{1/2}| ≤ (1 − R^{2})^{−1/2}

and ||z| − |w|| ≤ |z − w| to see that

sup_{|z|,|w|≤R} |(1 − |z|^{2})^{1/2} − (1 − |w|^{2})^{1/2}| ≤ (1 − R^{2})^{−1/2}|z − w| (7.3)

We need to note that |\sqrt{a}−\sqrt{b} | ≤ \sqrt{|a − b|} and ||α|^{2}−|β|^{2}| ≤ 2|α−β|, so

|(1 − |z|^{2})^{1/2} − (1 − |w|^{2})^{1/2}| ≤ \sqrt{2} |z − w|^{1/2}  (7.4)

Thus,

Theorem 7.2. Let \{α_{n}\}_{n=0}^{∞} and \{β_{n}\}_{n=0}^{∞} be two sequences in \overline{𝔻}^{∞} and let ρ_{n} = (1 − |α_{n}|^{2})^{1/2}, σ_{n} = (1 − |β_{n}|^{2})^{1/2}. Then

(a) If sup_{n}|α_{n}| ≤ R < 1 and sup_{n}|β_{n}| ≤ R, then

∥C({α_{n}}_{n=0}^{∞})−C({β_{n}}_{n=0}^{∞})∥_{p} ≤ 6^{1−1/p}[1+(1−R^{2})^{−p/2}]^{1/p}( ∑_{n=0}^{∞} |α_{n}−β_{n}|^{p})^{1/p}

(7.5)

(b) In general, for 1 ≤ p ≤ ∞,

∥C({α_{n}}_{n=0}^{∞})−C({β_{n}}_{n=0}^{∞})∥_{p} ≤ 6^{1−1/p}( ∑_{n=0}^{∞} |α_{n}−β_{n}|^{p}+2^{p/2}|α_{n}−β_{n}|^{p/2})^{1/p}

(7.6)

One thing made possible by CMV matrices is scattering theory because all CMV matrices act on the same space (ℓ^{2}({0, 1, 2, … })). This is an important tool made possible by the CMV matrix. Thus,

Theorem 7.3. Suppose sup_{n}|α_{n}| ≤ R ≤ 1, sup_{n}|β_{n}| ≤ R < 1, and

∑_{n=0}^{∞} |α_{n} − β_{n}| < ∞  (7.7)

Then, if P_{ac}(·) is the projection onto the absolutely continuous subspace of an operator and C = C({α_{n}}_{n=0}^{∞}), C̃ = C({β_{n}}_{n=0}^{∞}), then

lim_{n→±∞} C^{n}C̃^{−n}P_{ac}(C̃)

exists and is a partial isometry with range P_{ac}(C). In particular, C ↾ P_{ac}(C) and C̃ ↾ P_{ac}(C̃) are unitarily equivalent.

Remarks. 1. This follows from the fact that C − C̃ is trace class and from the Kato–Birman theorem [65].

2. If {α_{n}}_{n=0}^{∞} corresponds to

dμ = f(θ) \frac{dθ}{2π} + dμ_{δ}  (7.8)

and {β_{n}}_{n=0}^{∞} corresponds to

dν = g(θ) \frac{dθ}{2π} + dν_{δ}  (7.9)

then this theorem implies that up to sets of dθ-measure 0,

{θ | f(θ) ≠ 0} = {θ | g(θ) ≠ 0}  (7.10)

(also see Theorem 9.3).

3. For the case β_{n} ≡ 0, this holds if only ∑_{n=0}^{∞}|α_{n}|^{2} < ∞; see [70, Section 10.7].

Finally, following Simon [69, Section 4.2], we want to state the connection of C to the Szegő function, defined for |z| < 1 by

D(z) = lim_{n→∞} φ_{n}^{∗}(z)^{−1}  (7.11)

which exists and is nonzero if (and only if)

∑_{j=0}^{∞} |α_{j}|^{2} < ∞  (7.12)

(see Section 2.4 of [69]). We will let C_{0} be the free CMV matrix corresponding to dμ = \frac{dθ}{2π}; equivalently, α_{n} ≡ 0.

Theorem 7.4. Suppose

∑_{n=0}^{∞} |α_{n}| < ∞  (7.13)

Then C − C_{0} is trace class and

D(z)^{−1}D(0) = det(\frac{1 − zC̄}{1 − zC̄_{0}} )  (7.14)

If (7.12) holds, then C − C_{0} is Hilbert–Schmidt, and

D(z)^{−1}D(0) = det_{2}(\frac{1 − zC̄}{1 − zC̄_{0}} )e^{−zw_{1}}  (7.15)

where

w_{1} = α_{0} − ∑_{j=1}^{∞} α_{j}ᾱ_{j−1}  (7.16)

Remarks. 1. Alas, (4.2.53) of [69] has a sign error: it is e^{−zw_{1}} as we have here, not e^{zw_{1}} as appears there!

2. By det(\frac{1−zC̄}{1−zC̄_{0}} ), we mean det((1 − zC̄)(1 − zC̄_{0})^{−1}). Since

(1 − zC̄)(1 − zC̄_{0})^{−1} = 1 − z(C̄ − C̄_{0})(1 − zC̄_{0})^{−1}

we see that this is 1+ trace class (resp., Hilbert–Schmidt) if C − C_{0} is trace class (resp., Hilbert–Schmidt).

3. For a proof, see Section 4.2 (Theorem 4.2.14) of [69].

4. C̄ is the complex conjugate of C, that is, (C̄)_{ij} = \overline{(C_{ij})}.

3. det(·) is defined on operators of the form 1+A with A trace class, and then det_{2} on 1 + A with A Hilbert–Schmidt by

det(1 + A) = det_{2}((1 + A)e^{−A})  (7.17)

When A is trace class,

det(1 + A) = det_{2}(1 + A)e^{Tr(A)}  (7.18)

If (7.13) holds, −zw_{1} = Tr((1 − zC̄)/(1 − zC̄_{0})) and (7.14)/(7.15) are consistent by (7.18). See [39] or [71] for a discussion of det(·) and det_{2}(·).

5. The connection for one-dimensional Schrödinger operators of the Jost function and Fredholm determinants goes back to Jost–Pais [46]. For Jacobi matrices, under the name “perturbation determinant,” they were used by Killip–Simon [52].

