# Beurling-Lax type theorems and Cuntz relations

Daniel Alpay  (DA) Faculty of Mathematics, Physics, and Computation
Schmid College of Science and Technology
Chapman University
One University Drive Orange, California 92866
USA<EMAIL_ADDRESS>,  Fabrizio Colombo  (FC) Politecnico di Milano
Dipartimento di Matematica
Via E. Bonardi, 9
20133 Milano
Italy<EMAIL_ADDRESS>,  Irene Sabadini  (IS) Politecnico di
Milano
Dipartimento di Matematica
Via E. Bonardi, 9
20133 Milano
Italy<EMAIL_ADDRESS>and  Baruch Schneider  (BS) University of
Ostrava
Department of Mathematics
30.dubna 22, 70200 Ostrava
Czech Republic<EMAIL_ADDRESS>
###### Abstract.

We prove various Beurling-Lax type theorems, when the classical backward-shift
operator is replaced by a general resolvent operator associated with a
rational function. We also study connections to the Cuntz relations. An
important tool is a new representation result for analytic functions, in terms
of composition and multiplication operators associated with a given rational
function. Applications to the theory of de Branges-Rovnyak spaces, also in the
indefinite metric setting, are given.

AMS Classification: 47B32, 30C10

Keywords: de Branges-Rovnyak spaces, Cuntz relations, rational functions,
Beurling-Lax theorem, backward-shift operator, structure theorems today

###### Contents

  1. 1 Introduction
  2. 2 Preliminary results
  3. 3 The associated symmetric matrix
  4. 4 A representation theorem for analytic function
  5. 5 Cuntz relations
  6. 6 $R_{\alpha}^{(r)}$-invariant subspaces
  7. 7 Operator models
  8. 8 $\mathfrak{P}(\Theta)$ spaces
  9. 9 $\mathfrak{P}(S)$ spaces
  10. 10 $\mathfrak{P}(E_{+},E_{-})$ spaces
  11. 11 $\mathfrak{L}({\mathsf{N}})$ spaces

##  1\. Introduction

In this paper we discuss decompositions of spaces associated with positive
definite kernels and with kernels having a finite number of negative squares,
based on a new representation theorem for analytic functions. To set the
framework and motivate our results, let us consider a $\mathbb{C}^{p\times
p}$-valued function $K(z,w)$ positive definite on a set
$\Omega\subset\mathbb{C}$; let $r$ be a map from a set
$\Omega_{1}\subset\mathbb{C}$ into $\Omega$, and let $A(z)$ be a
$\mathbb{C}^{1\times p}$-valued function defined on $\Omega_{1}$. The function
$A(z)K(r(z),r(w))A(w)^{*}$ is positive definite on $\Omega_{1}$; elements of
the reproducing kernel Hilbert space with reproducing kernel
$A(z)K(r(z),r(w))A(w)^{*}$ are of the form

(1.1) $f(z)=A(z)F(r(z)),$

where $F$ varies in the reproducing kernel Hilbert space with reproducing
kernel $K(z,w)$. Depending on various assumptions, one can insure that the
above decomposition is unique, and one can then relate the norms of $f$ and
$F$. In the present paper we assume that $r$ is a rational function and that
the entries of $A(z)$ span the backward-shift invariant space (also know as
state space) $\mathfrak{L}(r)$ generated by $r$ (see Definition 1.2 for the
latter). We focus in particular on de Branges and de Branges-Rovnyak spaces
(see [25, 26, 27, 33, 34, 35]), in view of their important role in operator
theory, linear system theory and related topics; see e.g. [2, 32]. The
properties of these spaces are related to properties of the classical
backward-shift operators

(1.2)
$(R_{\alpha}F)(z)=\begin{cases}\,\,\dfrac{F(z)-F(\alpha)}{z-\alpha},\quad
z\not=\alpha,\\\ \,\,F^{\prime}(\alpha),\hskip 48.93877ptz=\alpha,\end{cases}$

where $F$ is assumed analytic in a neighborhood of $\alpha$. Moreover, various
structure theorems characterizing de Branges-Rovnyak spaces in terms of the
backward-shift operators can be seen as Beurling-Lax type theorems. For
instance, given a signature matrix $J\in\mathbb{C}^{s\times s}$ (i.e. $J$
satisfies $J=J^{*}=J^{-1}$) and given a reproducing kernel Hilbert space
$\mathfrak{H}$ of $\mathbb{C}^{s}$-valued functions analytic in an open subset
of the complex plane, symmetric with respect to the real line, the equality

(1.3)
$\left[R_{\alpha}F,G\right]-\left[F,R_{\beta}G\right]-(\alpha-\overline{\beta})\left[R_{\alpha}F,R_{\beta}G\right]=iG(\beta)^{*}JF(\alpha)$

holds for all $\alpha,\beta\in\Omega$ and all $F,G\in\mathfrak{H}$ if and only
if the reproducing kernel of $\mathfrak{H}$ is of the form

(1.4) $K(z,w)=\frac{J-\Theta(z)J\Theta(w)^{*}}{-i(z-\overline{w})},$

see [21, 24, 46]. The links with operator models stem from the following: When
the operators $R_{\alpha}$ are bounded and with zero kernel, a result of Stone
(see [47, Theorem 4.10, p. 137]) asserts that there is a closed linear
operator $\mathsf{A}$ such that

(1.5) $((\mathsf{A}-\alpha I)^{-1}f)(z)=zf(z)+c_{f},\quad\alpha\in\Omega,$

where $c_{f}\in\mathbb{C}$. Properties of $\mathsf{A}$ and of the space
$\mathfrak{H}$ are closely related.

de Branges-Rovnyak spaces come in all shapes and sizes, depending on the
application in mind, and the domain of analyticity where they are defined. We
here focus on three types of spaces, usually denoted by the symbols
$\mathfrak{P}(\Theta)$, $\mathfrak{P}(E_{+},E_{-})$ and
$\mathfrak{L}(\mathsf{N})$, associated respectively with closed to self-
adjoint (or close to unitary), Hermitian (or isometric) and self-adjoint (or
unitary) operators.

Although different in aim and approach, the present paper has connections with
the works of Kailah and Lev-Ari [42], Nudelman [44], and the approach
developed in the papers [6, 13, 14]. There, one is given two functions
analytic in a connected open set $\Omega$, such that the sets

$\begin{split}\Omega_{+}&=\left\\{z\in\Omega\,;\,|b(z)|<|a(z)|\right\\}\\\
\Omega_{-}&=\left\\{z\in\Omega\,;\,|b(z)|>|a(z)|\right\\}\end{split}$

are not empty. Then the set

$\Omega_{0}=\left\\{z\in\Omega\,;\,|b(z)|=|a(z)|\right\\}$

is also not empty (see e.g. [4, Exercise 6.1.9, p. 295]), and the function

$k(z,w)=\frac{1}{a(z)\overline{a(w)}-b(z)\overline{b(w)}}$

is positive definite in $\Omega_{+}$. The reproducing kernels studied are then
of the form

(1.6)
$\frac{J-\Theta(z)J\Theta(w)^{*}}{a(z)\overline{a(w)}-b(z)\overline{b(w)}}$

when $z,w$ vary in some open subset of $\Omega_{+}$, where $J$ is a signature
matrix. When such a kernel has a finite number of negative squares, the
function $\Theta(z)$ in (1.6) can be written as $M(\sigma(z))$, where
$\sigma(z)=\frac{b(z)}{a(z)}$, and where $M(z)$ is analytic in some open
subset of the open unit disk. Our rational function $r$ (or $1-r/1+r$) plays
the role of $\sigma$.

de Branges-Rovnyak spaces have been furthermore generalized to numerous
settings, ranging from the polydisk (and the theory of Schur-Agler classes) to
non-commutative settings and real compact Riemann surfaces setting, and here
we take a different avenue and study the counterparts of de Branges-Rovnyak
spaces when the classical resolvent operator (see (1.2) below) is replaced by
a general resolvent operator associated with a rational function $r$.
Motivated by the Riemann surface case (see [19, §4]), a generalization of
(1.2) has been introduced in [17]. More precisely (in [17] only the polynomial
case was considered) we will assume, unless otherwise stated, that $r$
satisfies:

######  Hypothesis 1.1.

We will assume that the rational function $r(z)=\frac{p(z)}{q(z)}$, where $p$
and $q$ are coprime polynomials satisfying ${\rm deg}\,p\,\geq\,{\rm deg}\,q$.
When this hypothesis is not in force we can replace $r$ by $1/r$.

We set $N={\rm deg}\,p$, and define

(1.7) $\Omega(r)=\left\\{\alpha\in\mathbb{C}\,\mbox{\text{\,such that the
equation $r(z)=\alpha$ has $N$ pairwise different roots}}\right\\}.$

We do not write explicitly the dependence of the roots, say
$w_{1}(\alpha),\ldots,w_{N}(\alpha)$, on $r$ and $\alpha$, but when it makes
the reading easier (such as in the statement and proof of Lemma 3.1). For
$\alpha\in\Omega(r)$ we set

(1.8)
$(R^{(r)}_{\alpha}f)(z)=\begin{cases}\,\,\dfrac{f(z)}{r(z)-\alpha}-\sum_{n=1}^{N}\dfrac{f(w_{n})}{r^{\prime}(w_{n})(z-w_{n})},\quad
r(z)\not=\alpha,\\\ \\\
\,\,\sum_{n=1}^{N}\dfrac{f^{\prime}(w_{n})}{r^{\prime}(w_{n})},\hskip
110.96556ptz\in
r^{-1}\left\\{\alpha\right\\}=\left\\{w_{1},\ldots,w_{N}\right\\},\end{cases}$

where $f$ is assumed analytic in a neighborhood of the set
$r^{-1}\left\\{\alpha\right\\}$ (the use of a lower case $f$ for (1.8) and of
an upper case $F$ for (1.2) is on purpose, and is motivated by expressions
such as (1.1)). We note that the first formula in (1.8) can be rewritten as
(for $\alpha\in\mathbb{C}$ such that $r(\infty)\not=\alpha$ and $z\not\in
r^{-1}\left\\{\alpha\right\\}$):

(1.9)
$\begin{split}(R^{(r)}_{\alpha}f)(z)&=\frac{f(z)}{r(\infty)-\alpha}+\sum_{n=1}^{N}\frac{1}{r^{\prime}(w_{n})}\dfrac{f(z)-f(w_{n})}{z-w_{n}}\\\
&=\frac{f(z)}{r(\infty)-\alpha}+\sum_{n=1}^{N}\frac{1}{r^{\prime}(w_{n})}(R_{w_{n}}f)(z).\end{split}$

The operators (1.8) satisfy the resolvent identity

(1.10)
$R_{\alpha}^{(r)}-R_{\beta}^{(r)}=(\alpha-\beta)R_{\alpha}^{(r)}R_{\beta}^{(r)},\quad\alpha,\beta\in\Omega(r),$

see [17]. Another slightly different proof can be obtained using (1.9) since
the classical resolvent operators satisfy the resolvent identity; see
Corollary 2.2 below. When these operators have a trivial kernel, the
underlying operator models are now of the form (7.5) below, that is

$(Tf)(z)=r(z)f(z)+h_{f}(z)$

where $h_{f}(z)$ is a rational function of $z$, linear in $f$.

In the present paper we study the counterpart of the de Branges and de Branges
-Rovnyak spaces when the classical backward-shift operators are replaced by
the operators (1.8). For instance (1.3) will now be replaced by

(1.11)
$\left[R^{(r)}_{\alpha}f,g\right]-\left[f,R_{\beta}^{(r)}g\right]-(\alpha-\overline{\beta})\left[R^{(r)}_{\alpha}f,R^{(r)}_{\beta}g\right]=i(\alpha-\overline{\beta})\sum_{i,j=1}^{N}\frac{g(v_{j})^{*}}{\overline{r^{\prime}(v_{j})}}J\frac{f(u_{i})}{r^{\prime}(u_{i})}\frac{1}{u_{i}-\overline{v_{j}}}$

where the $u_{i}$ (resp. the $v_{j}$) are the pre-images of $\alpha$ (resp. of
$\beta$) under $r$. We note that the right hand side of (1.11) is adapted from
the Riemann surface case; see [19, Theorem 5.1, p. 316].

To describe our results we need some more notation and definitions.

######  Definition 1.2.

The linear span of the functions $z\mapsto\frac{r(z)-r(a)}{z-a}$ where $a$
runs through the points of analyticity of $r$, will be called the state space
associated with $r$ and will be denoted by $\mathfrak{L}(r)$.

The space $\mathfrak{L}(r)$ is finite dimensional, as we recall in Lemma 2.6,
and we denote by $Z_{r}(z)$ a $1\times N$ matrix with entries a basis of
$\mathfrak{L}(r)$:

(1.12) $Z_{r}(z)=\begin{pmatrix}e_{1}(z)&\cdots&e_{N}(z)\end{pmatrix}.$

######  Example 1.3.

For $r(z)=z+\frac{1}{z}$, the space $\mathfrak{L}(r)$ is spanned by the
functions $1$ and $1/z$ since

$\frac{r(z)-r(a)}{z-a}=1-\frac{1}{za}$

for this $r$.

######  Remark 1.4.

When $r(z)=z$, the corresponding $Z_{r}(z)$ can be taken to be $1$, and the
operator reduces to the classical backward shift operators (1.2).

We will be interested in functions of the form

(1.13) $f(z)=(Z_{r}(z)\otimes I_{s})F(r(z)),$

where $F$ is analytic in a neighborhood of the origin, and a characterization
of these functions is given in Theorem 4.1.

Condition (1.11) is now equivalent (see Theorem 4.13 for a precise statement)
to the reproducing kernel of the space to be of the form

(1.14) $(Z_{r}(z)\otimes
I_{s})Y^{-1}\frac{J_{0}-\Theta(r(z))J_{0}\Theta(r(w))^{*}}{-i(r(z)-\overline{r(w)})}Y^{-*}(Z_{r}(w)\otimes
I_{s})^{*},$

where $\Theta$ is a $\mathbb{C}^{rs\times rs}$-valued function analytic in a
neighborhood of the origin, $Y\in\mathbb{C}^{rs\times rs}$ is invertible,
$J_{0}\in\mathbb{C}^{rs\times rs}$ is a signature matrix, and the latter two
depend only on $Z_{r}$ and $J$.

Another question which we are considering here is related to the Cuntz
relations. With the above notation, we look for pairs of positive definite
kernels $(K,K_{0})$ such that, with $K_{0}(r)(z,w)=K_{0}(r(z),r(w))$, we have
the orthogonal decomposition (the functions $e_{j}$ have been defined by
(1.12))

(1.15) $\mathfrak{H}(K)=\oplus e_{j}\mathfrak{H}(K_{0}(r)).$

The weighted multiplication operators
$T_{j}:\,\,\mathfrak{H}(K_{0})\longrightarrow\mathfrak{H}(K)$, $j=1,\ldots,N$,
defined by

(1.16) $(T_{j}g)(z)=e_{j}(z)g(r(z)),\quad j=1,\ldots,N$

satisfy then the Cuntz relations.

