Refined interlacing properties for zeros of paraorthogonal polynomials on the unit circle

TL;DR

This paper extends known results on the interlacing of zeros of paraorthogonal polynomials on the unit circle, providing a unified approach that relates to matrices similar to unitary upper Hessenberg matrices with positive subdiagonals.

Contribution

It offers a simple, unified extension of interlacing properties for zeros of paraorthogonal polynomials on the unit circle.

Findings

Extended interlacing results for zeros of paraorthogonal polynomials.

Unified approach applicable to matrices similar to unitary upper Hessenberg matrices.

Clarified the relationship between polynomial zeros and matrix characteristics.

Abstract

The purpose of this note is to extend in a simple and unified way the known results on interlacing of zeros of paraorthogonal polynomials on the unit circle. These polynomials can be regarded as the characteristic polynomials of any matrix similar to an unitary upper Hessenberg matrix with positive subdiagonal elements.

Equations

$$\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\},{\mathbb{S}}^1:=\{z\in \mathbb{C}:|z|=1\}.$$

$${\Theta }_j:=\Theta ({\alpha }_j),\Theta (\alpha ):=\left(\begin{array}{cc}\overline{\alpha }& \rho \\ \rho & -\alpha \end{array}\right),\rho :={\left(1-|\alpha {|}^2\right)}^{1/2}.$$

$$\mathcal{C}:=\mathcal{L}\mathcal{M},$$

$$\mathcal{L}$$

$$\mathcal{M}$$

$$\left(\begin{array}{cccccccc}{\overline{\alpha }}_0& {\rho }_0{\overline{\alpha }}_1& {\rho }_0{\rho }_1& & & & & \\ {\rho }_0& -{\alpha }_0{\overline{\alpha }}_1& -{\alpha }_0{\rho }_1& & & & & \\ & {\rho }_1{\overline{\alpha }}_2& -{\alpha }_1{\overline{\alpha }}_2& {\rho }_2{\overline{\alpha }}_3& {\rho }_2{\rho }_3& & & \\ & {\rho }_1{\rho }_2& -{\alpha }_1{\rho }_2& -{\alpha }_2{\overline{\alpha }}_3& -{\alpha }_2{\rho }_3& & & \\ & & & {\rho }_3{\overline{\alpha }}_4& -{\alpha }_3{\overline{\alpha }}_4& {\rho }_4{\overline{\alpha }}_5& {\rho }_4{\rho }_5& \\ & & & {\rho }_3{\rho }_4& -{\alpha }_3{\rho }_4& -{\alpha }_4{\overline{\alpha }}_5& -{\alpha }_4{\rho }_5& \\ & & & & & {\rho }_5{\overline{\alpha }}_6& -{\alpha }_5{\overline{\alpha }}_6& {\rho }_6{\overline{b}}_7\\ & & & & & {\rho }_5{\rho }_6& -{\alpha }_5{\rho }_6& -{\alpha }_6{\overline{b}}_7\end{array}\right)$$

$$\displaystyle P_{n+1}(z):=\det\big{(}z\mathcal{I}-\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})\big{)}$$

$$\mathcal{C}({\alpha }_0,\dots ,{\alpha }_{n-1},b_n)=\left(\begin{array}{cc}{\mathcal{C}}_{11}& {\mathcal{C}}_{12}\\ {\mathcal{C}}_{21}& {\mathcal{C}}_{22}\end{array}\right),$$

$$b_n(\zeta ):=b_n,b_j(\zeta ):=\frac{\overline{\zeta }{\alpha }_j+b_{j+1}(\zeta )}{{\overline{\alpha }}_j{b}_{j+1}(\zeta )+\overline{\zeta }}(j=n-1,\dots ,1,0).$$

$$A:=C\cap \sigma (\mathcal{C}({\alpha }_0,\dots ,{\alpha }_{m-1},b_m)),B:=\sigma (\mathcal{N})\backslash A,$$

$${\gamma }_m:=\frac{{\alpha }_m-b_m}{{\overline{\alpha }}_m{b}_m-1}.$$

$$A=\sigma (\mathcal{N})\cap \sigma (\mathcal{C}({\alpha }_0,\dots ,{\alpha }_{m-1},b_m)).$$

$$\mathcal{C}(0,0,1)=\left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right),\mathcal{C}(0,0,0,0,0,1)=\left(\begin{array}{cccccc}0& 0& 1& 0& 0& 0\\ 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 1& 0& 0\end{array}\right),$$

$$\sigma (\mathcal{C}(0,0,1))=\left\{1,e^{\pm i2\pi /3}\right\},\sigma (\mathcal{C}(0,0,0,0,0,1))=\left\{\pm 1,e^{\pm i2\pi /3},e^{\pm i\pi /3}\right\}.$$