8. Essential Spectra

The discrete spectrum of an operator is the set of isolated points of finite multiplicity. The complement of the discrete spectrum in the spectrum is called the essential spectrum. Since a CMV matrix has a cyclic vector, the essential spectrum is just the set of nonisolated points in the support of the spectral measure, dμ, often called the derived set of supp(dμ). Last–Simon [54] have a general result for the essential spectrum of a CMV matrix C({α_{n}}_{n=0}^{∞}). Definition. A right limit of {α_{n}}_{n=0}^{∞} is any two-sided sequence \{β_{n}\}_{n=−∞}^{∞} in \overline{𝔻}^{ℤ} for which there exists n_{ℓ} → ∞ so lim_{ℓ→∞} α_{n_{ℓ}+j} = β_{j} for each j ∈ ℤ. R({α_{n}}_{n=0}^{∞}) is the set of right limits of {α_{n}}_{n=0}^{∞}.

Since \overline{𝔻}^{ℤ} is compact, R is nonempty. Indeed, if β̃_{0} is any limit point of α_{n}, there is a right limit with β_{0} = β̃_{0}.

Theorem 8.1 (Last–Simon [54]). For any \{α_{n}\}_{n=0}^{∞} ∈ \overline{𝔻}^{∞}, we have

σ_{ess}(C(\{α_{n}\}_{n=0}^{∞})) = \overline{⋃_{β∈R(\{α_{n}\}_{n=0}^{∞})} σ(E(\{β_{n}\}_{n=0}^{∞}))}  (8.1)

Remarks. 1. The proof [54] uses a Weyl trial sequence argument. The key is that because C has finite width, for any λ_{0} ∈ σ_{ess}(C) and ε, there exist L, n_{j} → ∞ and φ_{j} supported in (n_{j} − L, n_{j} + L) with ∥φ_{j}∥ = 1 and

lim sup_{j→∞} ∥(C − λ_{0})φ_{j}∥ ≤ ε  (8.2)

2. Right limits of Verblunsky coefficients were considered earlier by Golinskii–Nevai [41], motivated by earlier work on Schrödinger operators by Last–Simon [53]. This work was in the context of a.c. spectrum (see Theorem 10.9.11(ii) of [70]).

3. [54] used the same methods to study Jacobi and Schrödinger operators. Earlier results of the form (8.1) for Schrödinger operators (but not for CMV matrices) are due to Georgescu–Iftimovici [28], Măntoiu [56], and Rabinovich [63]. These rely on what I regard as elaborate machines (connected with C^{∗}-algebras or with Fredholm operators) although, no doubt, their authors regard them as very natural.

One can combine this with Theorem 5.3 to obtain Theorem 8.2. Let {α_{j}}_{j=0}^{∞} and {β_{j}}_{j=0}^{∞} be two sequences of Verblunsky coefficients. Suppose there exist λ_{j} ∈ ∂𝔻 so that

(i)  β_{j}λ_{j} − α_{j} → 0  (8.3)
(ii)  λ_{j+1}λ̄_{j} → 1  (8.4)

Then

σ_{ess}(C({α_{j}}_{j=0}^{∞})) = σ_{ess}(C({β_{j}}_{j=0}^{∞}))  (8.5)

Proof. Let {γ_{j}}_{j=−∞}^{∞} be a right limit of {β_{j}}_{j=0}^{∞}. By passing to a subsequence, we can suppose λ_{n_{j}} → λ_{∞} and β_{n_{j}+k} → γ_{k}. Since λ_{n_{j}+k}λ_{n_{j}}^{−1} → 1, we see that α_{n_{j}+k} → λ_{∞}γ_{k}. By Theorem 5.3, σ(E({γ_{k}}_{k=−∞}^{∞})) = σ(E({λ_{∞}γ_{k}}_{k=−∞}^{∞})). It follows (using symmetry) that (8.5) holds. □

Remark. This proof is from [54], but the result appears earlier as Theorem 4.3.8 in [69], motivated by a special case of Barrios–López [9].

Example 8.3. (This is due to Golinskii [40]; the method of proof is due to [54]. See the discussion in [54] for earlier related results.) Suppose |α_{n}| → 1 as n → ∞. Then σ_{ess}(C({α_{n}})_{n=0}^{∞}) is the set of limit points of {−ᾱ_{j+1}α_{j}}_{j=0}^{∞}. For any limit point has E({β_{j}}_{j=0}^{∞}) diagonal (since (1 − |β_{j}|^{2})^{1/2} ≡ 0) with diagonal values −β̄_{j+1}β_{j}, and by compactness, any limit point of −ᾱ_{j+1}α_{j} occurs as some −β̄_{1}β_{0}. In particular, this (plus an extra argument) implies σ_{ess}(E) = {λ_{0}} if and only if |α_{n}| → 1 and ᾱ_{n+1}α_{n} → −λ_{0}. See [40] and [54] for a discussion of when σ_{ess}(C) is a finite set. □

It is well known (see Example 1.6.12 of [69] and Example 11.1.4 of [70]) that if α_{n} ≡ a ∈ 𝔻, then σ_{ess}(C) = Δ_{|a|} = {z ∈ ∂𝔻 | |arg z| ≥ 2 arcsin(|a|)}, which increases as |a| decreases. It follows:

Example 8.4. (Theorem 7.8 of [54]; one direction was proven in [14], which motivated Theorem 7.8 of [54].) Suppose

\frac{α_{j+1}}{α_{j}} → 1  lim inf|α_{j}| = a  (8.6)

Then

σ_{ess}(C({α_{n}}_{n=0}^{∞})) = Δ_{a}  (8.7)

For \frac{α_{j+1}}{α_{j}} → 1 implies that each limit is of the form β_{j} ≡ b for some b ∈ 𝔻, so

σ_{ess}(C) = ⋃_{b=limits of α_{j}} Δ_{|b|} = Δ_{a}

since Δ_{|b|} ⊆ Δ_{a} if |b| ≥ a. □

9. Spectral Consequences

Section 4.3 of [69] describes joint work of Golinskii–Simon [42] that uses CMV matrices to obtain spectral results that relate properties of {α_{n}}_{n=0}^{∞} to the associated measures. Here, in brief, are some of their main results:

Theorem 9.1 (≡ Theorem 4.3.5 of [69]; subsumed in Theorem 8.2). If |α_{n} − β_{n}| → 0, then σ_{ess}(C({α_{n}}_{n=0}^{∞})) = σ_{ess}(C({β_{n}}_{n=0}^{∞})).

Remark. Of course, Theorem 9.1 also follows from Theorem 8.1.

