RANK ONE PERTURBATIONS AND THE ZEROS OF PARAORTHOGONAL POLYNOMIALS ON THE UNIT CIRCLE

BARRY SIMON*

Abstract. We prove several results about zeros of paraorthogonal polynomials using the theory of rank one perturbations of unitary operators. In particular, we obtain new details on the interlacing of zeros for successive POPUC.

1. Introduction

This note concerns an aspect of the theory of orthogonal polynomials on the unit circle (OPUC); for background, see [11, 4, 7, 8, 9]. Given a nontrivial probability measure, dμ, on ∂𝔻 = {z ∈ ℂ | |z| = 1}, we let Φ_{n}(z) (we use Φ_{n}(z, dμ) when dμ needs to be explicit) be the monic orthogonal polynomials. They obey the Szegő recursion relations

Φ_{n+1}(z) = zΦ_{n}(z) − ᾱ_{n}Φ_{n}^{∗}(z)  (1.1)
Φ_{n}^{∗}(z) = z^{n} \overline{Φ_{n}(1/z̄)}  (1.2)

where {α_{n}}_{n=0}^{∞} ∈ 𝔻^{∞} are the Verblunsky coefficients. dμ ↔ {α_{n}}_{n=0}^{∞} sets up a oneone correspondence between 𝔻^{∞} and nontrivial probability measures (Verblunsky’s theorem).

Given β ∈ ∂𝔻, the paraorthogonal polynomials (POPUC) are defined by (Note:

[7] uses β where (1.3) uses −β̄; (1.3) is the right convention.)

Φ_{n}(z, dμ; β) = zΦ_{n−1}(z, dμ) − β̄Φ_{n−1}^{∗}(z, dμ)  (1.3)

More generally, we will consider a sequence {β_{n}}_{n=1}^{∞} ∈ ∂𝔻 and

Φ̃_{n}(z) = Φ_{n}(z, dμ; β_{n})  (1.4)

POPUC were introduced at least as early as Jones, Njåstad, and Thron [6]. About five years ago, Cantero, Moral, and Velázquez [1] and Golinskii [5] realized that zeros of POPUC shared many properties of zeros of OPRL and independently proved a number of basic results about these zeros. Cantero et al. [3] recently proved additional results. The basic tool in [1, 5] is the Christoffel–Darboux formula; [3] also exploits the CMV matrix. Our goal in this paper is to use the theory of rank one perturbations of unitary matrices to recover many of the basic results about zeros of POPUC and prove some new results. In particular, we will illuminate the issue of interlacing of the zeros of successive POPUC.

Date: June 29, 2006.

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

First, some notation. Given distinct z, w ∈ ∂𝔻, (z, w) is the set of points, ζ, in ∂𝔻 with

Arg(z) < Arg(ζ) < Arg(w)  (1.5)

where a branch of Arg is chosen so 0 < Arg(w) − Arg(z) < 2π. An ordered set of points (z_{1}, …, z_{ℓ}) ∈ ∂𝔻^{ℓ} is called cyclicly ordered if each (z_{j}, z_{j+1})_{j=1}^{ℓ} and (z_{ℓ}, z_{1}) contain no other z_{j}’s. The ordering is fixed by such cyclicity up to a single choice. We will always assume zeros of POPUC are cyclicly ordered.

Two cyclicly ordered sets (z_{1}, …, z_{ℓ}) and (w_{1}, …, w_{ℓ}) in ∂𝔻^{ℓ} are said to strictly interlace if after a cyclic permutation of the w’s, we have w_{j} ∈ (z_{j}, z_{j+1}), j = 1, 2, …, ℓ − 1, w_{ℓ} ∈ (z_{ℓ}, z_{1}). This, of course, implies z_{j} ∈ (w_{j−1}, w_{j}), j = 2, 3, …, ℓ and z_{1} ∈ (w_{ℓ}, w_{1}).