$$\mathcal{U}=\left(\begin{array}{cc}{\mathcal{U}}_{11}& {\mathcal{U}}_{12}\\ {\mathcal{U}}_{21}& {\mathcal{U}}_{22}\end{array}\right),$$

$$U_1\cap U_2=U_1\cap U=U_2\cap U.$$

$${\chi }_{{}_{\mathcal{U}}}(\zeta )={\chi }_{{}_{\mathcal{U}\mathcal{S}}}(\zeta )+v^T\mathrm{Adj}(\zeta \mathcal{I}-\mathcal{U}\mathcal{S})u.$$

$$\mathrm{Adj}({\lambda }_j\mathcal{I}-\mathcal{U}\mathcal{S})={\chi }_{{}_{\mathcal{U}\mathcal{S}}}^{\prime }({\lambda }_j)z_j{z}_j^{\ast },$$

$${\chi }_{{}_{\mathcal{U}}}({\lambda }_j)=\left({\chi }_{{}_{{\mathcal{U}}_1}}^{\prime }({\lambda }_j){\chi }_{{}_{{\mathcal{U}}_2}}({\lambda }_j)+{\chi }_{{}_{{\mathcal{U}}_1}}({\lambda }_j){\chi }_{{}_{{\mathcal{U}}_2}}^{\prime }({\lambda }_j)\right)z_j^{\ast }u{v}^T{z}_j.$$

$${\mathcal{U}}_{11}{v}_j={\lambda }_j{v}_j,$$

$${\chi }_{{}_{\mathcal{U}\mathcal{S}}}({\lambda }_j)={\chi }_{{}_{\mathcal{U}}}^{\prime }({\lambda }_j)z_j^{\ast }u{v}^T{z}_j.$$

$$z_j^{\ast }u{v}^T{z}_j=z_j^{\ast }\mathcal{U}(\mathcal{I}-\mathcal{S})z_j={\lambda }_j(1-\beta ){\left|a_j\right|}^2\ne 0.$$

$$\displaystyle\det\big{(}\zeta\mathcal{I}-\mathcal{C}_{n}\big{)}=\det\big{(}\zeta\mathcal{I}-\mathcal{C}(\alpha_{0},\dots,\alpha_{m-1},b_{m}(\zeta))\big{)}\det\big{(}\zeta\mathcal{I}-\mathcal{C}_{22}\big{)}$$

$$\mathcal{C}({\alpha }_0,\dots ,{\alpha }_{m-1},b_m(\zeta ))={\mathcal{C}}_{11}-{\mathcal{C}}_{12}({\mathcal{C}}_{22}-\zeta \mathcal{I}{)}^{-1}{\mathcal{C}}_{21},$$

$$\left(\begin{array}{cc}\overline{\beta }& 0\\ 0& 1\end{array}\right)\Theta (\alpha )\left(\begin{array}{cc}1& 0\\ 0& \beta \end{array}\right)=\Theta (\alpha \beta ).$$

$$\displaystyle\mathcal{D}^{*}\,\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})\,\mathcal{D}\,\mathcal{S}=\big{(}\mathcal{D}^{*}\,\mathcal{L}\,\mathcal{V}\big{)}\,\big{(}\mathcal{V}^{*}\,\mathcal{M}\,\mathcal{D}\,\mathcal{S}\big{)}=\mathcal{C}_{m}^{\beta}\,,$$

$$\displaystyle\mathcal{S}\,\mathcal{D}\,\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})\,\mathcal{D}^{*}=\big{(}\mathcal{S}\,\mathcal{D}\,\mathcal{L}\,\mathcal{V}^{*}\big{)}\,\big{(}\mathcal{V}\,\mathcal{M}\,\mathcal{D}^{*}\big{)}=\mathcal{C}_{m}^{\beta}\,,$$

$$\mathcal{Z}={\Theta }_m^{\ast }\left(\begin{array}{cc}{\overline{b}}_m& 0\\ 0& {\gamma }_m\end{array}\right).$$

$$\mathcal{C}({\alpha }_0,\dots ,{\alpha }_{m-1},b_m)\oplus \mathcal{N}=\mathcal{C}({\alpha }_0,\dots ,{\alpha }_{n-1},b_n)\mathcal{S},$$

$$\mathcal{C}({\alpha }_0,\dots ,{\alpha }_{m-1},b_m)\oplus {\mathcal{N}}^T={\mathcal{S}}^T\mathcal{C}({\alpha }_0,\dots ,{\alpha }_{n-1},b_n),$$