An important tool in our results is a representation theorem for analytic
functions, characterizing functions of the form $Z_{r}(z)F(r(z))$ (the case
where $r(z)$ is a polynomial has been considered in [17]).

The paper contains ten sections besides the introduction. Sections 2-7 form
the first part of the paper. After the preliminary Section 2, in which we
discuss some aspects of realizations of rational functions, the main general
results are presented in Sections 3-7. Specifically, in Section 3 we study,
given a rational function, an associated symmetric matrix also proving a
technical result which may be of independent interest. Section 4 contains a
main result, namely a representation theorem, see Theorem 4.1, which
generalizes the polynomial case and some particular cases of rational
functions already studied in the literature. Section 5 discusses how our
results are connected with Cuntz relations. Section 6 studies subspaces
invariant under the action of suitable generalized backward shift operators,
while Section 7 moves to related operator models. Sections 8-11 are devoted to
the counterpart of de Branges-Rovnyak spaces in the present setting. We study
the Pontryagin spaces with assigned reproducing kernels in the line and in the
circle case, we present two structure theorems, and we consider also the case
of negative squares and generalized Nevanlinna functions.

##  2\. Preliminary results

In this section, which is of a preliminary nature, we discuss various aspects
of realization theory of rational functions. We mention in particular
Corollary 2.13, which may be of independent interest, and Corollary 2.15,
which allows us to make the connection with the classical case where $r(z)=z$.
The first result follows from a partial fraction expansion; we write out the
details since (2.1) plays a central role in the sequel. We note that it may be
that $r(\infty)\in{\rm ran}\,r$, for example $r(z)=1+\dfrac{z-1}{z^{2}+1}$. We
have $r(1)=r(\infty)=1$.

######  Proposition 2.1.

Let $r$ be a rational function of degree $N$ and let $\alpha\in\Omega(r)$. We
have:

(2.1)
$\frac{1}{r(z)-\alpha}=\frac{1}{r(\infty)-\alpha}+\sum_{n=1}^{N}\frac{1}{r^{\prime}(w_{n})(z-w_{n})},$

which reduces to

(2.2)
$\frac{1}{r(z)-\alpha}=\sum_{n=1}^{N}\frac{1}{r^{\prime}(w_{n})(z-w_{n})},$

when $r(\infty)=\infty$.

###### Proof.

The zeros $w_{1},\ldots,w_{N}$ cannot be zeros of $q$, even if we interpret
$r(w_{n})=\alpha$ as $p(w_{n})=\alpha q(w_{n})$. If $q(w_{n})=0$ then
$p(w_{n})=0$, but $p$ and $q$ are assumed prime together. From
$\alpha=\frac{p(w_{n})}{q(w_{n})}$ we get

(2.3) $\begin{split}\frac{p^{\prime}(w_{n})-\alpha
q^{\prime}(w_{n})}{q(w_{n})}&=\frac{p^{\prime}(w_{n})-\dfrac{p(w_{n})}{q(w_{n})}q^{\prime}(w_{n})}{q(w_{n})}\\\
&=\frac{p^{\prime}(w_{n})q(w_{n})-p(w_{n})q^{\prime}(w_{n})}{q(w_{n})^{2}}\\\
&=r^{\prime}(w_{n}).\end{split}$

Partial fraction expansion gives

$\begin{split}\frac{1}{r(z)-\alpha}&=\frac{1}{r(\infty)-\alpha}+\frac{q(z)}{p(z)-q(z)\alpha}\\\
&=\frac{1}{r(\infty)-\alpha}+\sum_{n=1}^{N}\frac{q(w_{n})}{p^{\prime}(w_{n})-\alpha
q^{\prime}(w_{n})}\frac{1}{z-w_{n}}\\\
&=\frac{1}{r(\infty)-\alpha}+\sum_{n=1}^{N}\frac{q(w_{n})}{p^{\prime}(w_{n})-\frac{p(w_{n})}{q(w_{n})}q^{\prime}(w_{n})}\frac{1}{z-w_{n}}\\\
&=\frac{1}{r(\infty)-\alpha}+\sum_{n=1}^{N}\frac{1}{r^{\prime}(w_{n})(z-w_{n})},\end{split}$

where we have used (2.3). ∎

We note that if $r(\infty)=0$ then $0\not\in\Omega(r)$.

From the previous result we deduce:

######  Corollary 2.2.

Let $r$ be a rational function of degree $N$ and $\alpha,\beta\in\Omega(r)$.
Then (1.10) follows from (1.9).

###### Proof.

Let $w_{1},\ldots,,w_{N}$ be the roots (which are pairwise different since
$\alpha\in\Omega(r)$) of the equation $r(z)=\alpha$, and let
$v_{1},\ldots,v_{N}$ be the roots (pairwise different since
$\beta\in\Omega(r)$) of $r(v_{m})=\beta$. With

$c=\frac{1}{r(\infty)-\alpha},\quad d=\frac{1}{r(\infty)-\beta},\quad
c_{n}=\frac{1}{r^{\prime}(w_{n})},\quad
d_{n}=\frac{1}{r^{\prime}(v_{n})},\,\,n=1,\ldots,N,$

we can write (since the operator (1.2) satisfy the resolvent identity, and
using (2.1)):

$\begin{split}R^{(r)}_{\alpha}R^{(r)}_{\beta}&=\left(c+\sum_{n=1}^{N}c_{n}R_{w_{n}}\right)\left(d+\sum_{m=1}^{N}d_{m}R_{v_{m}}\right)\\\
&=cR_{\beta}^{(r)}+d\sum_{n=1}^{N}c_{n}dR_{w_{n}}+\sum_{n,m=1}^{N}c_{n}d_{m}R_{w_{n}}R_{v_{m}}\\\
&=cR_{\beta}^{(r)}+d(R^{(r)}_{\alpha}-c)+\sum_{n,m=1}^{N}c_{n}d_{m}\frac{R_{w_{n}}-R_{v_{m}}}{w_{n}-v_{m}}\\\
&=cR_{\beta}^{(r)}+d(R^{(r)}_{\alpha}-c)+\sum_{n=1}^{N}c_{n}R_{w_{n}}\left(\sum_{m=1}^{N}d_{m}\frac{1}{w_{n}-v_{m}}\right)-\\\
&\hskip
14.22636pt-\sum_{m=1}^{N}d_{m}R_{v_{m}}\left(\sum_{n=1}^{N}c_{n}\frac{1}{w_{n}-v_{m}}\right)\\\
&=cR_{\beta}^{(r)}+d(R^{(r)}_{\alpha}-c)+(R_{\alpha}^{(r)}-c)\left(\frac{1}{\alpha-\beta}-d\right)-\\\
&\hskip
14.22636pt-(R^{(r)}_{\beta}-d)\left(-\frac{1}{\beta-\alpha}+c\right)\\\
&=\frac{R_{\alpha}^{(r)}-R_{\beta}^{(r)}}{\alpha-\beta}+cd+\frac{d-c}{\alpha-\beta}\\\
&=\frac{R_{\alpha}^{(r)}-R_{\beta}^{(r)}}{\alpha-\beta}\end{split}$

since, by direct computation, we have $cd+\dfrac{d-c}{\alpha-\beta}=0.$ ∎

For the next lemma, we use realization theory for rational functions; see e.g.
[22, 23, 36]. We follow the analysis in [14]. We will content ourselves with
recalling the following facts. First, a $\mathbb{C}^{r\times s}$-valued
function $M$ is rational (i.e. has rational components) if and only if the
linear span of the functions $R_{a}Mc$ is finite dimensional, where $a$ runs
through the regular points of $M$ and $c$ runs through $\mathbb{C}^{s}$. Next,
every rational function regular at the origin can be written in the form

(2.4) $M(z)=H+zG(I-zT)^{-1}F,$

where $H=M(0)$ and $G,T,F$ are matrices of appropriate sizes. One way to build
a realization (in terms of linear operators rather than matrices) uses the
space $\mathfrak{L}(M)$ and is called the backward-shift realization. More
generally, if $M$ is regular at a point $a\in\mathbb{C}$ there exists a
realization of the form

(2.5) $M(z)=H+(z-a)G(T_{1}-zT_{2})^{-1}F$

with $\det(T_{1}-aT_{2})\not=0$. The realization is minimal if the size of the
matrices $T_{1}$ and $T_{2}$ is minimal. Assuming
$T_{1},T_{2}\in\mathbb{C}^{N\times N}$, minimality is equivalent to having
both controllability,

(2.6) $\bigcap_{z\in\mathcal{U}}\ker G(T_{1}-zT_{2})^{-1}=\left\\{0\right\\}$

and observability

$\bigcup_{z\in\mathcal{U}}{\rm ran}\,(T_{1}-zT_{2})^{-1}F=\mathbb{C}^{N},$

where $\mathcal{U}$ is a neighborhood of $a$. One can in fact consider a
finite number of different points $w_{1},\ldots,w_{N}$, since we are in the
rational dimensional case. For instance, condition (2.6) can be rewritten as

$c\in\mathbb{C}^{N}\,\,:\,\,G(T_{1}-wT_{2})^{-1}c=0,\quad
z\in\mathcal{U}\,\,\Longrightarrow\,\,c=0$

but the function $G(T_{1}-wT_{2})^{-1}c$ is rational of degree less or equal
to $N$.

We now explicit the backward shift realization:

######  Proposition 2.3.

Let $M$ be a ${\mathbb{C}}^{p\times q}$–valued rational function with
associated space ${\mathfrak{L}}(M)$ and let $a$ be a point of analyticity of
$M$. Then,

(2.7) $M(z)=H+(z-a)G(I_{\mathfrak{L}(M)}-(z-a)A)^{-1}F$

where the operator matrix

(2.8) $\left(\begin{array}[]{cc}A&F\\\
G&H\end{array}\right)\quad:\quad\left(\begin{array}[]{c}{\mathfrak{L}}(M)\\\
{\mathbb{C}}^{q}\end{array}\right)\Longrightarrow\left(\begin{array}[]{c}{\mathfrak{L}}(M)\\\
{\mathbb{C}}^{p}\end{array}\right)$

is defined by (with $c\in\mathbb{C}^{q}$)

(2.9) $\displaystyle Af(z)$ $\displaystyle=$
$\displaystyle\frac{f(z)-f(a)}{z-a}$ (2.10) $\displaystyle Fc$
$\displaystyle=$ $\displaystyle\frac{M(z)-M(a)}{z-a}c$ (2.11) $\displaystyle
Gf$ $\displaystyle=$ $\displaystyle f(a)$ (2.12) $\displaystyle Hc$
$\displaystyle=$ $\displaystyle M(a)c.$

It is called a realization centered at $a$. It corresponds to

$T_{1}=I_{\mathfrak{L}(M)}+aA\quad{\rm and}\quad T_{2}=A$

in (2.5). When $a=0$ the above realization is called the backwards shift
realization.

For completeness we mention and prove:

######  Proposition 2.4.

Assume the realization (2.5) minimal. Then, the set of points $\mathcal{U}(M)$
where $M$ is defined coincides with the set of points $U$ for which
$\det(T_{1}-zT_{2})\not=0$.

###### Proof.

One direction is clear, $U\subset\mathcal{U}(M)$, and does not use minimality.
If $\det(T_{1}-zT_{2})\not=0$, then $M(z)$ is well defined at that point. To
study the converse we note the formula

(2.13)
$\frac{M(z)-M(w)}{z-w}=G(T_{1}-zT_{2})^{-1}(T_{1}-aT_{2})(T_{1}-wT_{2})^{-1}F,$

valid a priori for $z,w\in U$.

Taking now $z\in\mathcal{U}(M)$ and $w_{1},\ldots,w_{N}$ such that the union
of the ranges of $(T_{1}-w_{n}T_{2})^{-1}F$ span $\mathbb{C}^{N}$ we see that

$G(T_{1}-zT_{2})^{-1}(T_{1}-aT_{2})\begin{pmatrix}(T_{1}-w_{1}T_{2})^{-1}F&\cdots&(T_{1}-w_{N}T_{2})^{-1}F\end{pmatrix}$

is well defined at $z$, and so is $G(T_{1}-zT_{2})^{-1}$. Take now
$w_{1},\ldots,w_{N}$ such that the left range of the matrices

$G(T_{1}-w_{n}T_{2})^{-1},\quad n=1,\ldots,N,$

is all of $\mathbb{C}^{N}$, we see that $(T_{1}-zT_{2})^{-1}$ is well defined.
∎

######  Remark 2.5.

For the interested reader, we mention that the case of functions analytic at
$\infty$ corresponds to realizations of the form

(2.14) $M(z)=H+G(zI-T)^{-1}F.$

In the statement of next result, observability of the pair $(G,T)$ reduces to

(2.15) $\bigcap_{\ell=0}^{\infty}\ker GT^{\ell}=\left\\{0\right\\}.$

We give a proof of the lemma for completeness.

######  Lemma 2.6.

The linear span $\mathfrak{L}(r)$ of the functions
$z\mapsto\dfrac{r(z)-r(a)}{z-a}$ where $a$ runs through $\mathcal{U}(r)$, is
finite dimensional, of dimension $N$, where $N={\rm deg}\,r$. Assume $r$
analytic at the origin, and let $e_{1},\ldots,e_{N}$ be a basis of
$\mathfrak{L}(r)$. There exists an observable pair
$(G,T)\in\mathbb{C}^{1\times N}\times\mathbb{C}^{N\times N}$ such that

(2.16) $Z_{r}(z)\stackrel{{\scriptstyle\rm
def.}}{{=}}\begin{pmatrix}e_{1}(z)&\cdots&e_{N}(z)\end{pmatrix}=G(I_{N}-zT)^{-1}$

Furthermore, if $r$ is real, meaning $r(z)=\overline{r(\overline{z})}$, the
matrices $G$ and $T$ may be chosen to have real components.

###### Proof.

The claim is invariant under a Moebius transform, and we will assume that $0$
is not a pole of $r$. We can write $r$ in a realized form as

(2.17) $r(z)=d+zG(I_{N}-zT)^{-1}b,$

and thus

(2.18) $\frac{r(z)-r(w)}{z-w}=G(I_{N}-zT)^{-1}(I-wT)^{-1}b,$

from which the result follows. For the last claim see [5]. A more direct way
is to write real realizations of $p$ and $q$ and get a realization for $p/q$.
∎

######  Remark 2.7.

We note that two different choices of basis will amount to two different
$Z_{r}$ differing by an invertible multiplicative matrix on the left,
corresponding to the change of basis. In the paper we always stick to a pre-
assigned choice, unless stated otherwise.

######  Example 2.8.

When $r(z)=p(z)$ we can choose $e_{n}(z)=z^{n-1}$, $n=1,\ldots,N$ and then in
(2.17), we have $G=\begin{pmatrix}1&1&\cdots&1\end{pmatrix}$ and $T$ is the
nilpotent matrix

$T=\begin{pmatrix}0&1&0&&\cdots&0\\\ 0&0&1&0&\cdots&0\\\ 0&&&\ddots&&\\\
\vdots&&&&&\vdots\\\ 0&0&&\cdots&0&1\\\ 0&0&&\cdots&0&0\end{pmatrix}$

.