For {α_{j}}_{j=0}^{∞}, the second kind polynomials, Ψ_{n}(z, dμ) are defined, as usual, to be the Φ_{n}’s associated to α̃_{j} = −α_{j}(dμ). We define

Ψ_{n}(z, dμ; β) = zΨ_{n−1}(z, dμ) − β̄Ψ_{n−1}^{∗}(z, dμ)  (1.6)

We can now state our main results:

Theorem 1.1 ([1, 5]). If (w_{0}, w_{1}) is an interval disjoint from supp(dμ), then for any choice of β and any n, Φ_{n}(z, dμ; β) has at most one zero in (w_{0}, w_{1}).

The following has also been proven by Wong [12]:

Theorem 1.2. Let (z_{1}, …, z_{n}) be the zeros of some Φ_{n}(z, dμ; β) and (w_{1}, …, w_{n}) of Ψ_{n}(z, dμ; −β). (Note: −β, not β.) Then these zeros strictly interlace.

Theorem 1.3 ([1, 5]). Fix dμ and n and distinct β, β′ in ∂𝔻. Then the zeros of Φ_{n}(z, dμ; β) and Φ_{n}(z, dμ; β′) strictly interlace.

The power of our approach is shown by the refined version we obtain relating zeros of Φ̃_{n+1} and Φ̃_{n}. We will need the following computed sequence in ∂𝔻:

λ_{n} = β̄_{n+1}β̄_{n}(\frac{β_{n}α_{n} − 1}{β̄_{n}ᾱ_{n} − 1})  (1.7)

Theorem 1.4. For each n, one of two possibilities holds:

(i) Φ̃_{n} and Φ̃_{n+1} have no zeros in common. In that case, λ_{n} is not a zero of either, and {zeros of Φ̃_{n}} ∪ {λ_{n}} strictly interlace {zeros of Φ̃_{n+1}}.

(ii) Φ̃_{n} and Φ̃_{n+1} have a single zero in common. In that case, λ_{n} is that zero and {zeros of Φ̃_{n}} strictly interlace {zeros of Φ̃_{n+1}}\{λ_{n}}.

Corollary 1.5. If Φ̃_{1}, Φ̃_{2}, Φ̃_{3}, … have a common zero at λ, then β_{n} are given inductively by

β_{1} = λ̄  (1.8)
β_{n+1} = λ̄β̄_{n}(\frac{β_{n}α_{n} − 1}{β̄_{n}ᾱ_{n} − 1})  (1.9)

Example 1.6. α ≡ 0. Then β_{n} = λ̄^{n} and Φ̃_{n}(z) = z^{n} − λ^{n} precisely the POPs with a zero at λ for all n. □

The key to our proofs is the connection of Φ̃_{n} to CMV matrices [2, 7, 10]. Φ̃_{n} is the determinant of a suitable finite CMV matrix, and so its zeros are the eigenvalues. All our results concern what happens to eigenvalues of unitary matrices under rank one perturbations. Section 2 discusses general rank one perturbations, and Section 3 the application to POPUC.

It is a pleasure to thank María José Cantero and Lilian Wong for useful discussions.

2. Rank One Perturbations

Rank one perturbations of unitaries are discussed in Sections 1.3.9, 1.4.16, 3.2, 4.5, and 10.16 of [7, 8], and some of the results in this section are spread through that material.

If U and V are two unitaries on a finite- or infinite-dimensional Hilbert space and U − V is rank one, we pick a unit vector φ ∈ ker(U − V)^{⊥} and note there must be a λ ∈ ∂𝔻 with

V φ = λUφ  (2.1)

and thus

V − U = (λ − 1)⟨φ, · ⟩Uφ  (2.2)

It is convenient to define for z ∈ 𝔻 and A unitary:

F_{A,φ}(z) = ⟨φ, \frac{A + z}{A − z} φ⟩  (2.3)
f_{A,φ}(z) = z^{−1}(1 − F (z))(1 + F (z))^{−1}  (2.4)