Full text

1. Introduction and main result

The study of zeros of orthogonal polynomials on the real line (OPRL) can be regarded as an eigenvalue problem for Jacobi matrices111A symmetric tridiagonal matrix whose next-to-diagonal elements are positive (cf. [27, p. $36$]).. This allows us to go back to one of the most important single books in the nineteenth century, Cours d’analyse de l’École royale polytechnique (1821) by Cauchy to deduce, at least in the weak sense, the zero interlacing property of consecutive OPRL from the simplest form of the nowadays called Cauchy interlacing theorem. The search of more refined eigenvalue interlacing properties of Jacobi matrices was probably initiated by Cauchy himself in his work Sur l’ Équation à l’ Aide de Laquelle on Détermine les Inegalitées Séculaires des Mouvements des Planètes (1829) and later continued by several authors, including in the second half of the last century Wilkinson [45], Kahan [29], Golub [20], Hill and Parlett [26], and Bar-On [6]. In the same spirit, this work recovers one of the earliest approaches used to study zeros of paraorthogonal polynomials on the unit circle (POPUC), which is based on an eigenvalue problem for certain unitary matrices which bear many similarities with Jacobi matrices (cf. [31, 3, 23, 1, 25, 9, 16, 44, 7, 10, 11, 35, 36, 30, 39, 38, 40]).

Without wishing to delve into a historical discussion222The weakened orthogonality condition that POPUC satisfy appeared in [13, Equation $4.10$] as far as we can tell. While it is true that in Geronimus’ 1944 paper [18, Theorem IV] such polynomials were presented., as far as we know, the POPUC333In [13], Delsarte and Genin called to these polynomials (symmetric) predictor polynomials and its weakened orthogonality property quasiorthogonality. In [14], they refer to these polynomials as quasiorthogonal polynomials on the unit circle. This denomination could be also supported by the fact that in 1946 Geronimus regarding to these polynomials wrote that they “…play the same role here as the quasi-orthogonal polynomials of M. Riesz in the Hamburger problem.” (cf. [17, Remark I]). The denomination POPUC was coined in [28]. were introduced (in a somewhat hidden form) and successfully developed in a serie of papers by Delsarte and Genin at the end of the 1980’s [13, 15, 16], when they were working in signal processing. In [16], the authors focuses on the problem of computing the zeros of POPUC regarded as an eigenvalue problem for an unitary upper Hessenberg matrix with positive subdiagonal elements. Elegant and recent proofs of most interlacing properties of zeros of POPUC shared with OPRL are due to Simon [39] (cf. [40, Theorem $2.14.4$]) where the theory of rank one perturbations plays a central role. However, before such work (and references therein) the zeros of POPUC were studied by the Linear Algebra community based on ideas close to those of Simon but supported on more elementary facts. Further analysis of these ideas will allow us to easily extend the known results. Indeed, our main purpose is to prove and improve, in connection with the works of Delsarte and Genin on the subject, the known zero interlacing properties of POPUC, based on the development of the ideas discussed by Arbenz and Golub in [4, Section $6$]444Such ideas were pioneering employed in the present context by Bohnhorst in her Ph.D. thesis [7] defended in 1993 at the Bielefeld University under the supervision of Elsner..

Here and below, we mainly follow the notation of [35, 36, 40]. Denote by $\mathbb{D}$ the open unit disk and by $\mathbb{S}^{1}$ its boundary, i.e.,

[TABLE]

Let $(\alpha_{0},\dots,\alpha_{n-1},b_{n})$ with $\alpha_{j}\in\mathbb{D}$ ($j=0,1,\dots,n-1$) and $b_{n}\in\mathbb{S}^{1}$. Set

[TABLE]

Define the $(n+1)$-by-$(n+1)$ matrix

[TABLE]

where $\mathcal{L}$ and $\mathcal{M}$ are given explicitly by

[TABLE]

Any unitary $(n+1)$-by-$(n+1)$ upper Hessenberg matrix with positive subdiagonal elements is uniquely parameterized by $2n+1$ real numbers that compose the parameters of the array $(\alpha_{0},\dots,\alpha_{n-1},b_{n})$ [22] (cf. [24] and [2, Proposition $1$]). The resulting matrix after this process is referred as the Schur parametric form of the original matrix. The factorization (1), which is unitarily similar to the Schur parametric form of an upper Hessenberg matrix with positive subdiagonal elements, was presented by Bunse-Gerstner and Elsner [9] (cf. [21, Section $12.2.10$] and [7, Definition $3.3$ and Lemma $3.4$]). The explicit unitary pentadiagonal or double-staircase form of $\mathcal{C}$ (referred as Doppel-Treppen-Matrix in the original German source) was studied extensively by Bohnhorst [7], see Figure 1 for an $8$-by-$8$ example (cf. [7, Equation $3.9$] and [30, Figure $1.1$]). The matrix $\mathcal{C}$ becomes a very popular object in the Mathematical Physics and Orthogonal Polynomials communities after the work [11], specially after Simon’s monographs [35, 36] where it was called (improper) CMV matrix (cf. [38, 40]).