######  Example 2.9.

When $r(z)$ is a finite Blaschke product of the open unit disk, with pairwise
different zeros, we can write

$r(z)=\prod_{n=1}^{N}\frac{z-a_{n}}{1-z\overline{a_{n}}}.$

We have, as sets, $\mathfrak{L}(r)=\mathbf{H}_{2}(\mathbb{D})\ominus
r\mathbf{H}_{2}(\mathbb{D})$, and we can choose
$e_{n}(z)=\frac{1}{1-z\overline{a_{n}}}$, so that here too
$G=\begin{pmatrix}1&1&\cdots&1\end{pmatrix}$ and $T$ is the diagonal matrix
${\rm diag}\,(\overline{a_{1}},\overline{a_{2}},\ldots,\overline{a_{N}})$.

######  Lemma 2.10.

Let $r(z)=\frac{p(z)}{q(z)}$ be a rational function of degree $N\geq 1$.
Elements of $\mathfrak{L}(r)$ are exactly rational functions of the form
$\frac{m(z)}{q(z)}$, where $m$ is a polynomial of degree at most $N-1$. In
particular, if $m(z)\not\equiv 0$, then it cannot have more than $N-1$ zeros.

###### Proof.

This follows from

$\begin{split}\frac{r(z)-r(a)}{z-a}&=\frac{\left(\dfrac{p(z)q(a)-q(z)p(a)}{z-a}\right)}{q(z)q(a)}.\end{split}$

∎

######  Corollary 2.11.

Let $r(z)$ be a rational function of degree $N$, with numerator of the form
$\prod_{i=1}^{M}(z-w_{i})^{m_{i}}$ with $\sum_{i=1}^{M}m_{i}=N$, and let
$\xi\in\mathbb{C}^{N}$ be such that

(2.19) $Z_{r}^{(j)}(w_{i})\xi=0,\quad i=1,\ldots,M,\,\,j=0,\ldots,m_{i}-1.$

Then $\xi=0$.

###### Proof.

Indeed, the function $z\mapsto Z_{r}(z)\xi$ belongs to $\mathfrak{L}(r)$.
Condition (2.19) expresses that it has zeros of total multiplicity
$\sum_{i=1}^{M}m_{i}=N$, and so by the previous result we have
$Z_{r}(z)\xi\equiv 0$, and so $\xi=0$ since the entries of $Z_{r}(z)$ are
linearly independent. ∎

From this corollary we obtain:

######  Theorem 2.12.

Let $r(z)$ be a rational function of degree $N$, with numerator of the form
$\prod_{i=1}^{M}(z-w_{i})^{m_{i}}$ with $\sum_{i=1}^{M}m_{i}=N$. Assume that

(2.20) $Z_{r}(z)F(r(z))\equiv 0.$

Then, $F\equiv 0$.

###### Proof.

We have

$\displaystyle(Z_{r}(z)F(r(z)))^{(1)}$ $\displaystyle=$
$\displaystyle(Z_{r}(z))^{(1)}F(r(z))+Z_{r}(z)r^{(1)}(z)F^{(1)}(r(z))$
$\displaystyle(Z_{r}(z)F(r(z)))^{(2)}$ $\displaystyle=$
$\displaystyle(Z_{r}(z))^{(2)}F(r(z))+2(Z_{r}(z))^{(1)}r^{(1)}(z)F^{(1)}(r(z))+$
$\displaystyle\hskip
2.84526pt+Z_{r}(z)r^{(2)}(z)F^{(1)}(r(z))+Z_{r}(z)(r^{(1)}(z))^{2}F^{(2)}(r(z)),$

and similarly for derivatives of higher order. Using that $r^{(j)}(w_{i})=0$
for $i=1,\ldots,M,\,\,j=0,\ldots,m_{i}-1$, we get

(2.21) $Z_{r}^{(j)}(w_{i})F(0)=0,\quad i=1,\ldots,M,\,\,j=0,\ldots,m_{i}-1,$

so that $F(0)=0$ in view of Corollary 2.11. We write $F(z)=zF_{1}(z)$.
Equation (2.20) becomes

$Z_{r}(z)F_{1}(r(z))\equiv 0.$

Reiterating the previous argument on $Z_{r}(z)F_{1}(r(z))$ we have that
$F^{(1)}(0)=F_{1}(0)=0.$ Iterating we get that the coefficients of the Taylor
series expansion of $F$ at the origin all vanish, and hence $F\equiv 0$. ∎

The results in the next corollary play a key role in the sequel; in particular
see the proof of Theorem 5.1 in Section 5 on the Cuntz relations. Note that
the left hand side of (2.23) below is of the form (1.13).

######  Corollary 2.13.

In the previous notation, let $\alpha\in\Omega(r)$. Then, the matrix
$\mathsf{G}\in\mathbb{C}^{N\times N}$ defined by

(2.22) $g_{nj}=(e_{n}(w_{j}))_{n,j=1}^{N}$

is invertible. Let $F_{1},\ldots,F_{N}$ be analytic in a connected
neighborhood $\Omega$ of $\alpha$. Then

(2.23) $\sum_{n=1}^{N}e_{n}(z)F_{n}(r(z))\equiv 0,\quad z\in r^{-1}(\Omega),$

if and only if $F_{n}(z)\equiv 0$ for $z\in\Omega$ and $n=1,\ldots,N$.

###### Proof.

Let
$c=\begin{pmatrix}c_{1}&c_{2}&\cdots&c_{N}\end{pmatrix}^{t}\in\mathbb{C}^{N}$,
and assume that $\mathsf{G}c=0$. Let $g(z)=\sum_{n=1}^{N}c_{n}e_{n}(z)$. The
function $g\in\mathfrak{L}(r)$, and the condition $\mathsf{G}c=0$ implies that
$g(w_{i})=0$ for $i=1,\ldots,N$. By the previous lemma $g(z)\equiv 0$ and
hence $c=0$ since $e_{1},\ldots,e_{N}$ are linearly independent.

Assume now that (2.23) holds. Setting $z=w_{i}$, $i=1,\ldots,N$ we have that

$\sum_{n=1}^{N}e_{n}(w_{i})F_{n}(\alpha)=0,\quad i=1,\ldots,N$

and so $F_{1}(\alpha)=\cdots=F_{N}(\alpha)$ in view of the first part of the
corollary. For $n=1,\ldots,N$ write now $F_{n}(z)=(z-\alpha)G_{n}(z)$, where
$G_{n}$ is analytic in $\Omega$. We have

$\sum_{n=1}^{N}e_{n}(z)r(z)G_{n}(r(z))\equiv 0,\quad z\in r^{-1}(\Omega),$

and so

$\sum_{n=1}^{N}e_{n}(z)G_{n}(r(z))\equiv 0,\quad z\in r^{-1}(\Omega).$

Reiterating the previous argument we get
$G_{1}(\alpha)=\cdots=G_{N}(\alpha)=0$. We then get that the Taylor series
around $\alpha$ for $F_{1},\ldots,F_{N}$ vanish identically, and hence the
functions $F_{1},\ldots,F_{N}$ vanish identically in $\Omega$ since the latter
is connected. ∎

######  Proposition 2.14.

Let $r$ be a rational function of degree $N$ and let $\alpha\in\Omega(r)$.
Then, for $f\in\mathfrak{L}(r)$, we have

(2.24)
$\frac{f(z)}{r(z)-\alpha}=\sum_{n=1}^{N}\frac{f(w_{n})}{r^{\prime}(w_{n})(z-w_{n})}$

and, in particular,

(2.25)
$\frac{Z_{r}(z)}{r(z)-\alpha}=\sum_{n=1}^{N}\frac{Z_{r}(w_{n})}{r^{\prime}(w_{n})(z-w_{n})}.$

###### Proof.

It is enough to prove (2.24) for $f(z)=\dfrac{r(z)-r(\beta)}{z-\beta}$ since
$\mathfrak{L}(r)$ is finite dimensional and spanned by such functions. Since
$\dfrac{q(\beta)p(z)-p(\beta)q(z)}{z-\beta}$ is a polynomial of degree
strictly less that $N$, we then have the partial fraction expansion:

$\begin{split}\frac{f(z)}{r(z)-\alpha}&=\frac{\left(\dfrac{q(\beta)p(z)-p(\beta)q(z)}{z-\beta}\right)}{q(\beta)(p(z)-\alpha
q(z))}\\\ \\\
&=\sum_{n=1}^{N}\frac{\left(\dfrac{q(\beta)p(w_{n})-p(\beta)q(w_{n})}{w_{n}-\beta}\right)}{q(\beta)(p^{\prime}(w_{n})-\alpha
q^{\prime}(w_{n}))}\\\ \\\
&=\sum_{n=1}^{N}\frac{\left(\dfrac{q(\beta)p(w_{n})-p(\beta)q(w_{n})}{q(w_{n})(w_{n}-\beta)}\right)}{\dfrac{q(\beta)(p^{\prime}(w_{n})-\alpha
q^{\prime}(w_{n}))}{q(w_{n})}}\\\ \\\
&=\sum_{n=1}^{N}\frac{f(w_{n})}{r^{\prime}(w_{n})(z-w_{n})},\\\ &\end{split}$

where we have used another time (2.3). ∎

The following corollary shows that the operators $R^{(r)}_{\alpha}$ are well
suited for functions of the form (1.13)

######  Corollary 2.15.

Let $r$ be a rational function of degree $N$. Let
$w_{1}(\alpha),\ldots,w_{N}(\alpha)$ the roots, assumed simple, of the
equation $r(z)=\alpha$, and let $f$ be analytic in a neighborhood of
$\left\\{w_{1}(\alpha),\ldots,w_{N}(\alpha)\right\\}$ and of the form
$f(z)=Z_{r}(z)F(r(z))$, where the function $F$ is $\mathbb{C}^{N}$-valued and
analytic in a neighborhood of the point $\alpha$. Then,

(2.26) $(R^{(r)}_{\alpha}f)(z)=Z_{r}(z)(R_{\alpha}F)(r(z)).$

###### Proof.

We can write

$\begin{split}(R^{(r)}_{\alpha}f)(z)&=\frac{Z_{r}(z)F(r(z))}{r(z)-\alpha}-\sum_{n=1}^{N}\frac{Z_{r}(w_{n})F(r(w_{n}))}{r^{\prime}(w_{n})(z-w_{n})}\\\
&=\frac{Z_{r}(z)F(r(z))}{r(z)-\alpha}-\sum_{n=1}^{N}\frac{Z_{r}(w_{n})F(\alpha)}{r^{\prime}(w_{n})(z-w_{n})}\\\
&=\frac{Z_{r}(z)F(r(z))}{r(z)-\alpha}-\sum_{n=1}^{N}\frac{Z_{r}(w_{n})}{r^{\prime}(w_{n})(z-w_{n})}F(\alpha)\\\
&=\frac{Z_{r}(z)F(r(z))}{r(z)-\alpha}-\frac{Z_{r}(z)}{r(z)-\alpha}F(\alpha)\\\
&=Z_{r}(z)(R_{\alpha}F)(r(z)),\end{split}$

where we have used (2.25) to conclude. ∎

##  3\. The associated symmetric matrix

To a given rational function we can associate a symmetric matrix which plays
an important role in the sequel. The following lemma seems to be new and
possibly of independent interest.

######  Lemma 3.1.

Let $r$ be a rational function of degree $N$, $\alpha\in\Omega(r)$ and let
$w_{1}(\alpha),\ldots,w_{N}(\alpha)$ be the (simple) roots of $r(z)=\alpha$.
Let $f,g\in\mathfrak{L}(r)$, then the sum

(3.1)
$\sum_{n=1}^{N}\frac{f(w_{n}(\alpha))g(w_{n}(\alpha))}{r^{\prime}(w_{n}(\alpha))}$

does not depend on $\alpha$.

###### Proof.

It is enough to consider $f$ and $g$ respectively equal to

$f(z)=\frac{r(z)-r(a)}{z-a}\quad{\rm and}\quad g(z)=\frac{r(z)-r(b)}{z-b},$

where $a,b$ are points of analyticity of $r$. We note that the poles of $r$
are removable singularities of

$\frac{f(z)g(z)}{(r(z)-\alpha)r(z)}=\frac{\left(\dfrac{p(z)q(a)-q(z)p(a)}{z-a}\right)\left(\dfrac{p(z)q(b)-q(z)p(b)}{z-b}\right)}{(p(z)-\alpha
q(z))p(z)q(a)q(b)},$

and so, for $R>0$ large enough (depending on $\alpha$), such that all the
poles of the integrand are inside $|z|\leq R$,

$\frac{1}{2\pi i}\int_{|z|=R}\frac{f(z)g(z)}{(r(z)-\alpha)r(z)}dz=0$

since the difference between the degree of the denominator and of the
numerator is at least $2$. So, for such $R$:

(3.2) $\frac{1}{2\pi
i}\int_{|z|=R}\frac{f(z)g(z)}{r(z)-\alpha}dz=\frac{1}{2\pi
i}\int_{|z|=R}\frac{f(z)g(z)}{r(z)}dz.$

Furthermore

(3.3)
$\frac{f(z)g(z)}{r(z)-\alpha}=\frac{\left(\dfrac{p(z)q(a)-q(z)p(a)}{z-a}\right)\left(\dfrac{p(z)q(b)-q(z)p(b)}{z-b}\right)}{(p(z)-\alpha
q(z))q(z)q(a)q(b)}$

and in particular, for $\alpha=0$,

$\frac{f(z)g(z)}{r(z)}=\frac{\left(\dfrac{p(z)q(a)-q(z)p(a)}{z-a}\right)\left(\dfrac{p(z)q(b)-q(z)p(b)}{z-b}\right)}{p(z)q(z)q(a)q(b)}.$

It follows from these two equalities that to evaluate both sides of (3.2) we
need to compute the residues of the corresponding functions at the points
$w_{n}(\alpha),w_{n}(=w_{n}(0))$ and at the zeros of $q$. Since
$\alpha\in\Omega(r)$, we have

${\rm
Res\,}\left(\frac{f(z)g(z)}{r(z)-\alpha},w_{n}(\alpha)\right)=\frac{f(w_{n}(\alpha))g(w_{n}(\alpha))}{r^{\prime}(w_{n}(\alpha))},\quad
n=1,\ldots,N.$

The key to the proof of the lemma is that the residues at the zeros of $q$ do
not depend on $\alpha$, as we now show. Let $z_{1},\ldots,z_{U}$ be the zeros
of $q$, and assume $z_{u}$ be a zero of order $m_{u}$:
$q(z)=(z-z_{u})^{m_{u}}\pi_{u}(z)$, where $\pi_{u}(z)$ is a polynomial not
vanishing at $z_{u}$. We rewrite (3.3) as

(3.4) $\frac{f(z)g(z)}{r(z)-\alpha}=\frac{G(z)}{(p(z)-\alpha
q(z))(z-z_{u})^{m_{u}}}$

where $G$ is independent of $\alpha$ and is analytic at the point $z_{u}$ and
$G(z_{u})\not=0$. From the formula

${\rm
Res\,}\left(\frac{f(z)g(z)}{r(z)-\alpha},z_{u}\right)=\frac{1}{(m_{u}-1)!}\left(\frac{G(z)}{(p(z)-\alpha
q(z))}\right)^{(m_{u}-1)}(z_{u})$

we see that the residue involves only derivatives of $\alpha q$ of order less
or equal to $m_{u}-1$, evaluated at the point $z_{u}$, and hence does not
depend on $\alpha$ since

$q(z_{u})=\cdots=q^{(m_{u}-1)}(z_{u})=0.$

Thus, we have for large $R$

$\frac{1}{2\pi
i}\int_{|z|=R}\frac{f(z)g(z)}{r(z)-\alpha}dz=\sum_{n=1}^{N}\frac{f(w_{n}(\alpha))g(w_{n}(\alpha))}{r^{\prime}(w_{n}(\alpha))}+M_{1}$

where $M_{1}$ denotes the sum of the residues of the function
$\frac{f(z)g(z)}{r(z)-\alpha}$ evaluated at the zeros of $q$, and

$\frac{1}{2\pi i}\int_{|z|=R}\frac{f(z)g(z)}{r(z)}dz=\frac{1}{2\pi
i}\int_{|z|=R}\frac{f(z)g(z)}{r(z)}dz+M_{1}.$

It follows that the sum

$\sum_{n=1}^{N}\frac{f(w_{n})g(w_{n})}{r^{\prime}(w_{n})}=\frac{1}{2\pi
i}\int_{|z|=R}\frac{f(z)g(z)}{r(z)}dz$

is independent of $\alpha$.