F is a Carathéodory function (Re F (z) > 0 on 𝔻, F (0) = 1) and f a Schur function (|f(z)| < 1 on 𝔻). The spectral measure for A, φ is given by

F_{A,φ}(z) = ∫ \frac{e^{iθ} + z}{e^{iθ} − z} dμ_{A,φ}(z)  (2.5)

It is not hard to see that

Proposition 2.1. Let φ be a cyclic vector for a unitary A (i.e., {A^{k}φ}_{k=−∞}^{∞} is total). An interval (w_{0}, w_{1}) in ∂𝔻 is disjoint from σ_{ess}(A) if and only if f has an analytic continuation through (w_{0}, w_{1}) with |f(z)| = 1 on that interval. In that case,

(a) Arg(f) is strictly monotone increasing on (w_{0}, w_{1}).

(b) The only spectra of A on (w_{0}, w_{1}) are simple eigenvalues precisely at the points

z where

zf(z) = 1  (2.6)

Arg(f) is increasing since |f(z)| < 1 in 𝔻 and |f(z)| = 1 on (w_{0}, w_{1}) implies ∂|f(re^{iθ})|/∂r ≥ 0 on (w_{0}, w_{1}). So by the Cauchy–Riemann equations, ∂Arg(f(e^{iθ}))/∂θ ≥ 0. (b) holds since

F (z) = \frac{1 + zf(z)}{1 − zf(z)}  (2.7)

has poles at points where (2.6) holds.

When (2.2) holds, a direct calculation (see (1.4.90) and the end of Section 3.2 in [7]) shows that

Proposition 2.2. If (2.2) holds, then

f_{V,φ}(z) = λ^{−1}f_{U,φ}(z)  (2.8)

We immediately have Theorem 2.3. Let (2.2) hold. If (w_{0}, w_{1}) ∩ σ(U) = ∅, then V has at most one eigenvalue in [w_{0}, w_{1}] and no other spectrum there.

Proof. Let K be the cyclic subspace for U and φ. Since U = V on K^{⊥} which is invariant for both, we can suppose φ is cyclic. In that case, picking z_{0} ∈ (w_{0}, w_{1}) and then Arg(z(f(z))) so Arg(z_{0}f_{0}(z_{0})) ∈ (0, 2π), we see Arg(z(f(z)) ∈ (0, 2π) on all of (w_{0}, w_{1}) since (2.6) has no solution there. By the strict monotonicity of Arg(f), zf(z) = λ has at most one solution in [w_{0}, w_{1}], so by Propositions 2.1 and 2.2, V that at most one eigenvalue there. □

Proposition 2.4. Let U, V be unitaries on ℂ^{n} so (2.2) holds for λ ≠ 1 and for φ cyclic for U. Then the eigenvalues of U and V strictly interlace.

Proof. Since U has a cyclic vector, its spectrum is simple so zf(z) = 1 has n solutions. Since Arg(z(f)) is strictly monotone, zf(z) = λ has n solutions which interlace the solutions of zf(z) = 1. □

One can say something about the case where φ is not cyclic.

Proposition 2.5. Let U, V be unitaries on ℂ^{n} so (2.2) holds. Let z_{0}, z_{1} be two eigenvalues of U. Then V has an eigenvalue in [z_{0}, z_{1}] (= (z_{0}, z_{1}) ∪ {z_{0}, z_{1}}).

Proof. Let K be the cyclic subspace of (U, φ) which is invariant for U. If z_{0} and z_{1} are eigenvalues of U ↾ K, V has an eigenvalue in (z_{0}, z_{1}) by Proposition 2.4. If not, since U ↾ K^{⊥} = V ↾ K^{⊥}, either z_{0} or z_{1} is an eigenvalue of V. □

Finally, we have a specialized result that is precisely what we need to prove Theorem 1.4:

Proposition 2.6. Let U = U_{1} ⊕ U_{2} on K_{1} ⊕ K_{2}, two finite-dimensional subspaces of H, a space of dimension n. Let φ_{j} (j = 1, 2) be cyclic vectors for U_{j} on K_{j}. Let φ = aφ_{1} ⊕ bφ_{2} where (a, b) ≠ (0, 0) and |a|^{2} + |b|^{2} = 1. Let V be given by (2.2) with λ ≠ 1. If U_{1} and U_{2} have ℓ eigenvalues in common, then V has these ℓ common values as eigenvalues and its other n − ℓ eigenvalues strictly interlace those of U.