In order to make the notation more transparent, we write $\mathcal{C}(\alpha_{0},\dots,$ $\alpha_{n-1},b_{n})$ instead of $\mathcal{C}$. We choose the representation (1) instead of their unitary similar upper Hessenberg matrix for a technical reason related to the manner in which Lemma 2.1 below is presented. In the next definition and subsequently, $\mathcal{I}$ denotes the identity matrix, whose order is made explicit or may be inferred from the context.

Definition 1.1** (cf. [39, Proposition $3.2$]).**

Let $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})$ be the matrix given by (1), where $\alpha_{j}\in\mathbb{D}$ ($j=0,1,\dots,n-1$) and $b_{n}\in\mathbb{S}^{1}$. The (monic) polynomial $P_{n+1}$ defined by

[TABLE]

is the POPUC of degree $n+1$ associated with the array $(\alpha_{0},\dots,\alpha_{n-1},b_{n})$.

It is not difficult to see that the eigenvalues of $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})$ are simple. This fact was observed in $1944$ by Geronimus [18, Theorem IV] (cf. [17, Theorem III], [19, Theorem $9.1$] and [5, Theorem $7.2.2$]) using the connection between POPUC and OPUC. Note that if $b_{n}$ were in $\mathbb{D}$, then the corresponding characteristic polynomial would be an OPUC and their zeros would be in $\mathbb{D}$. A remarkable property of the eigenvectors of $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})$ is the fact that all their components are nonzero (cf. [35, Chapter $4$] and references therein). This property is clearly valid also for the corresponding unitarily similar Hessenberg matrix.

Definition 1.2**.**

Two finite subsets $\{\zeta_{1},\zeta_{2},\dots,\zeta_{m}\}$ and $\{\xi_{1},\xi_{2},\dots,\xi_{n}\}$ $(1\leq m\leq n)$ of $\mathbb{S}^{1}$ interlace (resp. strictly interlace) whenever there exist $n-m$ points $\zeta_{m+1},$ $\zeta_{m+2},\dots,\zeta_{n}\in\mathbb{S}^{1}$ such that any closed arc (resp. open arc) on $\mathbb{S}^{1}$ connecting two distinct elements of $\{\zeta_{1},\zeta_{2},\dots,\zeta_{n}\}$ contains at last one element of $\{\xi_{1},\xi_{2},\dots,\xi_{n}\}$, and vice versa.

We can now formulate our main result.

Theorem 1.1**.**

Let $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})$ be a matrix given by (1), where $\alpha_{j}\in\mathbb{D}$ ($j=0,1,\dots,n-1$) and $b_{n}\in\mathbb{S}^{1}$. The following sentences hold:

1. (i)

Let $\beta\in\mathbb{S}^{1}\setminus\{1\}$ and define $\mathcal{C}_{m}^{\beta}:=\mathcal{C}(\alpha_{0},\dots,\alpha_{m-1},\beta\alpha_{m},\dots,$ $\beta\alpha_{n-1},\beta b_{n})$ ($0\leq m<n$) and $\mathcal{C}_{n}^{\beta}:=\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},\beta b_{n})$. Then the eigenvalues of $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})$ and $\mathcal{C}_{m}^{\beta}$ strictly interlace on $\mathbb{S}^{1}$ for each $0\leq m\leq n$. 2. (ii)

For each $0\leq m<n$, let $b_{m}\in\mathbb{S}^{1}$, and let $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})$ be partitioned as

[TABLE]

$\mathcal{C}_{11}$* being the $(m+1)$-by-$(m+1)$ leading principal submatrix of $\mathcal{C}(\alpha_{0},\dots,$ $\alpha_{n-1},b_{n})$. For each $\zeta\in\mathbb{S}^{1}$, define recursively the numbers*555In [15] (cf. [16, Equation $2.6$]), Delsarte and Genin have shown that if the $b_{j}(\zeta)$’s (known as pseudo reflection coefficients) are given by (3), then the corresponding POPUC satisfy a three-term recurrence relation (cf. [12]). Bunse-Gerstner and He [10] have provided an illuminating discussion of the works of Delsarte and Genin on POPUC in matrix terms.**

[TABLE]

Set666$\sigma(\mathcal{A})$ denotes the spectrum of $\mathcal{A}$.**

[TABLE]

where ${C}:=\{\zeta\in\mathbb{S}^{1}:b_{m}(\zeta)=b_{m}\}$, $\mathcal{N}:=\mathcal{C}(\alpha_{m+1},\dots,\alpha_{n-1},b_{n})\,\mathcal{D}$ with $\mathcal{D}:=\operatorname{diag}\,(\gamma_{m},\mathcal{I})$, and

[TABLE]