Proof. By (7.6) and a limiting argument, C({α_{n}}_{n=0}^{∞}) − C({β_{n}}_{n=0}^{∞}) is compact. The result follows from Weyl’s theorem on the invariance of essential spectrum under compact perturbation. □

Theorem 9.2 (≡ Theorem 4.3.4 of [69]). If lim sup|α_{n}(dμ)| = 1, then dμ is purely singular.

Remark. This result is called Rakhmanov’s lemma, after [64]. The proof is motivated by earlier results for Jacobi matrices of Dombrowski [18] and Simon–Spencer [73].

Proof. Let α̂_{n} be defined by

α̂_{n} = \begin{cases}1 & if α_{n} = 0 \\ \frac{α_{n}}{|α_{n}|} & if α_{n} ≠ 0 \end{cases}

Since lim sup|α_{n}| = 1, we can find a sequence n_{j} → ∞, so

∑_{j=0}^{∞} |α_{n_{j}} − α̂_{n_{j}}|^{1/2} < ∞

Let

β_{n} = \begin{cases}α̂_{n} & if n = n_{j} for some j \\ α_{n} & otherwise \end{cases}

Then C({β_{n}}_{n=0}^{∞}) − C({α_{n}}_{n=0}^{∞}) is trace class by (7.6). By the Kato– Birman theorem [65],

σ_{ac}(C({α_{n}}_{n=0}^{∞})) = σ_{ac}(C({β_{n}}_{n=0}^{∞}))

Since |α̂_{n}| = 1, C({β_{n}}_{n=0}^{∞}) is a direct sum of finite matrices of size n_{j+1} − n_{j}, and so it has no a.c. spectrum. □

Theorem 9.3 (≡ Theorem 4.3.6 of [69]). If {α_{n}}_{n=0}^{∞} and {β_{n}}_{n=0}^{∞} are the Verblunsky coefficients of dμ and dν given by (7.8) and (7.9), and (7.7) holds, then (7.10) holds.

Proof. If lim sup|α_{n}| < 1, then lim sup|β_{n}| < 1, and by Theorem 7.3, (7.10) holds. If lim sup|α_{n}| = lim sup|β_{n}| = 1, then σ_{ac}(C({α_{n}}_{n=0}^{∞})) = σ_{ac}(C({β_{n}}_{n=0}^{∞})) = ∅ by Theorem 9.2. □

10. The AGR Factorization of GGT Matrices

This section is primarily preparatory for the next and discusses GGT matrix representations (for Geronimus [34], Gragg [43], and Teplyaev [78]) associated to a measure on ∂𝔻:

G_{kℓ}({α_{n}}_{n=0}^{M}) = ⟨φ_{k}, zφ_{ℓ}⟩  (10.1)

and discussed in Section 4.1 of [69]. If μ is nontrivial, M in (10.1) is ∞ and α_{n} ∈ 𝔻 for all n. If μ is supported on exactly N points, M = N −1 and α_{0}, …, α_{N−2} ∈ 𝔻, α_{N−1} ∈ ∂𝔻. There is an explicit calculation (see Proposition 1.5.9 of [69]):

G_{kℓ} = \begin{cases}−ᾱ_{ℓ}α_{k−1} ∏_{m=k}^{ℓ−1} ρ_{m} & 0 ≤ k ≤ ℓ \\ ρ_{ℓ} & k = ℓ + 1 \\ 0 & k ≥ ℓ + 2 \end{cases}  (10.2)

We present a remarkable factorization of GGT matrices due to Ammar, Gragg, and Reichel [6], use it to provide a result about cosets in 𝕌(N)/𝕌(N − 1), and then present an alternate proof of Theorems 9.2 and 9.3 using GGT rather than CMV matrices. For the special case of orthogonal matrices (all α_{j} ∈ (−1, 1)), AGR found this factorization earlier [5].

We defined Θ_{j}(α) before (3.18) as a 2 × 2 matrix acting on the span of δ_{j}, δ_{j+1}. We define Θ̃(α_{j}) to be this matrix viewed as an operator on

ℂ^{N} by 1_{j} ⊕ Θ_{j}(α) ⊕ 1_{N−j−2}. ˜ Θ̃_{N−1}(α) is the matrix 1_{N−1} ⊕ ᾱ.

Theorem 10.1 (AGR factorization). For any finite N × N GGT matrix,

G({α}_{n=0}^{N−1}) = Θ̃_{0}(α_{0}) … Θ̃_{N−2}(α_{N−2})˜Θ̃_{N−1}(α_{N−1})  (10.3)

For N = ∞,

G({α_{n}}_{n=0}^{∞}) = s-lim_{M→∞} Θ̃_{0}(α_{0}) … Θ̃_{M}(α_{M})  (10.4)

Remarks. 1. (10.4) follows from (10.3) by a simple limiting argument. We will only prove (10.3) below.

2. We will give three proofs which illustrate slightly different aspects of the formula.

3. As explained in Killip–Nenciu [50], the Householder algorithm lets one write any unitary as a product of N − 1 reflections; in many ways, the representation (10.4) is more useful.

First Proof. By a direct calculation using (10.2),

G({α_{n}}_{n=0}^{N−1}) = Θ_{0}(α_{0})[1_{1×1} ⊕ G({α_{n+1}}_{n=0}^{N−2})]  (10.5)

(10.3) follows by induction. □

Second Proof (that of AGR [6]). We will prove first that any unitary (upper) Hessenberg matrix H (i.e., H_{kℓ} = 0 if k ≥ ℓ + 1) with positive subdiagonal (i.e., H_{ℓ+1,ℓ} > 0 for all ℓ) has the form (10.3) for suitable α_{0}, α_{1}, …, α_{N−2} ∈ 𝔻 and α_{N−1} ∈ ∂𝔻. The first column of H has the form (ᾱ_{0}, ρ_{0}, 0, …, 0)^{t} for some α_{0} ∈ 𝔻. Then Θ_{0}(α_{0})^{−1}H is of the form 1_{1×1} ⊕H^{(1)}, where H^{(1)} is a unitary (N −1)×(N −1) Hessenberg matrix with positive subdiagonal. By induction, H has the form (10.3). One proves that {α_{n}}_{n=0}^{N−1} are the Verblunsky coefficients of the GGT matrix, either by using (10.2) or by deriving recursion relations. □

For the third proof, we need a lemma that is an expression of Szegő recursion.