∎

######  Remark 3.2.

When the zeros of $q$ are simple, we have

$M_{1}=\sum_{u}\frac{\left(\dfrac{p(z_{u})}{(z_{u}-a)(z_{u}-b)}\right)}{q^{\prime}(z_{u})},$

and when $0\in\Omega(r)$

$\frac{1}{2\pi
i}\int_{|z|=R}\frac{f(z)g(z)}{r(z)}dz=\sum_{n=1}^{N}\frac{f(w_{n})g(w_{n})}{r^{\prime}(w_{n})}.$

######  Corollary 3.3.

Let $r$ be a real rational function of degree $N$, $\alpha\in\Omega(r)$ and
let us assume that the roots $w_{1}(\alpha),\ldots,w_{N}(\alpha)$ of
$r(z)=\alpha$ are simple. Let $J$ be a $p\times p$ signature matrix with real
entries. The sum

(3.5) $X(J,r)=\sum_{n=1}^{N}\frac{(Z_{r}(w_{n})\otimes
I_{p})^{t}J(Z_{r}(w_{n})\otimes
I_{p})}{r^{\prime}(w_{n})}\in\mathbb{R}^{Np\times Np}$

is self-adjoint and real (i.e. symmetric), invertible, and independent of
$\alpha$.

###### Proof.

In the proof we use the fact that, for any (possibly non square) matrices $A$
and $B$ of possibly different sizes we have (see [37] for a discussion and
more information on the properties of the tensor, or Kronecker, product):

$\begin{split}(A\otimes B)^{t}&=A^{t}\otimes B^{t}\\\ \overline{A\otimes
B}&=\overline{A}\otimes\overline{B},\end{split}$

and that, in the case of invertible matrices,

(3.6) $(A\otimes B)^{-1}=A^{-1}\otimes B^{-1}.$

Using (1.12) we deduce

$\begin{split}(Z_{r}(w_{n})\otimes I_{p})^{t}J(Z_{r}(w_{n})\otimes
I_{p})&=\begin{pmatrix}e_{1}(w_{n})I_{p}\\\ \vdots\\\
e_{N}(w_{n})I_{p}\end{pmatrix}J\begin{pmatrix}e_{1}(w_{n})I_{p}&\cdots&e_{N}(w_{n})I_{p}\end{pmatrix}\\\
&=\left(e_{i}(w_{n})e_{j}(w_{n})J\right)_{i,j=1}^{N}\\\ &=X(1,r)\otimes
J,\end{split}$

and the fact that $X(1,r)$ does not depend on $\alpha$ follows from the
preceding lemma. To prove that $X(J,r)$ is invertible it is enough to show
that $X(1,r)$ is invertible. Using (3.6) we then have

$X(J,r)^{-1}=X(1,r)^{-1}\otimes J$

since $J=J^{-1}$. To prove that $X(1,r)$ is invertible we use the following
readily checked fact: a $N\times N$ matrix of the form
$\sum_{n=1}^{N}a_{n}^{t}b_{n}$, where the
$a_{1},\ldots,a_{N},b_{1},\ldots,b_{N}\in\mathbb{C}^{N}$, is invertible if and
only if both $a_{1},\ldots,a_{N}$ and $b_{1},\ldots,b_{N}\in\mathbb{C}^{N}$
are linearly independent vectors. In the case of $X(1,r)$, it is therefore
enough to check that the vectors $Z(w_{1}),\ldots,Z(w_{N})$ are linearly
independent. Assume not. There will then be a non-zero vector

$c=\begin{pmatrix}c_{1}\\\ \vdots\\\ c_{N}\end{pmatrix}\in\mathbb{C}^{N}$

such that

$Z(w_{n})c=0,\quad n=1,\ldots,N.$

Thus the rational function $Z(z)c$ has (at least) $N$ pairwise different
zeros. Since $Zc\in\mathfrak{L}(r)$, this cannot be in view of Lemma 2.10,
unless $Z(z)c\equiv 0$. But this implies $c=0$ since the entries of $Z(z)$
form a basis of $\mathfrak{L}(r)$.

We now show that $X(J,r)$ is self-adjoint. Since

$(X(1,r)\otimes J)^{t}=X(1,r)^{t}\otimes J^{t}=X(1,r)\otimes J,$

it is enough to show that the entries of $X(1,r)$ are real. This follows from
the fact that either the $w_{n}$ are real or they appear in pairs. In the
first case the matrix $\frac{Z(w_{n})^{t}JZ(w_{n})}{r^{\prime}(w_{n})}$ has
real entries while in the second case the matrix

$\frac{Z(w_{n})^{t}JZ(w_{n})}{r^{\prime}(w_{n})}+\frac{Z(\overline{w_{n}})^{t}JZ(\overline{w_{n}})}{r^{\prime}(\overline{w_{n}})}$

has real entries since $Z$ is real and
$r^{\prime}(\overline{w_{n}})=\overline{r^{\prime}(w_{n})}$. ∎

######  Definition 3.4.

$X(J,r)$ is called the associated symmetric matrix to $r$ and $J$. We will
write $X(J,r)=X(r)$ or $X$ when $J$ or $r$ are clear from the context.

Note that $X(1,r)$ can be written in term of a realization as:

(3.7)
$X(1,r)=\sum_{n=1}^{N}\frac{(I-w_{n}T)^{-t}G^{t}G(I-w_{n}T)^{-1}}{r^{\prime}(w_{n})}.$

######  Corollary 3.5.

For a Blaschke product with pairwise different zeros we have

(3.8)
$X(1,r)=\left(\sum_{n=1}^{N}\frac{1}{r^{\prime}(a_{n})(1-a_{n}\overline{a_{u}})(1-a_{n}\overline{a_{v}})}\right)_{u,v=1}^{N}.$

###### Proof.

We have $G=\begin{pmatrix}1&1&\cdots&1\end{pmatrix}$ and $T$ is the diagonal
matrix ${\rm
diag}\,(\overline{a_{1}},\overline{a_{2}},\ldots,\overline{a_{N}})$. and so

$(X(1,r))_{uv}=\sum_{n=1}^{N}\frac{1}{r^{\prime}(w_{n})(1-w_{n}\overline{a_{u}})(1-w_{n}\overline{a_{v}})}.$

Taking $\alpha=0$ we get $w_{n}=a_{n}$ for $n=1,\ldots,N$, from which (3.8)
follows by using Corollary 3.3. ∎

In case of a polynomial $r(z)=p(z)$ of degree $N$ and
$Z_{p}(z)=\begin{pmatrix}1&z&\cdots&z^{N-1}\end{pmatrix}$ the above result
takes the following special form:

######  Corollary 3.6.

Let $r(z)=p(z)=p_{0}+\cdots+p_{N}z^{N}$, with pairwise distinct zeros
$w_{1},\ldots,w_{N}$. Then, the matrix $X(1,p)=X$ is equal to

(3.9) $X=\begin{pmatrix}0&\cdots&0&0&0&h_{N-1}\\\
0&\cdots&0&0&h_{N-1}&h_{N}\\\ 0&\cdots&0&\iddots&&\vdots\\\
0&\iddots&h_{N-1}&&&\vdots\\\ 0&h_{N-1}&\iddots&&&\vdots\\\
h_{N-1}&h_{N}&\cdots&\cdots&&h_{2N-2}\end{pmatrix},$

where

(3.10) $h_{n}=\frac{1}{2\pi i}\int_{|z|=R}\frac{z^{n}dz}{p(z)},$

with $R>\max_{n}|w_{n}|$.

###### Proof.

We can take $Z_{p}(z)=\begin{pmatrix}1&z&\cdots&z^{N-1}\end{pmatrix}$, and so

$\begin{split}X&=\sum_{n=1}^{N}\frac{\begin{pmatrix}1\\\ w_{n}\\\ \vdots\\\
w_{n}^{N-1}\end{pmatrix}\begin{pmatrix}1&w_{n}&\cdots&w_{n}^{N-1}\end{pmatrix}}{p^{\prime}(w_{n})}\\\
&\\\
&=\left(\sum_{n=1}^{N}\frac{w_{n}^{k+\ell}}{p^{\prime}(w_{n})}\right)_{k,\ell=0}^{N-1},\end{split}$

which is a Hankel matrix. To prove (3.9) we now show that $X$ is lower with
respect to the main off-diagonal. Let $R_{0}$ be such that the roots
$w_{1},\ldots,w_{N}$ (assumed pairwise different) of $p(z)$ lie inside
$B(0,R_{0})$. For $R\geq R_{0}$ and $t=0,1,2,\ldots,$ we have

$\begin{split}\sum_{n=1}^{N}\frac{w_{n}^{t}}{p^{\prime}(w_{n})}&=\frac{1}{2\pi
i}\int_{|z|=R}\frac{z^{t}}{p(z)-\alpha}dz\\\
&=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{R^{t+1}e^{i(t+1)t}dt}{p(Re^{it})}\end{split}$

which will go to $0$ as $R\rightarrow\infty$ for $t+1<N$, that is $t\leq N-2$,
and hence the formula for $X$. ∎

As another example we consider the function $r(z)=z+\frac{1}{z}$ (see Example
1.3). Then we have:

######  Example 3.7.

Let $\alpha\in\mathbb{C}$ and let $\alpha_{+}$ and $\alpha_{-}$ be the roots
of

$z+\frac{1}{z}=\alpha.$

Then $Z_{r}(z)=\begin{pmatrix}1&\frac{1}{z}\end{pmatrix}$ and the
corresponding matrix $X=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$ and does not
depend on $\alpha$.

We give a direct proof of this fact. We note that the roots satisfy
$\alpha_{-}\alpha_{+}=1$. Hence

$\begin{split}X&=\frac{\begin{pmatrix}1\\\
\frac{1}{\alpha_{+}}\end{pmatrix}\begin{pmatrix}1&\frac{1}{\alpha_{+}}\end{pmatrix}}{1-\frac{1}{\alpha_{+}^{2}}}+\frac{\begin{pmatrix}1\\\
\frac{1}{\alpha_{-}}\end{pmatrix}\begin{pmatrix}1&\frac{1}{\alpha_{-}}\end{pmatrix}}{1-\frac{1}{\alpha_{-}^{2}}}\\\
&=\frac{\begin{pmatrix}1\\\
\alpha_{-}\end{pmatrix}\begin{pmatrix}1&\alpha_{-}\end{pmatrix}}{1-\alpha_{-}^{2}}+\frac{\begin{pmatrix}1\\\
\alpha_{+}\end{pmatrix}\begin{pmatrix}1&\alpha_{+}\end{pmatrix}}{1-\alpha_{+}^{2}}\\\
&=\begin{pmatrix}\dfrac{1}{1-\alpha_{-}^{2}}+\dfrac{1}{1-\alpha_{+}^{2}}&\dfrac{\alpha_{-}}{1-\alpha_{-}^{2}}+\dfrac{\alpha_{+}}{1-\alpha_{+}^{2}}\\\
\dfrac{\alpha_{-}}{1-\alpha_{-}^{2}}+\dfrac{\alpha_{+}}{1-\alpha_{+}^{2}}&\dfrac{\alpha_{-}^{2}}{1-\alpha_{-}^{2}}+\dfrac{\alpha_{+}^{2}}{1-\alpha_{+}^{2}}\end{pmatrix}\\\
&=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}\end{split}$

since

$\frac{1}{1-\alpha_{-}^{2}}+\frac{1}{1-\alpha_{+}^{2}}=\frac{1}{1-\alpha_{-}^{2}}+\frac{\alpha_{-}^{2}}{\alpha_{-}^{2}-1}=1,$

$\begin{split}\frac{\alpha_{-}}{1-\alpha_{-}^{2}}+\frac{\alpha_{+}}{1-\alpha_{+}^{2}}&=\frac{\alpha_{-}}{1-\alpha_{-}^{2}}+\frac{\alpha_{+}\alpha_{-}^{2}}{\alpha_{-}^{2}-1}\\\
&=\frac{\alpha_{-}}{1-\alpha_{-}^{2}}+\frac{\alpha_{-}}{\alpha_{-}^{2}-1}=0\end{split}$

and

$\frac{\alpha_{-}^{2}}{1-\alpha_{-}^{2}}+\frac{\alpha_{+}^{2}}{1-\alpha_{+}^{2}}=\frac{\alpha_{-}^{2}}{1-\alpha_{-}^{2}}+\frac{\alpha_{+}^{2}\alpha_{-}^{2}}{\alpha_{-}^{2}-1}=-1,$

where we have used repeatedly that $\alpha_{+}\alpha_{-}=1$.

We also compute directly the associate symmetric matrix for a cubic.

######  Example 3.8.

Let $r(z)=z^{2}+1/z$. Then,

$X=\begin{pmatrix}0&1&0\\\ 1&0&0\\\ 0&0&-1\end{pmatrix}.$

Indeed, we have $Z_{r}(z)=\begin{pmatrix}1&z&1/z\end{pmatrix}$, and so

$X=\sum_{z\,:\,r(z)=\alpha}\frac{\begin{pmatrix}1\\\ z\\\
\frac{1}{z}\end{pmatrix}\begin{pmatrix}1&z&\frac{1}{z}\end{pmatrix}}{r^{\prime}(z)}$

and we need to compute

$\sum_{n=1}^{3}\frac{w_{n}^{k}}{r^{\prime}(w_{n})},\quad k=0,\pm 1,\pm 2,$

with $\alpha\in\Omega(r)$ and $r(w_{n})=\alpha$ where $n=1,2,3$.