Then $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})$ and $\mathcal{C}(\alpha_{0},\dots,\alpha_{m-1},b_{m})$ have at most $\min\{m+1,n-m\}$ common eigenvalues. More precisely, $\mathcal{C}(\alpha_{0},\dots,$ $\alpha_{n-1},b_{n})$ and $\mathcal{C}(\alpha_{0},\dots,\alpha_{m-1},b_{m})$ have ${A}$ as the set of common eigenvalues, ${A}$ being also given by the alternative expression

[TABLE]

Furthermore, the elements of the sets \sigma\big{(}\mathcal{C} $(\alpha_{0},\dots,$ \alpha_{n-1},b_{n})\big{)}\setminus{A} and \sigma\big{(}\mathcal{C}(\alpha_{0}, $\dots,\alpha_{m-1},$ b_{m})\big{)}\,\cup\,{B} strictly interlace on $\mathbb{S}^{1}$.

Let $P_{n+1}$ be the POPUC of degree $n+1$ associated to the array $(0,\dots,0,1)$. Since $\mathcal{C}(0,\dots,0,1)$ is a permutation matrix, it follows that $P_{n+1}(z)=z^{n+1}-1$. The sequence $(P_{j})_{j\geq 1}$ (all of whose zeros are roots of unity) produce, by geometric intuition, illuminating examples that fall within Theorem 1.1.

Example 1.1**.**

Let $P_{3}$ and $P_{6}$ be the POPUC associated to the arrays $(0,0,1)$ and $(0,0,0,0,0,1)$, respectively. In this situation,

[TABLE]

and, therefore,

[TABLE]

In the notation of Theorem 1.1 we have $n=5$, $m=2$, $b_{j}(\zeta)=\zeta^{5-j}$ $(0\leq j\leq 5)$, ${A}={C}=\sigma(\mathcal{C}(0,0,1))$, and ${B}=\emptyset$, where ${A}$ is obtained by using any of the expressions outlined in Theorem 1.1. Clearly, $\mathcal{C}(0,0,0,0,0,1)$ and $\mathcal{C}(0,0,1)$ have ${A}$ as the set of common eigenvalues and the elements of the sets $\sigma(\mathcal{C}(0,0,0,0,0,1))\setminus{A}$ and $\sigma(\mathcal{C}(0,0,1))$ strictly interlace on $\mathbb{S}^{1}$, in concordance with sentence (ii) of Theorem 1.1.

Regarding Theorem 1.1, as far as we know, sentence (i) for $m=0$ was proved by Ammar, Gragg and Reichel [1, Proposition $4.2$], although the particular case $\beta=-1$ is known since Geronimus’ work [18, Theorem IV] (cf. [17, Theorem III]). The sentence (i) for $m=n$ was proved by Bohnhorst in [7, Theorem $3.19$] (cf. [8, Theorem $3.5$]). In [39, Theorem $3.4$], Simon proved a weaker version of sentence (ii) that reads as follows: Strictly between any pair of eigenvalues of $\mathcal{C}(\alpha_{0},\dots,\alpha_{m-1},b_{m})$ there is at least one eigenvalue of $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})$.

Corollary 1.1**.**

Let $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})$ be a matrix given by (1), where $\alpha_{j}\in\mathbb{D}$ ($j=0,1,\dots,n-1$) and $b_{n}\in\mathbb{S}^{1}$. Let $b_{n-1}\in\mathbb{S}^{1}$ and define $\gamma_{n-1}$ as in (4) for $m=n-1$. Then $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},$ $b_{n})$ and $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-2},b_{n-1})$ have at most one common eigenvalue. More precisely, either $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})$ and $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-2},b_{n-1})$ have $\overline{b}_{n}\gamma_{n-1}$ as (only) common eigenvalue and the elements of \sigma\big{(}\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})\big{)}\setminus\{\overline{b}_{n}\gamma_{n-1}\} and \sigma\big{(}\mathcal{C}(\alpha_{0},\dots,\alpha_{n-2}, b_{n-1})\big{)} strictly interlace on $\mathbb{S}^{1}$, or else $\mathcal{C}(\alpha_{0},\dots,$ $\alpha_{n-1},b_{n})$ and $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-2},b_{n-1})$ have no common eigenvalues, and in such case $\overline{b}_{n}\gamma_{n-1}$ is not an eigenvalue of either, and the elements of the sets \sigma\big{(}\mathcal{C}(\alpha_{0}, $\dots,\alpha_{n-1},$ b_{n})\big{)} and \sigma\big{(}\mathcal{C}(\alpha_{0}, \dots,\alpha_{n-2},b_{n-1})\big{)}\cup\{\overline{b}_{n}\gamma_{n-1}\} strictly interlace on $\mathbb{S}^{1}$.

Proof.