Lemma 10.2. We have that

⟨φ_{j}^{∗}, zφ_{j}⟩ = ᾱ_{j}  (10.6)
⟨φ_{j+1}, zφ_{j}⟩ = ρ_{j}  (10.7)
⟨φ_{j+1}, φ_{j+1}^{∗}⟩ = −ᾱ_{j}  (10.8)
⟨φ_{j}^{∗}, φ_{j+1}^{∗}⟩ = ρ_{j}  (10.9)

Remark. This says that a certain change of basis on a two-dimensional space is Θ(α_{j}).

Proof. φ_{j+1} ⊥ φ_{j}^{∗} since deg(φ_{n}^{∗}) ≤ j. Moreover, by (2.5) and (2.12),

zφ_{j} = ρ_{j}φ_{j+1} + ᾱ_{j}φ_{j}^{∗}
φ_{j+1}^{∗} = −α_{j}φ_{j+1} + ρ_{j}φ_{j}

from which (10.6)–(10.9) are immediate. □

Third Proof of Theorem 10.1. This is an analog of the proof of LM factorization in Section 2. There C is the matrix of overlap of the orthonormal bases {zχ_{ℓ}}_{ℓ=0}^{∞} and {χ_{ℓ}}_{ℓ=0}^{∞}. The LM factorization comes from inserting the basis {zx_{ℓ}}_{ℓ=0}^{∞}. Here we have the bases

e^{(0)} = (zφ_{0}, …, zφ_{N−1})  (10.10)
e^{(N)} = (φ_{0}, …, φ_{N−1})

and G is an overlap matrix. We introduce N − 1 intermediate bases:

e^{(1)} = (zφ_{0}, …, zφ_{N−2}, φ_{N−1}^{∗})
e^{(2)} = (zφ_{0}, …, zφ_{N−3}, φ_{N−2}^{∗}, φ_{N−1})
⋯
e^{(j)} = (zφ_{0}, …, zφ_{N−j−1}, φ_{N−j}^{∗}, φ_{N−j+1}, …, φ_{N−1})
⋯

where e^{(N)} is given by (10.10) since φ_{0}^{∗} = 1 = φ_{0}.

Thus

\begin{aligned}G_{kℓ} & = ⟨e_{k}^{(N)}, e_{ℓ}^{(0)}⟩ \\ & = ∑_{m_{1}...m_{N−1}}⟨e_{k}^{(N)}, e_{m_{1}}^{(N−1)}⟩ … ⟨e_{m_{j}}^{(N−j)}, e_{m_{j+1}}^{(N−j−1)}⟩ … ⟨e_{m_{N−1}}^{(1)}, e_{ℓ}^{(0)}⟩ \end{aligned}

(10.11)

is a product of N matrices. N −1 have a change from zφ_{j}, φ_{j+1}^{∗} to φ_{j}^{∗}, φ_{j} whose overlap matrix, by (10.6)–(10.9), is Θ̃(α_{j}) and the extreme right

has a change from zφ_{N−1} to φ_{N−1}^{∗}, which is ˜ Θ̃(α_{N−1}) since in L^{2}(∂𝔻),

zφ_{N−1} − ᾱ_{N−1}φ_{N−1}^{∗} = 0

Thus, (10.11) is (10.3). □

As a first application, we want to show that each finite unitary has an LM factorization without recourse to orthogonal polynomials. By taking limits, one obtains an LM factorization in general. This calculation fleshes out an argument given by AGR [5] in the orthogonal case by using induction to make the proof more transparent:

Theorem 10.3. Let U be a unitary matrix on ℂ^{N} with (1, 0, … 0)^{t} as cyclic vector. Then there exists a unitary V on ℂ^{N} with V (1, 0, … 0)^{t} = (1, 0, …, 0)^{t} so that V UV^{−1} has an LM factorization.

Remark. By LM factorization, we mean L = Θ̃(α_{0}) ⊕ Θ̃(α_{2}) ⊕ ⋯ and

M = 1_{1×1} ⊕ Θ̃(α_{1}) ⊕ Θ̃(α_{3}) ⊕ ⋯, with a ˜ Θ̃(α_{N−1}) at the end of L if N is odd and of M is N is even.

Proof. We use induction on N. N = 1, which says U = (˜Θ̃(α_{0})(1), is trivial. By the GGT representation and AGR factorization, we can find W (with W (1, 0, …, 0)^{t} = (1, 0, …, 0)^{t}) so

W UW^{−1} = Θ̃_{0}(α_{0})Θ̃_{1}(α_{1}) … ˜ Θ̃_{N−1}(α_{N−1})

Let

U_{1} = Θ̃_{0}(α_{1}) … ˜ Θ̃_{N−2}(α_{N−1})  (10.12)

on ℂ^{N−1}. By induction and adding 1_{1×1} ⊕ ⋯ everywhere, we can find L_{1}, M_{1}, and V_{1} so

(1 ⊕ V_{1})W UW^{−1}(1 + V_{1})^{−1} = Θ̃_{0}(α_{0})[1 ⊕ L_{1}][1 ⊕ M_{1}]  (10.13)

Define

V = [1 ⊕ M_{1}][1 ⊕ V_{1}]W

(note V maps (1 0 … 0)^{t} to itself). We have

V UV^{−1} = {[1 ⊕ M_{1}]Θ̃_{0}(α_{0})}{[1 ⊕ L_{1}]}

which precisely has the form LM. □

As a second application, we want to provide an explicit map that will be critical in the next section. Fix δ_{0} ∈ ℂ^{n} and U ∈ 𝕌(n), the n × n unitary matrices. Let 𝕌(n − 1) = {U ∈ U(n) | Uδ_{0} = δ_{0}}. The symbol 𝕌(n−1) is accurate since each such U defines and is defined by a unitary on {δ_{0}}^{⊥} ≅ ℂ^{n−1}. Let 𝕊ℂ^{2n−1} = 𝕌(n)/𝕌(n − 1), the group theoretic quotient. By mapping

π: U ∈ 𝕌(n) → Uδ_{0}  (10.14)

we see that 𝕊ℂ^{2n−1} ≅ {z ∈ ℂ^{n} | |z| = 1}, the sphere of real dimension 2n − 1. Here is the result we will need:

Theorem 10.4. There exists continuous maps g_{1} and g_{2} defined on {z ∈ 𝕊ℂ^{2n−1} | z ≠ δ_{0}} with g_{1} mapping to 𝕌(n) and g_{2} to 𝕊ℂ^{2n−3} = {z ∈ 𝕊ℂ^{2n−1} | ⟨δ_{0}, z⟩ = 0} so that (i)

π[g_{1}(z)] = z  (10.15)

(ii) V (U) ≡ g_{1}(π(U))^{−1}U ∈ 𝕌(n − 1) for all U ∉ 𝕌(n − 1) (iii) If δ_{0} is cyclic for U with Verblunsky coefficients α_{j}(U, δ_{0}), then

g_{2}(π(U)) is cyclic for V (U) ↾ ℂ^{n−1} and

α_{j}(V (U), g_{2}(π(U))) = α_{j+1}(U, δ_{0})  (10.16)

(iv)

⟨δ_{0}, Uδ_{0}⟩ = \overline{α_{0}(U, δ_{0})}  (10.17)

if δ_{0} is cyclic for U.