We have for $k=0$,

$\sum_{n=1}^{3}\frac{1}{r^{\prime}(w_{n})}=\frac{1}{2\pi
i}\int_{|s|=R}\frac{ds}{r(s)-\alpha}=\frac{1}{2\pi
i}\int_{|s|=R}\frac{sds}{s^{3}-\alpha s+1}=0$

by the exactity relation. For $k=1$ we have

$\sum_{n=1}^{3}\frac{w_{n}}{r^{\prime}(w_{n})}=\frac{1}{2\pi
i}\int_{|s|=R}\frac{sds}{r(s)-\alpha}=\frac{1}{2\pi
i}\int_{|s|=R}\frac{s^{2}ds}{s^{3}-\alpha s+1}=1,$

as is seen by either computing the residue at $\infty$ or letting
$R\rightarrow\infty$. For $k=2$ we have

$\sum_{n=1}^{3}\frac{w_{n}^{2}}{r^{\prime}(w_{n})}=\frac{1}{2\pi
i}\int_{|s|=R}\frac{s^{2}ds}{r(s)-\alpha}=\frac{1}{2\pi
i}\int_{|s|=R}\frac{s^{3}ds}{s^{3}-\alpha s+1}=-{\rm
Res}\,(\frac{s^{3}}{s^{3}-\alpha s+1},\infty)=0.$

On the other hand, for $k=-1$ we have:

$\frac{1}{2\pi i}\int_{|s|=R}\frac{ds}{s(r(s)-\alpha)}=\frac{1}{2\pi
i}\int_{|s|=R}\frac{ds}{s^{3}+1-\alpha s}=0,$

here too thanks to the exactity relation, and for $k=-2$ we have:

$\begin{split}\frac{1}{2\pi
i}\int_{|s|=R}\frac{ds}{s^{2}(r(s)-\alpha)}&=\frac{1}{2\pi
i}\int_{|s|=R}\frac{ds}{s(s^{3}+1-\alpha s)}=0\end{split}$

and

$\sum_{n=1}^{3}\frac{1}{w_{n}^{2}r^{\prime}(w_{n})}+{\rm
Res}\,\left(\frac{1}{s(s^{3}+1-\alpha s)},0\right)=0$

and so

$\sum_{n=1}^{3}\frac{1}{w_{n}^{2}r^{\prime}(w_{n})}=-1.$

One can also directly compute $X$ for $\alpha=0$. Then the equation
$r(z)=\alpha$ becomes $z^{3}=-1$, and with $j=\frac{1+i\sqrt{3}}{2}$, its
roots are $w_{1}=-1,w_{2}=j$ and $w_{3}=\overline{j}$. Furthermore
$r^{\prime}(w_{n})=2w_{n}-1/w_{n}^{2}=3w_{n}$. Hence,

$\sum_{n=1}^{3}\frac{w_{n}^{k}}{r^{\prime}(w_{n})}=\frac{\sum_{n=1}^{3}w_{n}^{k-1}}{3}=\begin{cases}\,0,\quad\hskip
5.69054ptk=-1,0,2\\\ \,1,\quad\hskip 5.69054ptk=1,\\\ \,-1,\quad
k=-2.\end{cases}$

More generally:

######  Example 3.9.

The function $r(z)=z^{N}+\frac{1}{z}$ leads to

(3.11) $X=\begin{pmatrix}&&&0\\\ &\mathcal{I}_{N}&&\vdots\\\ &&&0\\\
0&\cdots&0&-1\\\ \end{pmatrix}$

where $\mathcal{I}_{N}$ is the $N\times N$ matrix with antidiagonal elements
all equal to $1$, and other elements equal to $0$.

Indeed, we now have $r^{\prime}(z)=Nz^{N-1}-\frac{1}{z^{2}}$, and is equal to
$-\frac{N+1}{w_{n}^{2}}$ when $z=w_{n}$ is a root of $r(z)=0$, i.e. satisfies
$w_{n}^{N+1}+1=0$. We then have

$\begin{split}X&=\sum_{z\,:\,z^{N+1}+1=0}-z^{2}\frac{\begin{pmatrix}1\\\ z\\\
\vdots\\\ z^{N-1}\\\
\frac{1}{z}\end{pmatrix}\begin{pmatrix}1&z&\cdots&z^{N-1}&\frac{1}{z}\end{pmatrix}}{N+1}\\\
&=\sum_{z\,:\,z^{N+1}+1=0}\frac{-\begin{pmatrix}z^{2}&z^{3}&\cdots&\cdots&z^{N+1}&z\\\
z^{3}&z^{4}&\cdots&z^{N+1}&z^{N+2}&z^{2}\\\ &&\iddots&&&\\\
&\iddots&\cdots&&&\\\ z^{N+1}&z^{N+2}&\cdots&&&\\\
z&z^{2}&\cdots&&z^{N}&1\end{pmatrix}}{N+1},\end{split}$

from which (3.11) follows.

######  Lemma 3.10.

$X(J,r)$ is uniquely defined up to a real similarity matrix $X(J,r)\mapsto
SX(J,r)S^{t}$.

###### Proof.

Let $Z_{r}^{(1)}$ be constructed as $Z_{r}$ but with a different basis. We can
write

$Z_{r}^{(1)}(z)=Z_{r}(z)S$

where $S$ is an invertible matrix; $S$ has real entries since we choose a
basis of $\mathfrak{L}(r)$ with real coefficients. The result follows from

$\sum_{n=1}^{N}\frac{Z_{r}^{(1)}(w_{n})^{t}JZ_{r}^{(1)}(w_{n})}{r^{\prime}(w_{n})}=\sum_{n=1}^{N}\frac{S^{t}Z_{r}(w_{n})^{t}JZ_{r}(w_{n})S}{r^{\prime}(w_{n})}.$

∎

##  4\. A representation theorem for analytic function

Corollary 2.13 leads to the following natural question: Which functions can be
written in the form $Z_{r}(z)F(r(z))$? When $r$ is a polynomial the answer was
given in [17, Theorem 2.1, p.44] and in [16] when $r$ is a finite Blaschke
product. We present a result which encompasses both cases. The result is valid
in particular for rational functions $r$ with possibly less than $N$ pairwise
different zeros, but having limit infinity at infinity, and also for Blaschke
products, corresponding to results in [17] and [16] respectively. These cases
are considered after the proof of the lemma. Note that condition (4.2) will
not hold for functions $r$ such that $\lim_{z\rightarrow\infty}r(z)=0$.

######  Theorem 4.1.

Let $r$ be a rational function with non constant numerator, and let
$D_{1},\ldots,D_{L}$ be pairwise disjoint open disks around the distinct zeros
$w_{1},\ldots,w_{L}$ of $r$, not containing poles of $r$, and such that
$\bigcup_{\ell=1}^{L}\partial D_{\ell}$ does not contain poles of $r$ either.
We let

(4.1) $\rho=\min_{s\in\bigcup_{\ell=1}^{L}\partial D_{\ell}}|r(s)|.$

We assume

(4.2) $\Omega_{0}\stackrel{{\scriptstyle\rm
def.}}{{=}}\left\\{z\in\mathbb{C}\,:\,|r(z)|<\rho\right\\}\subset\bigcup_{\ell=1}^{L}D_{\ell}.$

Then, every function $f$ analytic in a neighborhood of
$\bigcup_{\ell=1}^{L}\overline{D_{\ell}}$ can be written in unique way as

(4.3) $f(z)=Z_{r}(z)F(r(z)),\quad z\in\Omega_{0},$

where $F$ is $\mathbb{C}^{N}$-valued and analytic in $|z|<\rho$.

###### Proof.

We proceed in a number of steps.

STEP 1: Let $z\in\bigcup_{\ell=1}^{L}D_{\ell}$. It holds that

$f(z)=\frac{1}{2\pi i}\sum_{\ell=1}^{L}\int_{\partial
D_{\ell}}\frac{f(s)}{s-z}ds.$

This is Cauchy’s formula because of the analyticity assumption on $f$.

STEP 2: The function

(4.4) $F(z)=\frac{1}{2\pi i}\sum_{\ell=1}^{L}\int_{\partial
D_{\ell}}\frac{(I_{N}-sT)^{-1}bf(s)}{r(s)-z}ds$

satisfies (4.3) and is analytic in $|z|<\rho$.

Indeed, using (2.18) and (2.16), we can write

$\begin{split}f(z)&=\frac{1}{2\pi i}\sum_{\ell=1}^{L}\int_{\partial
D_{\ell}}\frac{f(s)}{r(s)-r(z)}\frac{r(s)-r(z)}{s-z}ds\\\ &=\frac{1}{2\pi
i}G(I_{N}-zT)^{-1}\left(\sum_{\ell=1}^{L}\int_{\partial
D_{\ell}}\frac{(I_{N}-sT)^{-1}bf(s)}{r(s)-r(z)}ds\right)\\\
&=G(I_{N}-zT)^{-1}F(r(z))\end{split}$

with $F(z)$ as in (4.4). We now check that $F$ is analytic in $|z|<\rho$. This
follows from the fact that on $\bigcup_{\ell=1}^{L}\partial D_{\ell}$ we have
$|r(z)|\geq\rho$ by (4.2), and that $(I_{N}-sT)^{-1}$ is well defined on
$\bigcup_{\ell=1}^{L}\partial D_{\ell}$ since the realization of $r$ is
minimal; see Proposition 2.4.

STEP 3: The representation (4.3) is unique.

This follows from Corollary 2.13. ∎

######  Remark 4.2.

$F$ is in fact analytic in the set

(4.5)
$A(r)=\left\\{z\in\mathbb{C}\,;\,r(z)\not\in\bigcup_{\ell=1}^{L}r(\partial
D_{\ell})\right\\}$

and in particular at infinity. This fact will be used in particular in the
sequel in the proofs of various structure theorems.

######  Remark 4.3.

The space $A(r)$ need not be connected, and depends on the choice of the
disks, $D_{1},\ldots,D_{\ell}$ and so is not unique. In the various structure
theorems in the sequel, we fix each time a set $A(r)$. The function $f$ is
then extended analytically in the (possibly not-connected) set $r^{-1}(A(r))$.

######  Lemma 4.4.

Condition (4.2) holds for rational functions regular at the origin and having
a pole at infinity.

###### Proof.

We consider open disks $D_{L+1},\ldots,D_{M}$ around the poles of $r$ (if
any), all the disks $D_{1},\ldots,D_{M}$ are assumed pairwise non
intersecting. As above, we set

(4.6) $\rho=\min_{s\in\bigcup_{\ell=1}^{L}\partial D_{\ell}}|r(s)|.$

STEP 1: We can choose $D_{L+1},\ldots,D_{M}$ such that

(4.7) $\rho<\min_{s\in\bigcup_{\ell=L+1}^{M}\overline{D_{\ell}}}|r(s)|$

and in particular

(4.8) $\rho<\min_{s\in\bigcup_{\ell=L+1}^{M}\partial D_{\ell}}|r(s)|.$

Indeed, write $r(z)=\frac{h_{L+1}(z)}{(z-w_{L+1})^{m_{L+1}}}$, where
$m_{L+1}\in\mathbb{N}$, and $h_{L+1}(z)$ regular at $w_{L+1}$ with
$h(w_{L+1})\not=0$. We can write

$h_{L+1}(z)=h(w_{L+1})(1+t_{L+1}(z)),\quad{\rm with}\quad\lim_{z\rightarrow
w_{L+1}}t_{L+1}(z)=0.$

Let $\epsilon_{0}$ be such that

$|z-w_{L+1}|<\epsilon_{0}\quad\Longrightarrow\quad|t_{L+1}(z)|<1/2.$

Then, for $\epsilon\leq\epsilon_{0}$ we have

$s\in\partial
B(w_{L+1},\epsilon)\quad\Longrightarrow\quad|r(s)|\geq\frac{1-|t_{L+1}(s)|}{|s-w_{L+1}|^{m_{L+1}}}\geq\frac{1}{2\epsilon^{m_{L+1}}}\rightarrow\infty$

as $\epsilon\rightarrow 0$, an so $\lim_{\epsilon\rightarrow
0}\min_{|s-w_{L+1}|=\epsilon}|r(s)|=\infty$.

(4.8) follows by going over the above construction around each of the poles,
and taking the value of $\epsilon$ for which (for instance)

$\min_{|s-w_{\ell}|=\epsilon}|r(s)|=2\rho,\quad\ell=L+1,\ldots,M.$

We then get (4.7) by taking $\epsilon\rightarrow 0$.

STEP 2: There exists $R_{0}>0$ such that all the closed disks
$\overline{D_{1}},\ldots,\overline{D_{M}}$ are inside the open disk of radius
$R$ for $R\geq R_{0}$ and

(4.9) $\min\left\\{|r(z)|:|z|=R\right\\}\geq 2\rho,\quad\forall R\geq R_{0}.$

This is clear since $\lim_{z\rightarrow\infty}|r(z)|=\infty$ implies
$\lim_{z\rightarrow\infty}\min\left\\{|r(z)|:|z|=R\right\\}=\infty$. To check
the latter, write $r(z)=\frac{p_{1}(z)}{p_{2}(z)}$, where $p_{1}$ and $p_{2}$
are polynomials, of degrees respectively $n_{1}$ and $n_{2}$, with
$n_{1}>n_{2}$ since $r$ has a pole at infinity. We first assume that $p_{2}$
is not a constant. Then, with
$c_{1},c_{2}\in\mathbb{C}\setminus\left\\{0\right\\}$,

$r(z)=\frac{c_{1}z^{n_{1}}(1+t_{1}(z))}{c_{2}z^{n_{2}}(1+t_{2}(z))}=\frac{c_{1}}{c_{2}}z^{n_{1}-n_{2}}\frac{1+t_{1}(z)}{1+t_{2}(z)}.$

To conclude take $R_{1}$ such that for every $R\geq R_{1}$,

$|z|>R\quad\Longrightarrow\quad|t_{1}(z)|<1/2\quad{\rm and}\quad|t_{2}(z)|<1.$

For such $R$,

$|1+t_{1}(z)|\geq 1-|t_{1}(z)|\geq 1/2\quad{\rm and}\quad|1+t_{2}(z)|\leq 2,$

and so

$\left|\frac{1+t_{1}(z)}{1+t_{2}(z)}\right|\geq\frac{1-|t_{1}(z)|}{1+|t_{2}(z)|}\geq\frac{1}{4}.$

Hence, for $|z|\geq R\geq R_{1}$,

$|r(z)|\geq\frac{R^{n_{1}-n_{2}}}{4}\rightarrow\infty\quad{\rm as}\quad
z\rightarrow\infty.$

By taking $R_{1}$ large enough, we can therefore find $R_{0}$ such that (4.9)
holds when $p_{2}$ is not a constant polynomial. The case where $p_{2}$ is a
constant works the same, with considering only the term $1+t_{1}(z)$.

STEP 3: Let $K=\left\\{|z|\leq
R\right\\}\setminus\bigcup_{\ell=1}^{M}D_{\ell}$. We have $|r(z)|\geq\rho$ for
$z\in K$.