Take $m=n-1$ in Theorem 1.1. Hence, (3) and (4) yield ${C}=\{\overline{b}_{n}\gamma_{n-1}\}$ which, in turn, is equal to $\sigma(\mathcal{N})$. Then either ${A}={C}$ and ${B}=\emptyset$ if \overline{b}_{n}\gamma_{n-1}\in\sigma\big{(}\mathcal{C}(\alpha_{0},\dots,\alpha_{n-2},b_{n-1})\big{)}, or else ${A}=\emptyset$ and ${B}={C}$ otherwise. The result follows immediately from sentence (ii) of Theorem 1.1. ∎

Corollary 1.1 was proved by Bohnhorst [7, p. $48$] (cf. [8, p. $819$]) and rediscovered by Simon [39, Theorem $1.4$]. It is worth noting that in view of Corollary 1.1 and besides the several and well-known practical consequences, POPUC answered the following open-ended question proposed by Turán as far as 1974 [42, Problem LXVI, p. 60]: “It is known that the zeros of the $n$th orthogonal polynomial (with respect to a Lebesgue-integral function on an interval) separate the zeros of the $(n+1)$th polynomial. What corresponds to this fact on the unit circle?”777We quote the English translation provided by Szüsz [43, Problem LXVI]..

2. Proof of Theorem 1.1

2.1. Some preliminary lemmas

Theorem 1.1 will be proved through the following sequence of lemmas.

Lemma 2.1**.**

Let $\mathcal{U}$ and $\mathcal{S}$ be unitary matrices of the same order and suppose that $\operatorname{rank}\ (\mathcal{I}-\mathcal{S})=1\,.$ Then $\mathcal{U}$ and $\mathcal{US}$ have interlacing eigenvalues on $\mathbb{S}^{1}$. Moreover, assume that $\mathcal{U}\mathcal{S}$ admits a decomposition $\mathcal{U}\mathcal{S}=\mathcal{U}_{1}\,\oplus\,\mathcal{U}_{2}\,,$ and let $\mathcal{U}$ be partitioned as

[TABLE]

$\mathcal{U}_{11}$* and $\mathcal{U}_{1}$ being of the same order. Set $U_{1}:=\sigma(\mathcal{U}_{1})$, $U_{2}:=\sigma(\mathcal{U}_{2})$, and $U:=\sigma(\mathcal{U})$. Assume further that the eigenvalues of $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ are simple and $\sigma(\mathcal{U}_{11})\,\cap\,U_{1}=\sigma(\mathcal{U}_{22})\,\cap\,U_{2}=\emptyset$. Then the elements of the sets U\setminus\big{(}U_{1}\,\cap\,U_{2}\big{)} and U_{1}\,\cup\,\big{(}U_{2}\backslash(U_{1}\,\cap\,U_{2})\big{)} strictly interlace on $\mathbb{S}^{1}$.*

Proof.

The first sentence of the lemma is the simplest form of a result due to Arbenz and Golub [4, Section $6$] (cf. [7, Theorem $2.9$] and [8, Theorem $2.7$])888It can be deduced directly using [32, p. $222$] and [27, Corollary $4.3.9$].. In order to deduce the second one, we first claim that

[TABLE]

Indeed, since $\operatorname{rank}\,(\mathcal{US}-\mathcal{U})=1$, there exist nonzero vectors $u,v\in\mathbb{C}^{n}$ ($n$ being the common order of $\mathcal{U}$ and $\mathcal{S}$) such that $\mathcal{US}=\mathcal{U}+uv^{T}$. Using the formula for the determinant of a rank one perturbation (cf. [34, Proposition $3.21$]), we may write for each $\zeta\in\mathbb{C}$ 999$\chi_{{}_{\mathcal{A}}}$ denotes the characteristic polynomial of $\mathcal{A}$.

[TABLE]

Let $\mathcal{US}=\mathcal{Z}\Lambda\mathcal{Z}^{*}$ be the spectral decomposition of $\mathcal{US}$ in which $\Lambda=\operatorname{diag}\,(\lambda_{1},\dots,\lambda_{n})$ and $\mathcal{Z}=(z_{1}\dots z_{n})$. Thompson-McEnteggert’s formula for the adjugate [41] (cf. [33, Theorem $2.1$]) gives

[TABLE]

where the prime denotes the derivative. Combining (6) with (7) yields101010The eigenvalue interlacing already stated implies $U_{1}\,\cap\,U_{2}\subseteq U$, and so $U_{1}\,\cap\,U_{2}\subseteq U_{1}\,\cap\,U$ and $U_{1}\,\cap\,U_{2}\subseteq U_{2}\,\cap\,U$.