Proof. If z ≠ δ_{0}, a(z) = \overline{⟨δ_{0}, z⟩} ∈ 𝔻 and so

g_{2}(z) = \frac{z − ⟨δ_{0}, z⟩δ_{0}}{∥z − ⟨δ_{0}, z⟩δ_{0}∥}

is well defined and in 𝕊ℂ^{2n−3}. In particular, if p(z) = (1 − a(z)^{2})^{1/2} = ∥z − ⟨δ_{0}, z⟩δ_{0}∥, we have

z = p(z)g_{2}(z) + \overline{a(z)} δ_{0}  (10.18)

Define g_{1}(z) by

g_{1}(z)w = \begin{cases}w & if w ⊥ δ_{0}, g_{2}(z) \\ z & if w = δ_{0} \\ −a(z)g_{2}(z) + p(z)δ_{0} & if w = g_{2}(z) \end{cases}

and otherwise linear. Then g_{1}(z) is unitary since Θ(a(z)) is unitary.

(i) is obvious from g_{1}(z)δ_{0} = z. (ii) is a restatement of (i). (iii) follows from the fact that g_{2}(z) corresponds to δ_{1} in a δ_{j} = χ_{j}(z) basis and the AGR factorization. (iv) is a consequence of zφ_{0} − ᾱ_{0}φ_{0}^{∗} = φ_{1}, so ⟨φ_{0}, zφ_{0}⟩ = ᾱ_{0}⟨φ_{0}, φ_{0}^{∗}⟩ = ᾱ_{0}. □

We want to close this section by noting that the AGR factorization implies an estimate on the GGT matrices that is not obvious from (10.2). Indeed, in Section 4.1 of [69], an unnecessary condition, lim inf|α_{n}| > 0, is made because the estimate below is not obvious.

In essence, the AGR factorization plays the role for estimates of GGT matrices that the LM factorization does for CMV matrices. In some ways, it is more critical because CMV matrices are five-diagonal with matrix elements which are quadratic in α and ρ, so one can easily get estimates like (7.2) (but with a worse constant) without using the LM factorization. Since GGT matrices are not finite width and have matrix elements that are products of arbitrary orders, direct estimates from (10.2) are much harder.

Theorem 10.5. Let \{α_{n}\}_{n=0}^{∞} and \{β_{n}\}_{n=0}^{∞} be two sequences in \overline{𝔻}^{∞} and let σ_{n} = (1 − |β_{n}|^{2})^{1/2}. Then

∥G({α_{n}}_{n=0}^{∞}) − G({β_{n}}_{n=0}^{∞})∥_{1} ≤ 2 ∑_{n=0}^{∞} (|α_{n} − β_{n}| + |σ_{n} − ρ_{n}|) (10.19)

Proof. By (10.4) and standard trace class techniques [71], we need only prove for finite sequences that

∥Θ̃_{0}(α_{0}) … Θ̃_{N}(α_{N})−Θ̃_{0}(β_{0}) … Θ_{N}(β_{N})∥_{1} ≤ 2 ∑_{n=0}^{N} (|α_{n}−β_{n}|+|σ_{n}−ρ_{n}|)

(10.20)

Since ∥Θ̃_{j}(α_{j}) − Θ̃_{j}(β_{j})∥_{1} ≤ 2(|α_{j} − β_{j}| + |σ_{j} − ρ_{j}|), and ∥Θ̃(α)∥ = 1, writing the difference of products as a telescoping sum yields (10.20).

□

Notice also that AGR factorization shows that if |α_{j}| = 1, G decouples. This fact and (10.19) provide an alternate proof of Rakhmanov’s lemma (Theorem 9.2) by the same decoupling argument, but using G in place of C. If ∑_{n=0}^{∞}|α_{n}|^{2} = ∞, one also gets a proof of Theorem 9.3, using G in place of C and (10.19). If ∑_{n=0}^{∞}|α_{n}|^{2} < ∞, one must use the extended GGT matrix, F, of Section 4.1 of [69] (see also Constantinescu [15]). It is easy to prove that if ∑|α_{n}|^{2} < ∞ and ∑|α_{n} − β_{n}| < ∞, then F({α_{n}}_{n=0}^{∞})−F({β_{n}}_{n=0}^{∞}) is trace class since the difference of F’s differs from the difference of G’s by a rank one operator, which is always trace class!

11. CUE, Haar Measure, and the Killip–Nenciu Theorem

In [50], Killip and Nenciu proved the following result:

Theorem 11.1. Let dμ be a normalized Haar measure on 𝕌(n), the n × n unitary matrices. Then, for a.e. U, δ_{0} = (1, 0, …, 0)^{t} is cyclic, and the measure induced on 𝔻^{n−1}×∂𝔻 by U → α_{j}(U_{0}, δ_{0}) is the product measure:

\{∏_{j=0}^{n−2} [\frac{n − j − 1}{π} (1 − |α_{j}|^{2})^{n−j−2} d^{2}α_{j}]\} \frac{dθ(α_{n−1})}{2π}  (11.1)

where θ(α_{n−1}) is defined by

α_{n−1} = e^{iθ(α_{n−1})}  (11.2)

and d^{2}α is a two-dimensional Lebesgue measure on 𝔻.

Remark. By the “induced measure,” we mean the measure dν on 𝔻^{n−1}× ∂𝔻 given by ν(B) = μ(A^{−1}[B]), where A(U) = (α_{1}(U), …, α_{n−1}(U)).