Proof. Since φ_{j} is cyclic for U_{j}, any simple eigenvalue of U is in the cyclic subspace generated by U, φ. Moreover, any common eigenvalue is a simple eigenvalue for U ↾ K where K = cyclic subspace of φ. Thus U ↾ K has all the eigenvalues of U but with multiplicity 1. The eigenvalues of V ↾ K strictly interlace by Proposition 2.4. The eigenvalues of V ↾ K^{⊥} = U ↾ K^{⊥} are exactly the common eigenvalues. □

Remark. φ is cyclic if and only if ℓ = 0.

3. Zeros of POPUC and Finite CMV Matrices

Given a sequence \{γ_{n}\}_{n=0}^{∞} of elements in \overline{𝔻}, one defines the CMV matrix C({γ_{n}}_{n=0}^{∞}) on ℓ^{2} by

C = LM  (3.1)
L = Θ(γ_{0}) ⊕ Θ(γ_{2}) ⊕ ⋯  (3.2)
M = 1_{1×1} ⊕ Θ(γ_{1}) ⊕ Θ(γ_{3}) ⊕ ⋯  (3.3)

where 1_{1×1} is the one-dimensional identity matrix, and Θ is given by

Θ(γ) = ( \begin{matrix}γ̄ & τ \\ τ & −γ \end{matrix} )  (3.4)

τ = (1 − |γ|^{2})^{1/2}  (3.5)

It is a fundamental result of Cantero, Moral, and Velázquez [2] (see also [7, Section 4.2]) and see [10] for other references) that if dμ is a nontrivial probability measure on ∂𝔻, χ_{n} is the basis of L^{2}(∂𝔻, dμ) obtained by applying Gram–Schmidt to 1, z, z^{−1}, z^{2}, z^{−2}, …, and α_{n}(dμ) are the Verblunsky coefficients of dμ, then C({α_{n}(dμ)}_{n=0}^{∞}) is the matrix of multiplication by z on L^{2}(∂𝔻, dμ) in χ_{n} basis. Note in this case that γ_{n} ∈ 𝔻 (rather than some γ_{n} ∈ ∂𝔻), in which case we call C a proper CMV matrix.

If |γ| = 1, then τ = 0, and Θ(γ) is a direct sum of two 1 × 1 matrices, and so, if |γ_{n−1}| = 1, C breaks into a direct sum of an n × n matrix and an infinite piece. The finite piece, C_{n}({γ_{0}, …, γ_{n−1}}), is called a finite CMV matrix. It is not hard to show that (see, e.g., [10]):

Proposition 3.1. If γ_{n} ∈ 𝔻 for all n, then δ_{0} ≡ (1, 0, …)^{t} is a cyclic vector for C({γ_{n}}_{n=0}^{∞}). If γ_{0}, …, γ_{n−2} ∈ 𝔻, γ_{n−1} ∈ ∂𝔻, then δ_{0} is a cyclic vector for C_{n}({γ_{m}}_{m=0}^{n−1}).

Moreover (see [7, Section 4.2]),

Proposition 3.2. If α_{0}, …, α_{n−2} ∈ 𝔻 and β ≡ α_{n−1} ∈ ∂𝔻, then

Φ_{n}(z, dμ_{α}; β) = det(z − C_{n}({α_{j}}_{j=0}^{n−1}))  (3.6)

In particular, the zeros of Φ̃_{n} are the eigenvalues of a finite CMV matrix.