[TABLE]

We next claim that if $\lambda_{j}\in(U_{1}-U_{2})\,\cup\,(U_{2}-U_{1})$111111Given a set $E$ and $F,G\subseteq E$, we define $F-G:=F\cap(E\backslash G)$; if $G\subseteq F$, then $F-G=F\backslash G$., then $z_{j}^{*}uv^{T}z_{j}\not=0$. We only prove that $\lambda_{j}\in U_{1}-U_{2}$ implies $v^{T}z_{j}\not=0$. (To prove that $\lambda_{j}\in U_{1}-U_{2}$ implies $z_{j}^{*}u\not=0$, we proceed similarly, as well as for proving that $\lambda_{j}\in U_{2}-U_{1}$ implies $z_{j}^{*}uv^{T}z_{j}\not=0$.) Indeed, suppose that $\lambda_{j}\in U_{1}-U_{2}$ and $v^{T}z_{j}=0$. Since there is a normalized eigenvector $v_{j}$ of $\mathcal{U}_{1}$ associated with $\lambda_{j}$ such that $z_{j}=(v_{j}^{T},0,\dots,0)^{T}$, we deduce

[TABLE]

hence $\lambda_{j}\in\sigma(\mathcal{U}_{11})\,\cap\,U_{1}$, contrary to $\sigma(\mathcal{U}_{11})\,\cap\,U_{1}=\emptyset$. Consequently, (5) follows from (8). Finally, it follows from (5) that the sets U\setminus\big{(}U_{1}\,\cap\,U_{2}\big{)} and U_{1}\,\cup\,\big{(}U_{2}\backslash(U_{1}\,\cap\,U_{2})\big{)} have no common elements, thus the second sentence of the lemma follows from the first one. ∎

Lemma 2.2**.**

Let $\mathcal{U}$ be a unitary matrix and for a fixed $k$ let $\mathcal{S}$ be the diagonal matrix obtained from the identity matrix by replacing the $(k,k)$ entry with a number on $\mathbb{S}^{1}\setminus\{1\}$. Assume that $\mathcal{U}$ and $\mathcal{S}$ have the same order. Assume further that the eigenvalues of $\mathcal{U}$ are simple and all its eigenvectors have a nonzero component at the position $k$. Then $\mathcal{U}$ and $\mathcal{U}\mathcal{S}$ have strictly interlacing eigenvalues on $\mathbb{S}^{1}$.

Proof.

Without loss of generality we can assume that $k=1$, and so $\mathcal{S}=\operatorname{diag}\,(\beta,\mathcal{I})$ with $\beta\in\mathbb{S}^{1}\setminus\{1\}$. Let $\mathcal{U}=\mathcal{Z}\Lambda\mathcal{Z}^{*}$ be the spectral decomposition of $\mathcal{U}$ in which $\Lambda=\operatorname{diag}\,(\lambda_{1},\dots,\lambda_{n})$ and $\mathcal{Z}=(z_{1}\dots z_{n})$. Arguing as in the proof of Lemma 2.1 we have

[TABLE]

Let $a_{j}\neq 0$ be the first component of the vector $z_{j}$. Then

[TABLE]

Thus the result follows from (9) and the first sentence of Lemma 2.1. ∎

Lemma 2.3**.**

Let $\alpha_{j}\in\mathbb{D}$ ($j=0,1,\dots,n-1$) and $b_{n}\in\mathbb{S}^{1}$. The following sentences hold:

1. (i)

Let $\mathcal{S}$ be a diagonal matrix obtained from the $(n+1)$-by-$(n+1)$* identity matrix by replacing one of its diagonal entries with a number on $\mathbb{S}^{1}\setminus\{1\}$. Then $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})$ and $\mathcal{C}(\alpha_{0},\dots,$ $\alpha_{n-1},b_{n})\mathcal{S}$ have strictly interlacing eigenvalues on $\mathbb{S}^{1}$.* 2. (ii)

Let $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})$ be partitioned as in (2). Then, for each $0\leq m<n$, $\mathcal{C}_{22}$ has no eigenvalues on $\mathbb{S}^{1}$.

Proof.

(i) The result follows directly from Lemma 2.2 and the fact that all the components of the eigenvectors of $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})$ are nonzero.

(ii) Assume that $m$ is even. Note that $\mathcal{C}_{22}$ is the $(n-m)$-by-$(n-m)$ trailing principal submatrix of each of the matrices $\mathcal{C}(\alpha_{m},\dots,\alpha_{n-1},b_{n})$ and $\mathcal{C}(\alpha_{m},\dots,\alpha_{n-1},b_{n})\,\mathcal{S}$, where $\mathcal{S}:=\operatorname{diag}\,(\beta,\mathcal{I})$. Suppose the assertion (ii) is false. Since $\mathcal{C}(\alpha_{m},\dots,$ $\alpha_{n-1},b_{n})$ and $\mathcal{C}(\alpha_{m},\dots,\alpha_{n-1},b_{n})\,\mathcal{S}$ are unitary matrices, these matrices share all the eigenvalues of $\mathcal{C}_{22}$ on $\mathbb{S}^{1}$, which contradicts sentence (i). If $m$ is odd, we argue in the same way noting that $\mathcal{C}_{22}^{T}$ is the $(n-m)$-by-$(n-m)$ trailing principal submatrix of each of the matrices $\mathcal{C}(\alpha_{m},\dots,\alpha_{n-1},b_{n})$ and $\mathcal{S}\,\mathcal{C}(\alpha_{m},\dots,\alpha_{n-1},b_{n})$. ∎