This is really a result about Verblunsky coefficients, not CMV matrices, and both their proof and ours use the GGT, not the CMV, matrices. We provide this here because, first, Killip–Nenciu proved this result to provide a five-diagonal model for CUE (see below), and because the result was proven as part of the ferment stirred up by the CMV discovery. In this section, we will provide a partially new proof of Theorem 11.1 that is perhaps more natural from a group theoretic point of view, and then describe and sketch their somewhat shorter argument!

To understand where the factors in (11.1) come from:

Lemma 11.2. Let dμ_{𝕊ℂ^{2n−1}} be the measure on the 2n−1 real dimension manifold {z ∈ ℂ^{n} | |z| = 1}, which is normalized and invariant under rotations. Map 𝕊ℂ^{2n−1} ⟶^{Q} \overline{𝔻} by z ↦ z_{1}, the first component, and let dν be the measure on \overline{𝔻} given by ν(B) = μ_{𝕊ℂ^{2n−1}}(Q^{−1}[B]). Then

dν(w) = \frac{n − 1}{π} (1 − |w|^{2})^{n−2} d^{2}w  (11.3)

Proof. Since d^{2}w = \frac{1}{2}dθ d|w|^{2},

∫_{𝔻} dν(w) = \frac{2π(n − 1)}{π} \frac{1}{2} ∫_{0}^{1}(1 − x)^{n−2} dx = 1

so dν is normalized. Thus, we will not worry about constants. Using x_{1} + ix_{2}, x_{3} + ix_{4}, … for the n complex variables, the measure δ(1 − |x|^{2})dx_{1} … dx_{2n} is

\frac{dx_{1} … dx_{2n−1}}{2(1 − ∑_{j=1}^{2n−1} x_{j}^{2})^{1/2}}

Integrating out x_{3}, …, x_{2n−1} for fixed x_{1}, x_{2} with ρ = (1 − |x_{1}|^{2} − |x_{2}|^{2})^{1/2}, the measure is

\frac{1}{2} ∫_{|y|≤(1−ρ^{2})^{1/2}} \frac{d^{2n−3}y}{(1 − ρ^{2} − y^{2})^{1/2}}

Scaling x = y/(1 − ρ^{2})^{1/2}, we find

\frac{1}{2} (1 − ρ^{2})^{2n−4/2} ∫_{|x|≤1} \frac{d^{2n−3}x}{(1 − x^{2})^{1/2}}

so the measure is C(1 − w^{2})^{n−2} d^{2}w, proving (11.3). □

Theorem 11.4 below must be well known to experts on homogeneous spaces.

Lemma 11.3. Let dν_{1}, dν_{2} be two probability measures on compact spaces X and Y, and let dν be a probability measure on X ×Y. Suppose (i) π_{1}^{∗}(dν) = dν_{1}, that is, if π_{1}(x, y) = x, then ν_{1}(B) = ν(π_{1}^{−1}[B]). (ii) For any continuous f on X, ∫_{X} f dν = C_{f} dν_{2}, that is,

∫ f(x)g(y) dν = C_{f} ∫ g(y) dν_{2}  (11.4)

for all continuous g on Y. Then dν = dν_{1} ⊗ dν_{2}.

Proof. Taking g = 1 in (11.4),

C_{f} = ∫ f(x) dν = ∫ f(x) dν_{1}(x)

by (i). Thus,

∫ f(x)g(y) dν = (∫ f(x) dν_{1}(x))(∫ g(y) dν_{2}(y))

so dν = dν_{1} ⊗ dν_{2} integrated on product functions which are total in C(X × Y). □

Theorem 11.4. Let G be a compact group and H a closed subgroup. Let dν_{G}, dν_{H} be normalized Haar measures and π: G → G/H. Let dν_{G/H} be the measure induced by dν_{G} on G/H, that is,

ν_{G/H}(B) = ν_{G}(π^{−1}[B])

Let O be an open set in G/H and f: O → G a continuous crosssection, that is, π[f(x)] = x for all x ∈ O. Coordinatize π^{−1}[O] by O × H via

(x, h) ↦ f(x)h  (11.5)

Then, on π^{−1}[O],

dν_{G}(x, h) = dν_{G/H}(x) dν_{H}(h)  (11.6)

Proof. The existence of a cross-section implies that under the coordinates (11.5), π^{−1}[O] ≅ O × H. Clearly, π_{1}^{∗}(dμ_{G}) = dμ_{G/H} ↾ O, by construction of dμ_{G/H}. On the other hand, ∫_{O} f dμ_{G/H} is a measure on H invariant under right multiplication by any h ∈ H, so this is C_{f} dμ_{H}. Therefore, Lemma 11.3 applies and (11.6) holds. □

Proof of Theorem 11.1. We use induction on n. n = 1, that is, that for 𝕌(1), U = (e^{iθ_{0}}) has Haar measure \frac{dθ_{0}}{2π}, is immediate.

Note that U ∈ 𝕌(n) has δ_{0} as a cyclic vector if and only if U has simple spectrum, and for each eigenvector φ_{k} of U, we have ⟨φ_{k}, δ_{0}⟩ ≠ 0. As is well known, {U | U has a degenerate eigenvalue} has codimension 3 and so zero Haar measure. Similarly, ⟨φ_{k}, δ_{0}⟩ = 0 on a set of codimension 2 and so zero Haar measure. Thus, C_{n} = {E | δ_{0} is cyclic for U} has full Haar measure.

Let O = {η ∈ 𝕊ℂ^{2n−1} | η ≠ δ_{0}}. Then f(x) = g_{1}(z) given in Theorem 10.4 is a cross-section, and so dμ_{𝕌(n)} = dμ_{𝕊ℂ^{2n−1}} ⊗ dμ_{𝕌(n−1)} by Theorem 11.4.

By Theorem 10.4, g_{1}^{−1}(π_{1}[C_{n}]) is (z, V (U)) and V (U) has Verblunsky coefficients {α_{j+1}(U)}_{j=0}^{n−2}. Thus, by induction, dμ_{𝕌(n−1)} on these α’s is the product (11.6) without the α_{0} factor.