Indeed, the function $1/r(z)$ is analytic in the interior of the compact set
$K$ and continuous on its boundary, and so it attains its maximum. By the
maximum modulus principle and by the definition of $\rho$, this maximum is
equal to $1/\rho$. Hence, in the interior of $K$ we have $1/|r(z)|<1/\rho$,
that is $|r(z)|>\rho$.

STEP 4: We have $|r(z)|\geq\rho$ outside the $\bigcup_{\ell=1}^{L}D_{\ell}$,
and (4.2) holds.

That $|r(z)|\geq\rho$ outside the $\bigcup_{\ell=1}^{L}D_{\ell}$ follows from
the definition of $R_{0}$ for $|z|>R$, and from (4.7) for $z$ inside the
closed disks around the poles.
∎

######  Corollary 4.5.

In the notation of Theorem 4.1, one can assume the function $F$ analytic in a
neighborhood of any preassigned point $w_{0}$ different from a pole of $r$.

###### Proof.

Let $s_{1},\ldots,s_{u}$ be the roots (if any) of the equation $r(z)=w_{0}$.
In the proof of Theorem 4.1 we choose the disks $D_{1},\ldots,D_{M}$ so that
$s_{1},\ldots,s_{u}\not\in\cup_{\ell=1}^{M}\overline{D_{\ell}}$. Then the
various steps in the proof of the lemma do not change, neither in statement
nor in proof. To conclude we note that
$r(\cup_{\ell=1}^{M}\overline{D_{\ell}})$ is closed, and since $w_{0}\not\in
r(\cup_{\ell=1}^{M}\overline{D_{\ell}})$, there is a neighborhood of $w_{0}$
not intersecting $r(\cup_{\ell=1}^{M}\overline{D_{\ell}})$. The analyticity
claim follows from the formula for $F$.

∎

######  Definition 4.6.

We will denote by $A(r,w_{0})$ a set containing $A(r)$ constructed as in the
previous corollary.

We now consider the case of a finite Blaschke product.

######  Lemma 4.7.

Finite Blaschke products satisfy (4.2).

###### Proof.

We consider $D_{1},\ldots,D_{L}$ open disks around the zeros of $r$, and
assume them inside the open unit disk and non-intersecting, and define $\rho$
as in(4.1). Then $0<\rho<1$ and inside
$B(0,1)\setminus\bigcup_{\ell=1}^{L}D_{\ell}$ we have $|r(z)|\geq\rho$ by the
maximum modulus principle since $|r(z)|=1$ on the unit circle. The inequality
$|r(z)|\geq\rho$ then still holds outside the unit disk since $r$ is a
Blaschke product and $|r(z)|>1$ outside the closed unit disk. So (4.2) also
holds for finite Blaschke products. ∎

We will make use of the previous two results for $w_{0}=1$ in particular in
the proofs of Theorem 8.4 and Theorem 10.6.

The case of vector-valued functions is presented in the next corollary.

######  Corollary 4.8.

Assume $f$ $\mathbb{C}^{p}$-valued. Then representation (4.3) becomes

(4.10) $f(z)=(Z_{r}(z)\otimes I_{p})F(r(z))$

where now $F$ is $\mathbb{C}^{pN}$-valued.

###### Proof.

The proof follows the one of Theorem 4.1 with a modification in Step 2: we
define the function $F=\begin{pmatrix}F_{1}&\cdots&F_{p}\end{pmatrix})^{t}$ by

(4.11) $F_{\ell}(z)=\frac{1}{2\pi i}\sum_{\ell=1}^{L}\int_{\partial
D_{\ell}}\frac{(I_{N}-sT)^{-1}b}{r(s)-z}\,f_{\ell}(s)ds,$

where $f=\begin{pmatrix}f_{1}&\cdots&f_{p}\end{pmatrix}^{t}$.

In fact, reasoning as above to prove Theorem 4.1, we have

$\begin{split}f(z)&=\begin{pmatrix}f_{1}\\\ \vdots\\\
f_{p}\end{pmatrix}=\frac{1}{2\pi i}\sum_{\ell=1}^{L}\int_{\partial
D_{\ell}}\frac{f(s)}{r(s)-r(z)}\frac{r(s)-r(z)}{s-z}ds\\\ &=\frac{1}{2\pi
i}\begin{pmatrix}G(I_{N}-zT)^{-1}&0&\cdots&0\\\ 0&G(I_{N}-zT)^{-1}&\cdots&0\\\
\vdots&\cdots&\ddots&\vdots\\\
0&\dots&&G(I_{N}-zT)^{-1}\end{pmatrix}\left(\sum_{\ell=1}^{L}\int_{\partial
D_{\ell}}\frac{(I_{N}-sT)^{-1}b}{r(s)-r(z)}f(s)ds\right)\\\
&=(G(I_{N}-zT)^{-1}\otimes I_{p})F(r(z))\end{split}$

where $F(z)$ is $\mathbb{C}^{pN}$-valued, by its definition, and is analytic
in $|z|<\rho$. ∎

These representation results allow to make connections with some non-standard
interpolation problem, as we now explain.

######  Example 4.9.

With $r$ as in Example 1.3 we have

(4.12)
$f(z)=\begin{pmatrix}1&\frac{1}{z}\end{pmatrix}F\left(z+\frac{1}{z}\right).$

######  Corollary 4.10.

Let $f$ be analytic in a domain symmetric with respect to the unit circle and
assume $f(z)=f(1/z)$. Then $f(z)=F(z+1/z)$.

###### Proof.

We write

$f(z)=\begin{pmatrix}1&\frac{1}{z}\end{pmatrix}F\left(z+\frac{1}{z}\right)$

and

$f(1/z)=\begin{pmatrix}1&z\end{pmatrix}F\left(z+\frac{1}{z}\right)$

It follows that with $F=\begin{pmatrix}F_{1}\\\ F_{2}\end{pmatrix}$ we have
$F_{2}\equiv 0$ and hence the result. ∎

Considering the power series expansion of $f$ centered at the origin, a more
direct proof would be to prove first by induction that there exists a sequence
of polynomials $P_{1},P_{2},\ldots$ such that

$z^{n}+\frac{1}{z^{n}}=P_{n}\left(z+\frac{1}{z}\right),\quad n=1,2,\ldots$

Representation (4.12) should allow to solve interpolation problem with the
interpolation data invariant under the map $z\mapsto 1/z$. As a sample problem
consider the following:

######  Problem 4.11.

Given $w_{1},\ldots,w_{N}$ be $N$ points in the complex plane, and two sets of
complex numbers $\\{a_{1},\ldots,a_{N}\\}$ and $\\{b_{1},\ldots,b_{N}\\}$,
find the set of functions $f$ analytic in neighborhoods of the points $w_{i}$
and $1/w_{i}$, $i=1,\ldots,N$ and such that

$\begin{split}f(w_{i})&=a_{i},\\\ f(1/w_{i})&=b_{i},\quad
i=1,\ldots,N.\end{split}$

Representation (4.3) allows to transform this problem into the following
tangential interpolation problem with tangential constraints

$\begin{split}\begin{pmatrix}1&w_{i}\end{pmatrix}F(W_{i})&=a_{i},\\\
\begin{pmatrix}1&1/w_{i}\end{pmatrix}F(W_{i})&=b_{i}\quad
i=1,\ldots,N.\end{split}$

where $W_{i}=w_{i}+\frac{1}{w_{i}}$. We will not pursue this direction of
research in the present work. The next example is related to multipoint
interpolation; see [16, 20]. More generally, for given $w_{1},\ldots,w_{N}$
such that $r(w_{n})=\alpha$, $n=1,\ldots,N$, the above strategy suggests how
to solve multipoint interpolation problems; the case where $r$ is a Blaschke
product was considered in [16].

######  Example 4.12.

Given $w_{1},\ldots,w_{N},c_{1},\ldots,c_{N},\gamma\in\mathbb{C}$, describe
the set of functions analytic in a neighborhood of $w_{1},\ldots,w_{N}$ and
such that

(4.13) $\sum_{n=1}^{N}c_{n}f(w_{n})=\gamma.$

Using the representation (4.3) we rewrite (4.13) as

$\begin{split}\sum_{n=1}^{N}c_{n}f(w_{n})&=\gamma\\\ &\iff\\\
\sum_{n=1}^{N}c_{n}Z_{r}(w_{n})F(0)&=\gamma\\\ &\iff\\\ cF(0)&=\gamma,\quad
c=\sum_{n=1}^{N}c_{n}Z_{r}(w_{n}),\end{split}$

which is a one-point tangential interpolation problem with solutions
$F\in\mathfrak{H}(k)$ (if one can multiply on the left and stay in
$\mathfrak{H}(k)$) given by

(4.14)
$F(z)=k(z,0)k(0,0)^{-1}\frac{c^{*}}{cc^{*}}\gamma+\left(I-\frac{c^{*}c}{cc^{*}}\right)G(z),\quad
G\in\mathfrak{H}(k)$

so that

$f(z)=Z_{r}(z)\left(k(z,0)k(0,0)^{-1}\frac{c^{*}}{cc^{*}}\gamma+\left(I-\frac{c^{*}c}{cc^{*}}\right)G(z)\right).$

Note that the decomposition (4.14) is orthogonal. Let $d\in\mathbb{C}^{N}$. We
have:

$\begin{split}\langle\left(I-\frac{c^{*}c}{cc^{*}}\right)G(z),k(z,0)k(0,0)^{-1}\frac{c^{*}}{cc^{*}}d\gamma\rangle_{\mathfrak{H}(k)}&=\overline{\gamma}d^{*}ck(0,0)^{-1}k(0,0)\left(I-\frac{c^{*}c}{cc^{*}}\right)G(0)\\\
&=\overline{\gamma}d^{*}c\left(I-\frac{c^{*}c}{cc^{*}}\right)G(0)\\\
&=0\end{split}$

since $c\left(I-\frac{c^{*}c}{cc^{*}}\right)=0$.

We note that such multipoint constraints are considered in numerical analysis
to fix degrees of freedom; see [45, p. 1494].

######  Theorem 4.13.

Let $\mathfrak{H}(\mathbb{K})$ be a reproducing kernel Hilbert space of
$\mathbb{C}^{p}$-valued functions of the form (4.10), with reproducing kernel
$\mathbb{K}$. Assume that the corresponding space of functions $F$, with norm

(4.15) $\|F\|=\|f\|$

is a reproducing kernel Hilbert space $\mathfrak{H}(K)$ of functions analytic
in $B(0,\epsilon)$ for some $\epsilon>0$ of the origin. Then $\mathbb{K}$ and
$K$ are related by

(4.16) $\mathbb{K}(z,w)=(Z_{r}(z)\otimes I_{p})K(r(z),r(w))(Z_{r}(w)\otimes
I_{p})^{*}.$

###### Proof.

Consider the space of functions of the form $f(z)=(Z_{r}(z)\otimes
I_{p})F(r(z))$, where $F\in\mathfrak{H}(K)$. By Corollary 2.13 with
$\alpha=0$, we have

$f(z)\equiv 0\quad\Longrightarrow F(z)\equiv 0,\quad z\in B(0,\epsilon).$

It follows from Theorem 2.12 that the definition (4.15) makes sense and the
reproducing kernel is then as in (4.16). ∎

We conclude this section with the following remark and a proposition. It may
be of interest to look at functions $r$ of the form $r(z)=(r_{0}(z))^{m}$,
where $r_{0}$ is itself rational and $m\in\mathbb{N}$. The following
proposition connects the corresponding state spaces $\mathfrak{L}(r)$ and
$\mathfrak{L}(r_{0})$.

######  Proposition 4.14.

Let $r$ be a rational function and let $r_{0}(z)=(r(z))^{m}$ with
$m\in\mathbb{N}$. Let $\mathfrak{M}(r_{0})$ to be the linear span of the
functions $1,\ldots,r_{0}^{m-1}$.

(4.17) $\mathfrak{L}(r)=(\mathfrak{M}(r_{0}))\otimes\mathfrak{L}(r_{0}),$

meaning that $t\in\mathfrak{L}(r)$ if and only if it can be written as

(4.18) $t(z)=\sum_{u=0}^{m-1}t_{u}(z)r_{0}^{u}(z),\quad
t_{0},\ldots,t_{m-1}\in\mathfrak{(}r_{0}).$

###### Proof.

The result follows from

$\frac{r_{0}^{m}(z)-r_{0}^{m}(a)}{z-a}=\sum_{u=0}^{m-1}r_{0}^{u}(z)\frac{r_{0}(z)-r_{0}(a)}{z-a},$

where $z,a$ are points of analyticity of $r$. ∎

We conclude this section with a formula:

######  Proposition 4.15.

Let $F$ be given by (4.4), extended to the domain $A(r)$ (see (4.5)). Then,
for every $\alpha\in A(r)$

(4.19) $R_{\alpha}F(z)=\frac{1}{2\pi i}\sum_{\ell=1}^{L}\int_{\partial
D_{\ell}}\frac{(I_{N}-sT)^{-1}bf(s)}{(r(s)-\alpha)(r(s)-z)}ds.$

###### Proof.

This follows from

$\frac{1}{r(s)-z)}-\frac{1}{r(s)-\alpha}=\frac{z-\alpha}{(r(s)-\alpha)(r(s)-z)}.$

∎

##  5\. Cuntz relations

Representations of the Cuntz algebra (see [30]) play an important role in
analysis, wavelets and signal processing, in particular in the theory of
filter banks; see [28, 38, 39]. Connections between the Cuntz relations and
reproducing kernel spaces were considered in [15, 16]. In the present section
we show how the Cuntz relations are related to the present setting.

Let in (4.10)

$F=\begin{pmatrix}F_{1}\\\ \vdots\\\ F_{N}\end{pmatrix}$

where $F_{1},\ldots,F_{N}$ are $\mathbb{C}^{p}$-valued. We thus have

(5.1) $f(z)=\sum_{n=1}^{N}e_{n}(z)F_{n}(r(z)).$

######  Theorem 5.1.

If $K$ in (4.16) is block diagonal,

$K={\rm diag}\,(k_{1},\ldots,k_{N})$

with possibly different positive definite $\mathbb{C}^{p\times p}$-valued
kernels $k_{1},\ldots,k_{N}$, the formulas

$(T_{n}f)(z)=F_{n}(z),\quad n=1,\ldots,N,$

define bounded operators
$T_{n}\in\mathbf{L}(\mathfrak{H}(\mathbb{K}),\mathfrak{H}(k_{n}))$,
$n=1,\ldots,N$, with adjoint operators

$(T_{n}^{*}g)(z)=e_{n}(z)g(r(z))$

and $T_{1}^{*},\ldots,T_{N}^{*}$ satisfy the Cuntz relations.