Finally, we need the following, which generalizes Lemma 4.5.1 of [7]:

Lemma 3.3. Let α ∈ 𝔻 and β ∈ ∂𝔻. Then

Θ(α) − (\begin{matrix}β & 0 \\ 0 & x \end{matrix})  (3.7)

is rank one if and only if

x = β̄(\frac{βα − 1}{β̄ᾱ − 1})  (3.8)

Proof. A 2 × 2 matrix is rank one if and only if det(A) = 0. Since

det(Θ(α) − (\begin{matrix}β & 0 \\ 0 & x \end{matrix})) = (ᾱ − β)(−α − x) − (1 − |α|^{2})  (3.9)

we see (3.7) is rank one if and only if RHS of (3.9) = 0, which is solved by (3.8). □

Note: |x| = 1, so ( \begin{matrix}β & 0 \\ 0 & x \end{matrix} ) is unitary.

Proof of Theorem 1.1. Let C be the CMV matrix, C({α_{n}(dμ)}_{n=0}^{∞}), of dμ. Given n and β, pick x so Θ(α_{n−1})−( \begin{matrix}β & 0 \\ 0 & x \end{matrix} ) is rank one, and let C̃ be the matrix obtained from C by replacing Θ(α_{n−1}) by ( \begin{matrix}β & 0 \\ 0 & x \end{matrix} ). Then C̃ is unitary (by the note after Lemma 3.3) and C − C̃ is rank one. Thus, by Theorem 2.3, C̃ has at most one eigenvalue in (w_{0}, w_{1}). But C̃ is a direct sum of C_{n}({α_{0}, …, α_{n−2}, β}) and another matrix, so C_{n} has at most one eigenvalue in (w_{0}, w_{1}). By Proposition 3.2, zeros of Φ̃_{n} are eigenvalues of C_{n}. □

Proof of Theorem 1.2. Let α̃_{n−1} ≡ β. By Theorem 5.2 of [10] (see also [7, Theorem 4.2.9]), C_{n}({−α̃_{m}}_{m=0}^{n−1}) is unitarily equivalent to C̃_{n} ≡ L({α̃_{m}}_{m=0}^{n−1})M̃({α̃_{m}}_{m=0}^{n−1}) where M̃ differs from M by having −1_{1×1} in place of 1_{1×1}. Thus, C_{n}({α_{m}}_{m=0}^{n−1})−C̃ is rank one, and Theorem 1.2 follows from Propositions 2.4 and 3.1. □

Proof of Theorem 1.3. If

α_{j} = α′_{j} = α_{j}(dμ)  j = 0, …, n − 2  (3.10)
α_{n−1} = β  α′_{n−1} = β′  (3.11)

then C({α_{m}}_{m=0}^{n−1}) − C({α′_{m}}_{m=0}^{n−1}) is obviously rank one. Moreover, δ_{n−1} is a cyclic vector since C_{n} run backwards is essentially another C_{n} (with the initial 1_{1×1} replaced by ᾱ_{n−1}1_{1×1}) or C_{n}^{t}. Thus, Theorem 1.3 follows from Proposition 2.4. □

Proof of Theorem 1.4. Let C_{n+1} (resp. C_{n}) be the (n+1)×(n+1) (resp. n×n) finite CMV matrix whose characteristic polynomial is Φ̃_{n+1} (resp. Φ̃_{n}). By Lemma 3.3, a rank one perturbation turns C_{n+1} into C_{n} ⊕ λ_{n}1_{1×1} where λ_{n} is given by (1.7). The vector in the perturbation is aδ_{n−1} +bδ_{n}, so Proposition 2.6 applies and proves Theorem 1.4. □

As a final result:

Theorem 3.4. Let m > n. Then strictly between any pair of zeros of Φ̃_{n} is a zero of Φ̃_{m}.

Proof. Let C_{n}, C_{m} be as in the last proof. By a rank one perturbation, C_{m} can be changed to C_{n} ⊕ Q_{m−n}. Now apply Proposition 2.6. □

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