By (10.18) and Lemma 11.3, the α_{0} distribution generated by dμ_{𝕊ℂ^{2n−1}} is the α_{0} factor in (11.1). □

The proof in [50] differs in two ways: First, in place of the AGR factorization, Killip–Nenciu use a (Householder) factorization as a phase factor times a product of reflections. Instead of using induction on symmetric spaces as we do, they use an alternate that would work with the AGR factorization also. Starting with φ_{0} = δ_{0}, we let ψ_{0} = Uφ. There is a unique vector, φ_{1} (what we called g_{2}(π(U)) in Theorem 10.4), in the span of φ and ψ_{0}, so that ⟨ψ_{0}, φ_{1}⟩ > 0 and ⟨φ_{1}, δ_{0}⟩ = 0. φ_{1} is cyclic for V (U) and so, by induction, we obtain ψ_{0}, ψ_{1}, …, ψ_{n−1} an ON basis with |⟨δ_{j}, ψ_{j}⟩| < 1. It is not hard to see that, via the AGR factorization, this sets up a one-one map of ON basis with |⟨δ_{j}, ψ_{j}⟩| < 1 and U’s with δ_{0} cyclic for U. Haar measure induces on the ψ’s a measure as follows: ψ_{0} is uniformly distributed on 𝕊ℂ^{2n−1}; ψ_{1} uniformly on the copy of 𝕊ℂ^{2n−3} of unit vectors orthogonal to ψ_{1}; ψ_{2} uniformly on 𝕊ℂ^{2n−5}, etc. Since ⟨δ_{j}, ψ_{j}⟩ = ᾱ_{j}, we obtain the measure (11.1).

Since det(Θ̃_{j}(α_{j})) = −1, det(˜Θ̃_{N−1}(α_{N−1})) = ᾱ_{N−1}, we see

det(G({α_{n}}_{n=0}^{N−1}) = (−1)^{N−1}ᾱ_{N−1}

Thus, 𝕊𝕌(n) = {U ∈ 𝕌(n) | det(U)} is precisely these U with α_{N−1} = (−1)^{N−1}. The same inductive argument thus proves:

Theorem 11.5. Let dμ be normalized Haar measure on 𝕊𝕌(n). Then for a.e. U, δ_{0} = (1, 0, …, 0)^{t} is cyclic and the measure induced on 𝔻^{n−1} by U → α_{j}(U, δ_{0}) (with α_{n−1}(U, δ_{0}) ≡ (−1)^{n−1}) is the product measure given by (11.1) with the final dθ term dropped.

𝕊𝕆[n] is the n × n real unitary matrices (i.e., orthogonal matrices). If δ_{0} is cyclic, they have Verblunsky coefficients which are easily seen to lie in (−1, 1). Conversely, it is easy to see that if α_{j} ∈ (−1, 1) for j = 0, …, n − 2, there is an orthogonal matrix with those α_{j}’s. A similar analysis lets us compute the distribution on (−1, 1)^{n−1} induced by Haar measure on 𝕊𝕆[n]. We need only replace Lemma 11.3 by

Lemma 11.6. Let dη_{n−1} be the measure on the n − 1-dimensional unit sphere in ℝ^{n}. The induced measure on x_{1} is

\frac{Γ(\frac{n}{2})(1 − |x_{1}|^{2})^{(n−3)/2} dx_{1}}{\sqrt{π} Γ(\frac{n−1}{2})}

Proof. That the measure is C(1−|x_{1}|^{2})^{(n−3/2)} dx_{1} follows from the same calculation as in Lemma 11.3. The normalization is the inverse of the beta function 2^{2−n}Γ(n − 1)/Γ(\frac{n−1}{2}) which, as noted by [50] (there is a ( )^{−1} missing on the leftmost term in their (3.14)), can be written, using the duplication formula for beta functions, as Γ(\frac{n}{2})/\sqrt{π} Γ(\frac{n−1}{2}). □

We thus have

Theorem 11.7 ([50]). The measure on (−1, 1)^{n−1} induced by Haar measure on 𝕊𝕆[n] mapped to the real Verblunsky coefficients is

\frac{Γ(\frac{n}{2})}{π^{n/2}} ∏_{k=0}^{n−1}(1 − α_{k}^{2})^{(n−k−3)/2} dα_{k}

The CUE eigenvalue distribution [20, 21, 22] is the one for U ∈ 𝕌[n] induced by Haar measure. Weyl’s integration formula (see, e.g., [66]) says that if λ_{1}, …, λ_{n} with λ_{j} = e^{iθ_{j}} are the eigenvalues, this is C ∏_{i<j}|λ_{i} − λ_{j}|^{2} ∏_{j=1}^{n} \frac{dθ_{j}}{2π}. Theorem 11.1 says that CMV matrices with distribution of α’s given by (11.1) has the same distribution, and so gives a model for CUE by five-diagonal matrices. [50] find a similar model for the “β-distributions” given by C_{β} ∏_{i=j}|λ_{i} − λ_{j}|^{β} \frac{dθ_{j}}{2π}; see also Forrester–Rains [25].

12. CMV and the AL Flow

One of the great discoveries of the 1970’s ([19, 24, 26, 58, 80] and dozens of other papers) is that lurking within one-dimensional Schrödinger operators and Jacobi matrices is a completely integrable system (resp., KdV and Toda flows), natural “invariant” tori, and a natural symplectic structure in which the Schrödinger operator or Jacobi matrix is the dynamical half of a Lax pair.

Such structures occur also for Verblunsky coefficients, and the dynamical half of the Lax pair is the CMV matrix. While the CMV part obviously requires CMV matrices, the other parts do not, and it is surprising that it was only in 2003–04 that they were found. We will settle here for describing the two most basic structures, leaving further results to mentioning the followup papers: Geronimo–Gesztesy–Holden [30], Gesztesy–Zinchenko [36], Killip–Nenciu [51], Li [55], and Nenciu [61].

On 𝔻, introduce the symplectic form given by the Poisson bracket (where, as usual, \frac{∂}{∂z} and \frac{∂}{∂z̄} are \frac{1}{2}[\frac{∂}{∂x} ∓ i\frac{∂}{∂y}]),

\{f, g\} = iρ^{2}[\frac{∂f}{∂z̄} \frac{∂g}{∂z} − \frac{∂f}{∂z} \frac{∂g}{∂z̄}]  (12.1)

The ρ^{2} is natural as we will see below. Extend this to 𝔻^{p} (coordinatized by (α_{0}, …, α_{p−1}) by

\{f, g\} = i ∑_{j=0}^{p−1} ρ_{j}^{2}[\frac{∂f}{∂ᾱ_{j}} \frac{∂g}{∂α_{j}} − \frac{∂f}{∂α_{j}} \frac{∂g}{∂ᾱ_{j}} ]  (12.2)

Because of the ρ^{2},

\{ ∏_{j=0}^{p−1} ρ_{j}^{2}, g\} = − ∏_{j=0}^{p−1} ρ_{j}^{2}( ∑_{j=0}^{p+1} \frac{∂g}{∂θ_{j}} )

and functions of ρ_{0} … ρ_{p−1} generate simultaneous rotations of all phases. Nenciu–Simon [62] proved the following:

Theorem 12.1 ([62]). Let p be even and let Δ(z, {α_{j}}_{j=0}^{p−1}) be the discriminants (see (3.20)) for the periodic sequence with α_{j+kp} = α_{j} for j = 0, …, p − 1; k = 0, 1, 2, . . .. Then, with respect to the symplectic form (12.2),

{Δ(w), Δ(z)} = 0  (12.3)

for all w, z ∈ ℂ\{0}.