###### Proof.

The operators $T_{1},\ldots,T_{N}$ are well defined in view of Theorem 2.12.
They are bounded by definition of the norms in (4.15) since

$\|F\|^{2}_{\mathfrak{H}(K)}=\sum_{n=1}^{N}\|F_{N}\|^{2}_{\mathfrak{H}(k_{n})}.$

To compute the adjoint operators we note that

(5.2)
$T_{n}(\mathbb{K}(\cdot,w)d)=k_{n}(\cdot,r(w))\overline{e_{n}(w)}d_{n},\quad{\rm
with}\quad d=\begin{pmatrix}d_{1}\\\ \vdots\\\ d_{N}\end{pmatrix},\quad
d_{n}\in\mathbb{C}^{p}.$

Then it follows from a classical result on weighted composition operators that

$(T_{n}^{*}g)(z)=e_{n}(z)g(r(z)),\quad n=1,\ldots,N.$

Furthermore we have from (5.1) that

$\sum_{n=1}^{N}T_{n}^{*}T_{n}=I_{\mathfrak{H}(\mathbb{K})}.$

Finally for $n,m\in\left\\{1,\ldots,N\right\\}$, we have

$\begin{split}(T_{n}T_{m}^{*}g)(z)&=(T_{n}(e_{m}g\circ r))(z)\\\
&=\begin{cases}\,g(z),\,\,n=m\\\ \,0,\,\,\,\,\hskip
14.22636ptn\not=m,\end{cases}\end{split}$

since the function $e_{m}(z)g(r(z))$ has unique decomposition along the direct
sum corresponding to (5.1). ∎

In the case where $k_{1}=\cdots=k_{N}=k$ it is of interest to see when it
holds that

$k(z,w)=(Z_{r}(z)\otimes I_{p})k(r(z),r(w))(Z_{r}(w)\otimes I_{p})^{*},$

i.e.

(5.3) $k(z,w)=(\sum_{n=1}^{N}e_{n}(z)\overline{e_{n}(w)})k(r(z),r(w)).$

Such equations, and their connections with the Cuntz relations, have been
studied in [15].

##  6\. $R_{\alpha}^{(r)}$-invariant subspaces

Recall that $Z_{r}(z)$ has been defined by (2.16). We first consider the
finite dimensional case.

######  Theorem 6.1.

A finite dimensional vector space $\mathfrak{M}$ of $\mathbb{C}^{m}$-valued
functions is $R^{(r)}_{\alpha}$-invariant if and only if it is spanned by the
columns of a matrix of the form

(6.1) $\mathsf{M}(z)=(Z_{r}(z)\otimes I_{m})C(A-r(z)B)^{-1}$

where $C\in\mathbb{C}^{mN\times M}$ and $A,B\in\mathbb{C}^{M\times M}$
satisfying $\det(A+\alpha B)\not=0$ for some integer $M$. Furthermore, $M$ can
be chosen to be the dimension of $\mathfrak{M}$, and we can always assume that
the matrices $A$ and $B$ satisfy $A+\alpha B=I_{M}$.

For a fixed choice of $Z_{r}(z)$, the representation (6.1) is unique up to
similarity matrix $S$, meaning

(6.2) $\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle CS$ (6.3)
$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle S^{-1}AS$ (6.4)
$\displaystyle B_{1}$ $\displaystyle=$ $\displaystyle S^{-1}BS.$

###### Proof.

Let $M$ denote the dimension of $\mathfrak{M}$, and let $\mathsf{M}(z)$ be a
$m\times M$-valued function with columns made from a basis of $\mathfrak{M}$.
There exists $T\in\mathbb{C}^{M\times M}$ such that

$R^{(r)}_{\alpha}\mathsf{M}=\mathsf{M}T$

that is,

$\frac{\mathsf{M}(z)}{r(z)-\alpha}-\sum_{n=1}^{N}\frac{\mathsf{M}(w_{n})}{r^{\prime}(w_{n})(z-w_{n})}=G(z)T,$

and so

(6.5)
$\mathsf{M}(z)(I_{M}-(r(z)-\alpha)T)=\sum_{n=1}^{N}\frac{\mathsf{M}(w_{n})}{r^{\prime}(w_{n})}\frac{r(z)-\alpha}{z-w_{n}}.$

The functions

$\frac{r(z)-\alpha}{z-w_{n}}=\frac{r(z)-r(w_{n})}{z-w_{n}}$

belong to $\mathfrak{L}(r)$. So with a preassigned basis $e_{1},\ldots,e_{N}$
and $Z_{r}$ as in (2.16) we can write

(6.6) $\mathsf{M}(z)=(Z_{r}(z)\otimes I_{m})C(I_{M}-(r(z)-\alpha)T)^{-1}$

where $C\in\mathbb{C}^{mN\times M}$.

Conversely, let $\mathsf{M}$ be of the form (6.1), where
$C\in\mathbb{C}^{mN\times M}$ and $A,B\in\mathbb{C}^{M\times M}$ satisfying
$\det(A+\alpha B)\not=0$. We can write:

$\begin{split}(R^{(r)}_{\alpha}\mathsf{M})(z)&=\frac{(Z_{r}(z)\otimes
I_{m})C(A-r(z)B)^{-1}}{r(z)-\alpha}-\sum_{n=1}^{N}\frac{(Z_{r}(w_{n})\otimes
I_{m})C(A-r(w_{n})B)^{-1}}{r^{\prime}(w_{n})(z-w_{n})}\\\ &\\\
&=\frac{(Z_{r}(z)\otimes
I_{m})C(A-r(z)B)^{-1}}{r(z)-\alpha}-\left(\sum_{n=1}^{N}\frac{(Z_{r}(w_{n})\otimes
I_{m})}{r^{\prime}(w_{n})(z-w_{n})}\right)C(A-\alpha B)^{-1}\\\ &\\\
&=(Z_{r}(z)\otimes I_{m})\frac{C(A-r(z)B)^{-1}-C(A-\alpha
B)^{-1}}{r(z)-\alpha}\\\ &\\\ &=(Z_{r}(z)\otimes
I_{m})C(A-r(z)B)^{-1}B(A-\alpha B)^{-1}\\\ &\\\ &=\mathsf{M}(z)B(A-\alpha
B)^{-1},\end{split}$

where we have used (2.25) to go from the second line to the third one.

One can choose $M={\rm dim}\,\mathfrak{M}$ by taking the columns of $M(z)$
linearly independent. Finally, the last remark follows from replacing $C,A$
and $B$ by $C(A+\alpha B)^{-1}$, $A(A+\alpha B)^{-1}$ and $B(A+\alpha B)^{-1}$
respectively.

Assume that the space has two representations as the span of

$(Z_{r}(z)\otimes I_{m})C(A-r(z)B)^{-1}B(A-\alpha B)^{-1}\quad{\rm
and}\quad(Z_{r}(z)\otimes I_{m})C_{1}(A_{1}-r(z)B_{1})^{-1},$

with the properties that $C,C_{1}\in\mathbb{C}^{m\times N}$,
$A,B,A_{1},B_{1}\in\mathbb{C}^{N\times N}$, and such that $A+\alpha
B=A_{1}+\alpha B_{1}=I_{M}$. There exists an invertible matrix $S$ such that

$(Z_{r}(z)\otimes I_{m})C(A-r(z)B)^{-1}B(A-\alpha B)^{-1}=(Z_{r}(z)\otimes
I_{m})C_{1}(A_{1}-r(z)B_{1})^{-1}S.$

Using Corollary 2.15 it is sufficient to compare

$C(A-r(z)B)^{-1}B(A-\alpha B)^{-1}=C_{1}(A_{1}-r(z)B_{1})^{-1}S,$

i.e.

$C(I-(e-\alpha)B)^{-1}=C_{1}(I-(e-\alpha)B_{1})^{-1}S$

with $e\in\mathbb{C}$ and $e-\alpha$ small enough. But this is then classical
result from linear system theory. ∎

The case where $\mathfrak{M}$ has dimension $1$ will correspond to
eigenfunctions, and we get:

######  Corollary 6.2.

Eigenvectors of the operator $R^{(r)}_{\alpha}$ are exactly the functions of
the form

(6.7) $f(z)=\frac{Z_{r}(z)}{a-r(z)b}$

with corresponding eigenvalue $b$, where $a,b\in\mathbb{C}$ are such that
$a-\alpha b\not=0$.

Furthermore:

######  Corollary 6.3.

A finite dimensional $R^{(r)}_{\alpha}$-invariant reproducing kernel
Pontryagin space has a kernel of the form

(6.8) $K(z,w)=(Z_{r}(z)\otimes
I_{m})C(A-r(z)B)^{-1}P^{-1}(A-r(w)B)^{-*}C^{*}(Z_{r}(w)\otimes I_{m})^{*},$

for some Hermitian invertible matrix $P$.

###### Proof.

This is a direct consequence of the formula for the reproducing kernel of a
finite dimensional space. ∎

An important special case of (6.8), where

$C(A-r(z)B)^{-1}P^{-1}(A-r(w)B)^{-*}C^{*}={\rm
diag}\,(K(z,w),K(z,w),\ldots,K(z,w)).$

In the next results we connect properties of $P$ to the structure of the
kernel.

######  Theorem 6.4.

Assume $(C,A,B)$ minimal. Then the kernel (6.8) is of the form

(6.9) $(Z_{r}(z)\otimes
I_{m})\frac{J-\Theta(r(z))\Theta(r(w))^{*}}{1-r(z)\overline{r(w)}}(Z_{r}(w)\otimes
I_{m})^{*}$

if and only if $P$ solves the equation

(6.10) $A^{*}PA-B^{*}PB=C^{*}JC.$

###### Proof.

This follows from [13, (5.12)] or [12, Theorem 2.1, p. 37] with $a(z)=1$ and
$b(z)=r(z)$ in the notation of these papers. ∎

We refer to [3, Lemma 2.2, p. 415 and (2.4) p. 416] for the following result;
a proof is given for completeness.

######  Theorem 6.5.

Let $\mathfrak{M}$ be a space of finite dimension $d$ of
$\mathbb{C}^{p}$-valued functions analytic in some open set $\Omega$ and
assume that there is a point $a_{0}\in\Omega$ such that the linear span of the
vectors $f(a_{0})$ with $f$ varying in $\mathfrak{M}$ is equal to
$\mathbb{C}^{p}$. Then, $\mathfrak{M}$ satisfies for every $a\in\Omega$ (at
the possible exception of $a$ in a zero set) the condition

(6.11)
$f\in\mathfrak{M},\,\,and\,\,\,f(a)=0\quad\Longrightarrow\frac{f(z)}{z-a}\in\mathfrak{M},$

if and only if it is spanned by the columns of a matrix of the form

(6.12) $E(z)C(A-zB)^{-1}.$

where $E$ is $\mathbb{C}^{p\times p}$-valued analytic in $\Omega$ and such
that $\det E(z)\not\equiv 0$, and where $A,B$ and $C$ are matrices of
appropriate size and such that $\det(A-zB)\not\equiv 0$. Furthermore, $A$ and
$B$ can be chosen such that $A-a_{0}B=I_{d}$.

###### Proof.

We proceed in a number of steps.

STEP 1: Construction of the function $E(z)$.

The space $\mathbb{C}^{p}$ is spanned by the vectors $f(a_{0})$, and so
contains a basis made of such vectors, say $f_{1}(a_{0}),\ldots,f_{p}(a_{0})$,
where $f_{1},\ldots,f_{p}\in\mathfrak{M}$. The function $E(z)$ with columns
$f_{1}(z),\ldots,f_{p}(z)$ has a non-identically vanishing determinant since
$E(a_{0})$ is invertible by construction.

STEP 2: Construction of a basis for $\mathfrak{M}$.

Let $E(z)$ be a $\mathbb{C}^{p\times p}$-valued function $E(z)$ as in the
previous step, and let $d$ denote the dimension of $\mathfrak{M}$, Let $F(z)$
be a $\mathbb{C}^{p\times d}$-valued function, whose columns form a basis of
$\mathfrak{M}$. For every $c\in\mathbb{C}^{p}$ the function $z\mapsto
F(z)c-E(z)E^{-1}(a_{0})F(a_{0})c$ belongs to $\mathfrak{M}$ and vanishes at
$a_{0}$. By hypothesis (6.11), the function

$z\mapsto\frac{F(z)c-E(z)E^{-1}(a_{0})F(a_{0})c}{z-a_{0}}$

also belongs to $\mathfrak{M}$. It follows that there exists
$G\in\mathbb{C}^{d\times d}$ such that

$\frac{F(z)-E(z)E(a_{0})^{-1}F(a_{0})}{z-a_{0}}=F(z)G,$

and hence

$F(z)=E(z)E(a_{0})^{-1}F(a_{0})(I_{N}-(z-a_{0})G)^{-1},$

which is of the form (6.12) with $C=E(a_{0})^{-1}F(a_{0})$, $A=I_{d}+a_{0}G$
and $B=a_{0}G$.

STEP 3: Converse statement:

Conversely, let $F(z)$ be of the form (6.12) and let $a$ be such that $\det
E(a)\not=0$. We have for $c\in\mathbb{C}^{p}$ that $F(a)c=0$ if and only if
$C(A-aB)^{-1}c=0$. We can write

$\begin{split}\frac{F(z)c}{z-a}&=\frac{E(z)C\left((A-zB)^{-1}-(A-aB)^{-1}\right)c}{z-a}\\\
&=E(z)C\left((A-zB)^{-1}\left\\{\frac{A-aB-A+zB}{z-a}\right\\}(A-aB)^{-1}\right)c\\\
&=E(z)C(A-zB)^{-1}B(A-aB)^{-1}c\end{split}$

and so the function $z\mapsto\frac{F(z)c}{z-a}\in\mathfrak{M}$. ∎

We can now prove the following result, which characterizes the counterpart of
classical finite dimensional de Branges spaces in our setting.

######  Theorem 6.6.

Let $r$ be a real rational function satisfying the hypothesis of Theorem 4.1.
Under the notation of the previous theorem, let $\mathfrak{M}$ be a space of
finite dimension $d$ of $\mathbb{C}^{p}$-valued functions analytic in some
open set $\Omega$ and assume that there is a point $a_{0}\in\Omega$ such that
the linear span of the vectors $f(a_{0})$ with $f$ varying in $\mathfrak{M}$.
Assume furthermore that $\mathfrak{M}$ has the following property: Let
$\alpha\in\mathbb{C}$ for which the equation $r(z)=\alpha$ has $N$ distinct
solutions $w_{1},\ldots,w_{N}$, and such that

(6.13)
$f(w_{n})=0,\,\,\,\,n=1,\ldots,N\,\,\Longrightarrow\,\,z\mapsto\frac{f(z)}{r(z)-\alpha}\,\in\mathfrak{M}.$

Then $\mathfrak{M}$ has reproducing kernel of the form

$(Z_{r}(z)\otimes
I_{p})\frac{E_{+}(r(z))JE_{+}(r(w))^{*}-E_{-}(r(z))JE_{-}(r(w))^{*}}{-i(r(z)-\overline{r(w)})}(Z_{r}(w)\otimes
I_{p})^{*},$

if and only if $P$ satisfies (6.10).