Lemma 2.4**.**

Let $\alpha_{j}\in\mathbb{D}$ ($j=0,1,\dots,n-1$) and $b_{n}\in\mathbb{S}^{1}$. Let $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})$ be partitioned as in (2), where $0\leq m<n$. Let $b_{m}\in\mathbb{S}^{1}$ and define $b_{m}(\zeta)$ via (3) for each $\zeta\in\mathbb{S}^{1}$. Then $\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},b_{n})$ and $\mathcal{C}(\alpha_{0},\dots,\alpha_{m-1},b_{m})$ have at most $\min\{m+1,n-m\}$ common eigenvalues, which consist of the set of different solutions $\zeta$ of the equation $b_{m}(\zeta)=b_{m}$ on $\sigma(\mathcal{C}(\alpha_{0},\dots,\alpha_{m-1},b_{m}))$.

Proof.

We begin by noting that

[TABLE]

for each $\zeta\in\mathbb{S}^{1}$. Indeed, by sentence (ii) of Lemma 2.3, $\zeta\mathcal{I}-\mathcal{C}_{22}$ is nonsingular, hence (10) follows from the equality (cf. [7, Equation $3.41$])

[TABLE]

after taking into account that the Schur complement of $\zeta\mathcal{I}-\mathcal{C}_{22}$ in $\zeta\mathcal{I}-\mathcal{C}(\alpha_{0},\dots,\alpha_{n-1},$ $b_{n})$ is \zeta\mathcal{I}-\big{(}\mathcal{C}_{11}-\mathcal{C}_{12}(\mathcal{C}_{22}-\zeta\mathcal{I})^{-1}\mathcal{C}_{21}\big{)}. The result follows from (10) and the fact that for $\nu,\zeta\in\mathbb{S}^{1}$, with $\nu\neq\zeta$, $\mathcal{C}(\alpha_{0},\dots,\alpha_{m-1},\nu)$ and $\mathcal{C}(\alpha_{0},\dots,\alpha_{m-1},\zeta)$ have no common eigenvalues (see e.g. [40, Theorem $2.14.4$]; alternatively, apply sentence (i) of Lemma 2.3). ∎

2.2. Proof of Theorem 1.1

(i) Let $\mathcal{S}:=\operatorname{diag}\,(\mathcal{I}_{m},\overline{\beta},\mathcal{I}_{n-m})$, $\mathcal{D}:=\operatorname{diag}\,(\mathcal{I}_{m},$ $\mathcal{J}_{n-m+1}^{\beta})$, and $\mathcal{V}:=\operatorname{diag}\,(\mathcal{I}_{m+1},$ $\mathcal{J}_{n-m}^{\beta})$, where $\mathcal{J}_{k}^{\beta}:=\operatorname{diag}\,(\beta,1,\beta,1,\dots)$ is a $k$-by-$k$ diagonal matrix. Note that

[TABLE]

Using (11) it is easily seen that 121212A different proof is given in [37, Theorem $5.1$].

[TABLE]

when $m$ is even. Similarly, the transpose of (11) leads to

[TABLE]

when $m$ is odd. The result follows from sentence (i) of Lemma 2.3.

(ii) Define the block diagonal matrix $S:=\operatorname{diag}\,(\mathcal{I}_{m},\mathcal{Z},\mathcal{I}_{n-m-1})$, where

[TABLE]

Hence

[TABLE]

when $m$ is odd, and

[TABLE]

when $m$ is even. Note that $\mathcal{N}$ has simple eigenvalues (on $\mathbb{S}^{1}$) by sentence (i) of Lemma 2.3. The result follows from Lemma 2.1, sentence (ii) of Lemma 2.3, and Lemma 2.4.

Acknowledgment

The authors thank the Bielefeld University Library for kindly sending them a hard copy of Birgit Bohnhorst’s Ph.D. Thesis. KC is supported by the Portuguese Government through the Fundação para a Ciência e a Tecnologia (FCT) under the grant SFRH/BPD/101139/2014. This work is partially supported by the Centre for Mathematics of the University of Coimbra – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MCTES and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020. JP is also partially supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain under the project MTM2015–65888–C4–4–P.