Note: See [55, 51] for a discussion of symplectic forms on unitary matrices.

Since \overline{Δ(1/w̄)} = Δ(w), and the leading coefficient is real, Δ(z) has p real coefficients, that is, Δ(z) = ∑_{j=−p/2}^{p/2} a_{j}z^{j} with a_{−j} = ā_{j}, then a_{p/2}, Re a_{p/2−1}, Im a_{p/2−1}, …, Re a_{1}, Im a_{1}, a_{0} are the p real functions of {α_{j}}_{j=0}^{p−1} which Poisson commute. They are independent at a.e. points (in α) and define invariant tori. Each one generates flows that are completely integrable. The simplest is

−iα̇_{j} = ρ_{j}^{2}(α_{j+1} + α_{j−1})  (12.4)

which has been known as a completely integrable system for a long time under the name “defocusing Ablowitz–Ladik flow” (after [1, 2, 3]).

Nenciu has proven a beautiful result:

Theorem 12.2 ([59, 60, 61]). The flows generated by the coefficients of Δ can be put into Lax pair form with the dynamical element being the Floquet CMV matrix.

For details as well as extensions to some infinite CMV matrices, see the references above.

The flow generated by ∏_{j=0}^{p−1} ρ_{j}^{2} realizes the α_{j} → λα_{j} invariance of the isospectral tori. The flow (12.4) is generated by Re(a_{1}). The Im(a_{k}) generate flows that preserve the set of {α_{j}}_{j=0}^{p−1} where all α_{j} are real (as a set, not pointwise). The simplest of these, generated by Im(a_{1}), is

α̇_{n} = ρ_{n}^{2}(α_{n+1} − α_{n−1})  (12.5)

called the Schur flow. Via the Geronimus relations of the next section, these generate a flow on Jacobi parameters that is essentially the Toda flow. For further discussion, see [4, 23, 27, 59].

13. CMV Matrices and the Geronimus Relations

In a celebrated paper, Szegő [76] found a connection between orthogonal polynomials for measures on [−2, 2] (he had [−1, 1]; I use the scaling common in the Schrödinger operator community) and OPUC.

Given a measure dγ on [−2, 2], one defines the unique measure dξ on ∂𝔻 which is invariant under z → z̄ and obeys

∫ g(x) dγ(x) = ∫ g(2 cos θ) dξ(θ)  (13.1)

What Szegő showed is that the orthonormal polynomials p_{n} for dγ and the OPUC for φ_{n} for dξ are related by

p_{n}(z + \frac{1}{z}) = C_{n}z^{−n}(φ_{2n}(z) + φ_{2n}^{∗}(z))  (13.2)

The normalization constants C_{n} (see (13.1.14) in [70]) → 1 as n → ∞ if ∑_{n=0}^{∞}|α_{n}|^{2} < ∞. Motivated by this, Geronimus [35] found a relation between the Verblunsky coefficients, α_{n}, for dξ and the Jacobi parameters {a_{n}}_{n=1}^{∞}, {b_{n}}_{n=1}^{∞} for dγ (see Theorem 13.1.7 of [70]):

a_{n+1}^{2} = (1 − α_{2n−1})(1 − α_{2n}^{2})(1 + α_{2n+1})  (13.3a)
b_{n+1} = (1 − α_{2n−1})α_{2n} − (1 + α_{2n−1})α_{2n−2}  (13.3b)

In [50], Killip–Nenciu found a direct proof of (13.3) by finding a beautiful relation between CMV and some Jacobi matrices. We will sketch the idea, leaving the detailed calculations to [50] or the pedagogic presentation in Section 13.2 of [70].

A measure is invariant under z → z̄ if and only if all {α_{n}}_{n=0}^{∞} are real. Θ(α) with α real is selfadjoint and unitary with determinant −1, hence eigenvalues ±1, that is, a reflection on ℂ^{2}. Thus,

α_{n} = ᾱ_{n} all n ⇒ M^{2} = L^{2} = 1

Since χ_{n}(z) = \overline{x_{n}(1/z̄)}, we see that if μ is invariant and (Mf)(z) = f(z̄), then

⟨χ_{j}, Mχ_{ℓ}⟩ = M_{jℓ}

C + C^{∗} is selfadjoint and maps {f | Mf = f} to itself. Let us see in a natural basis that its restriction to this invariant subspace is a Jacobi matrix.

If α is real and

S(α) = \frac{1}{\sqrt{2}} (\begin{matrix}\sqrt{1 − α} & −\sqrt{1 + α} \\ \sqrt{1 + α} & \sqrt{1 − α} \end{matrix})

then

S(α)Θ(α)S(α)^{−1} = (\begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix})  (13.4)

Define

S = 1_{1×1} ⊕ S(α_{1}) ⊕ S(α_{3}) ⊕ ⋯

so

SMS^{−1} = R = ( \begin{matrix}1 \\ & −1 \\ & & 1 \\ & & & −1 \\ & & & & . & . & . \end{matrix} )

and define

B = SLS^{−1}

Then

S(C + C^{−1})S^{−1} = RB + BR

which commutes with R.

B is seven-diagonal as a product of three tridiagonal matrices. Moreover, since B commutes with R, its odd-even matrix elements vanish. It follows that

RB + BR = J_{e} ⊕ J_{o}

where J_{e} acts on {δ_{2n}}_{n=0}^{∞} and J_{o} on {δ_{2n+1}}_{n=0}^{∞}, and each is a Jacobi matrix. A calculation shows that the Jacobi parameters of J_{e} are given by (13.3) and that the spectral measures are related by (13.1). One can also analyze J_{o} which is related to another mapping of Szegő [76] and one gets two more Jacobi matrices by looking at C + C^{−1} restricted to the spaces where L = 1 or L = −1.

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