###### Proof.

We write the elements of the space in the form (4.10), $f(z)=(Z_{r}(z)\otimes
I_{p})F(r(z))$, where $F$ is analytic in $|z|<\rho$ where $\rho$ is
independent of $F\in\mathfrak{M}$; see Theorem 4.1. To proceed, we divide the
argument in a number of steps.

STEP 1: The space $\mathfrak{N}$ of functions $F(r(z))$ is finite dimensional.

By Corollary 2.13 the function $F$ is uniquely determined by $f$, and so
$\mathfrak{N}$ is finite dimensional of dimension $N$ since $\mathfrak{M}$ is
of this type.

We denote by $M(r(z))$ a matrix-valued functions whose columns are a basis of
this space. We define an inner product via

$[(Z_{r}(z)\otimes I_{p})M(r(z))\xi,(Z_{r}(z)\otimes
I_{p})M(r(z))\eta]=\eta^{*}P\xi,\quad\eta,\xi\in\mathbb{C}^{N},$

where $P$ is Hermitian and invertible.

STEP 2: The space of functions $M(r(z))\xi$ satisfies the invariance condition
(6.11).

Indeed, by assumption, $z\mapsto(Z_{r}(z)\otimes
I_{p})\frac{M(r(z))}{r(z)-\alpha}\in\mathfrak{M}$ since
$M(r(w_{n}))=M(\alpha)=0$.

and hence $M(z)$ is of the form (6.12). Applying Theorem 6.5 we see that the
reproducing of $\mathfrak{M}$ is therefore equal to

$(Z_{r}(z)\otimes
I_{p})E(r(z))C(A-r(z)B)^{-1}P^{-1}(A-r(w)B)^{-*}C^{*}(Z_{r}(w)^{*}\otimes
I_{p})$

and the proof is concluded by using Theorem 6.10. ∎

We now turn to the infinite dimensional case, and consider a Hilbert space
$\mathfrak{H}$ of $\mathbb{C}^{m}$-valued function analytic in some open set
$\Omega$, and $R^{(r)}_{\alpha}$-invariant. We assume $\mathfrak{H}$
separable. We choose an orthonormal basis $b_{1}(z),...$ of $\mathfrak{H}$ and
consider the unitary map $T$

$T(c_{1},...)=\sum_{j=1}^{\infty}c_{j}b_{j}$

from $\ell^{2}(\mathbb{N},\mathbb{C}^{m})$ onto $\mathfrak{H}$. Since the
space $\mathfrak{H}$ is assumed $R^{(r)}_{\alpha}$-invariant, for every
$c\in\ell^{2}(\mathbb{N},\mathbb{C}^{m})$ we have $R^{(r)}_{\alpha}Tc=Td$ for
some uniquely defined $d$ in $\ell^{2}(\mathbb{N},\mathbb{C}^{m})$. The map
$c\mapsto d$ is linear and bounded since

$||R^{(r)}Tc||_{\mathfrak{H}}\leq K||Tc||_{\mathfrak{H}},$

and

$||R^{(r)}Tc||_{\mathfrak{H}}=||Td||_{\mathfrak{H}},$

implying

$||Td||_{\mathfrak{H}}\leq K||Tc||_{\mathfrak{H}}.$

We write $d=Mc$, where $M:\,l^{2}\to l^{2}$ is bounded since

$||Td||_{\mathfrak{H}}=||d||_{\ell^{2}(\mathbb{N},\mathbb{C}^{m})},$

and

$||Tc||_{\mathfrak{H}}=||c||_{\ell^{2}(\mathbb{N},\mathbb{C}^{m})}.$

So, we obtain

$R^{(r)}_{\alpha}T\,c=T\,M\,c,\,\,\,\forall
c\in\ell^{2}(\mathbb{N},\mathbb{C}^{m}).$

Hence,

$\frac{(Tc)(z)}{r(z)-\alpha}-\sum_{j=1}^{N}\frac{(Tc)(w_{j})}{r^{\prime}(w_{j})(z-w_{j})}=(T\,M\,c)(z).$

So,

$(Tc)(z)-(r(z)-\alpha)(T\,M\,c)(z)=\sum_{j=1}^{N}\frac{(Tc)(w_{j})}{r^{\prime}(w_{j})}\frac{r(z)-\alpha}{z-w_{j}}.$

The argument is then the same as in the finite dimensional case.

##  7\. Operator models

In this section we assume that the kernel $K(z,w)$ in (4.16) is diagonal, in
the form

$K={\rm diag}\,(k,\ldots,k)$

where $k(z,w)$ is $\mathbb{C}^{p\times p}$-valued and positive definite.
Recall that $r$ is a rational function of degree $N$, with a pole at infinity,
and hence of the form $r(z)=p(z)/q(z)$, with ${\rm deg}\,p=N$ and ${\rm
deg}\,q<N$.

######  Proposition 7.1.

Let $\alpha\in\Omega(r)$ and consider $R^{(r)}_{\alpha}$ with domain of
definition the space function analytic in neighborhoods of the points
$w_{1},\ldots,w_{N}$. Then it holds that

(7.1) $\ker R^{(r)}_{\alpha}=\mathfrak{L}(r)\otimes\mathbb{C}^{p}$

###### Proof.

We assume $p=1$ to lighten the notation. Assume that $f\in\ker
R^{(r)}_{\alpha}$. Then,

$f(z)=\sum_{n=1}^{N}\frac{f(w_{n})}{r^{\prime}(w_{n})}\frac{r(z)-r(w_{n})}{z-w_{n}}\in\mathfrak{L}(r)$

where $r(w_{n})=\alpha$ and since the function

$z\mapsto\frac{r(z)-r(\beta)}{z-\beta}\in\mathfrak{L}(r)$

for every complex number which is not a pole of $r$. The functions $R_{w}r$
form a basis of $\mathfrak{L}(r)$ for any set of $N$ pairwise different
points, and so the kernel of $R^{(r)}_{\alpha}$ is exactly equal to
$\mathfrak{L}(r)$. More precisely, any function $f$ in $\mathfrak{L}(r)$ is of
the form

$f(z)=\sum_{n=1}^{N}\frac{c_{n}}{r^{\prime}(w_{n})}\frac{r(z)-r(w_{n})}{z-w_{n}}\in\mathfrak{L}(r),\quad
c_{1},\ldots,c_{N}\in\mathbb{C},$

and is such that $f(w_{m})=c_{m}$, $m=1,\ldots,N$, as is seen by plugging
$z=w_{m}$ in the above equation. So $f\in\ker R^{(r)}_{\alpha}$. ∎

We now use (2.26) to relate operators associated to the operators $R_{\alpha}$
and $R_{\alpha}^{(r)}$. The existence of the operators $\mathsf{A}$ and
$\mathsf{A}_{r}$ in the next theorem follow from [47, Theorem 4.10, p. 137]).

######  Theorem 7.2.

Assuming that the operators $R^{(r)}_{\alpha}$ and $R_{\alpha}$ are bounded in
some Hilbert space $\mathfrak{H}$, and have trivial kernels in $\mathcal{H}$
for every $\alpha\in\Omega(r)$. Then, there exist two possibly unbounded
operators $\mathsf{A}$ and $\mathsf{A}_{r}$ such that

(7.2) $R^{(r)}_{\alpha}f=(\mathsf{A}_{r}-\alpha I)^{-1}\quad{\it and}\quad
R_{\alpha}F=(\mathsf{A}-\alpha I)^{-1}.$

They are related by

$((\mathsf{A}_{r}-\alpha
I)^{-1}f)(z)=\sum_{n=1}^{N}e_{n}(z)((\mathsf{A}-\alpha I)^{-1}F_{n})(r(z))$

when $f$ is decomposed as (5.1),

$f(z)=\sum_{n=1}^{N}e_{n}(z)F_{n}(r(z)).$

######  Theorem 7.3.

In the notation of the previous theorem, assume that

(7.3) $\mathfrak{H}\cap\mathfrak{L}(r)=\left\\{0\right\\}.$

Then

(7.4) ${\rm Dom}\,\mathsf{A}_{r}=\left\\{f\in\mathfrak{H}\,\,\text{for which
there exists $h_{f}\in\mathfrak{L}(r)$ such that
}r(z)f(z)+h_{f}(z)\in\mathfrak{H}\right\\}$

and then

(7.5) $(\mathsf{A}_{r}f)(z)=r(z)f(z)+h_{f}(z).$

###### Proof.

By (7.3) we have $\ker R^{(r)}_{\alpha}=\left\\{0\right\\}$ for some (and
hence all) $\alpha$, and so the operator $\mathsf{A}_{r}$ exists; see [47,
Theorem 4.10, p. 137]). ∎

We remark that, with $(\mathsf{A}f)(z)=zf(z)+d_{f}$ we have

$\begin{split}\sum_{n=1}^{N}e_{n}(z)((A-\alpha
I)^{-1}f_{n})(r(z)&=\sum_{n=1}^{N}e_{n}(z)(r(z)f_{n}(r(z))+d_{f_{n}})\\\
&=r(z)\left(\sum)_{n=1}^{N}e_{n}(z)f_{n}(r(z))\right)+\sum_{n=1}^{N}e_{n}(z)d_{f_{n}}\\\
&=r(z)Z_{r}(z)F(r(z))+h_{f}(z)\\\ &=r(z)f(z)+h_{f}(z)\end{split}$

with $f(z)=Z_{r}(z)F(r(z))$ and $h_{f}(z)=\sum_{n=1}^{N}e_{n}(z)d_{f_{n}}$.

##  8\. $\mathfrak{P}(\Theta)$ spaces

We now study the counterparts of reproducing kernel Pontryagin spaces with
reproducing kernel of the form

(8.1) $\frac{J-\Theta(z)J\Theta(w)^{*}}{-i(z-\overline{w})},\quad{\rm
or}\quad\frac{J-\Theta(z)J\Theta(w)^{*}}{1-z\overline{w}},$

in the present setting. The first formula corresponds to the line case, while
the second corresponds to the circle case. These kernels give models for close
to self-adjoint and close to unitary operators in the classical case; see [29,
43] for the latter.

We begin with the line case. In the statement of the next theorem,
$J\in\mathbb{R}^{p\times p}$ is a signature matrix and $X(J,r)$ is the
corresponding associated symmetric matrix (see (3.5)). In the statements we
consider functions $r$ which satisfy the conditions of Theorem 4.1. This
includes in particular rational functions regular at the origin and having a
pole at infinity (see Lemma 4.4) and finite Blaschke products. Recall that
$\Omega(r)$ is defined by (2.6), that $A(r)$ is defined in (4.5) and that we
assume functions extended to $r^{-1}(A(r))$; see Remark 4.3.

######  Theorem 8.1.

Let $r$ be a real rational function satisfying the hypothesis of Theorem 4.1.
Let $\mathfrak{P}$ be a reproducing kernel Pontryagin space of
$\mathbb{C}^{p}$-valued functions analytic in $r^{-1}(A(r))$. Then, for each
$\alpha,\beta\in\Omega(r)\cap A(r)$ such that $\alpha\not=\overline{\beta}$,
it holds that

(8.2)
$\left[R^{(r)}_{\alpha}f,g\right]_{\mathfrak{P}}-\left[f,R_{\beta}^{(r)}g\right]_{\mathfrak{P}}-(\alpha-\overline{\beta})\left[R^{(r)}_{\alpha}f,R^{(r)}_{\beta}g\right]_{\mathfrak{P}}=i(\alpha-\overline{\beta})\sum_{i,j=1}^{N}\frac{g(v_{j})^{*}}{\overline{r^{\prime}(v_{j})}}J\frac{f(u_{i})}{r^{\prime}(u_{i})}\frac{1}{u_{i}-\overline{v_{j}}}$

(with $r(u_{i})=\alpha$ and $r(v_{j})=\beta$, $i,j=1,\ldots,N$) if and only if
the reproducing kernel of $\mathfrak{P}$ is of the form

(8.3) $(Z_{r}(z)\otimes
I_{p})\frac{X(J,r)^{-1}-\Theta(r(z))X(J,r)\Theta(r(w))^{*}}{-i(r(z)-\overline{r(w)})}(Z_{r}(w)\otimes
I_{p})^{*}$

where $\Theta$ is a $\mathbb{C}^{Np\times Np}$-valued function analytic in a
neighborhood of the origin.

###### Proof.

Using (2.26) we can write with $f=Z_{r}F(r)$ and $g=Z_{r}G(r)$, where $F$ and
$G$ are analytic in $A(r)$. By Corollary 2.13 applied to every connected
component of $r^{-1}(A(r))$ we see that $F$ and $G$ are uniquely determined,
and we can define an inner product by

(8.4) $[F,G]=[f,g]_{\mathfrak{P}}.$

Fo $\alpha,\beta\in A(r)$ we have

(8.5)
$\begin{split}\left[R^{(r)}_{\alpha}f,g\right]_{\mathfrak{P}}-\left[f,R_{\beta}^{(r)}g\right]_{\mathfrak{P}}-(\alpha-\overline{\beta})\left[R^{(r)}_{\alpha}f,R^{(r)}_{\beta}g\right]_{\mathfrak{P}}&=\\\
&=\left[R_{\alpha}F,G\right]-\left[F,R_{\beta}G\right]-(\alpha-\overline{\beta})\left[R_{\alpha}F,R_{\beta}G\right],\quad\end{split}$

Using (2.25) we can write

$\begin{split}(\alpha-\overline{\beta})\sum_{i,j=1}^{N}\frac{g(v_{j})^{*}}{\overline{r^{\prime}(v_{j})}}J\frac{f(u_{i})}{r^{\prime}(u_{i})}\frac{1}{u_{i}-\overline{v_{j}}}&=G(\beta)^{*}\left(\sum_{i,j=1}^{N}\frac{Z_{r}(v_{j})^{*}}{\overline{r^{\prime}(v_{j})}}J\frac{Z_{r}(u_{i})}{r^{\prime}(u_{i})}\frac{\alpha-\overline{\beta}}{u_{i}-\overline{v_{j}}}\right)F(\alpha)\\\
&=G(\beta)^{*}\left(\sum_{j=1}^{N}\frac{Z_{r}(v_{j})^{*}}{\overline{r^{\prime}(v_{j})}}J\left(\sum_{i=1}^{N}\frac{Z_{r}(u_{i})}{r^{\prime}(u_{i})}\frac{\overline{\beta}-\alpha}{\overline{v_{j}}-u_{i}}\right)\right)F(\alpha)\\\
&=G(\beta)^{*}\left(\sum_{j=1}^{N}\frac{Z_{r}(v_{j})^{*}}{\overline{r^{\prime}(v_{j})}}J\left(\sum_{i=1}^{N}\frac{Z_{r}(u_{i})}{r^{\prime}(u_{i})}\frac{\overline{\beta}-\alpha}{\overline{v_{j}}-u_{i}}\right)\right)F(\alpha)\\\
&=G(\beta)^{*}\left(\sum_{j=1}^{N}\frac{Z_{r}(v_{j})^{*}}{\overline{r^{\prime}(v_{j})}}J\left(Z_{r}(\overline{v_{j}})\frac{\overline{\beta}-\alpha}{\overline{\beta}-\alpha}\right)\right)F(\alpha)\\\
&=G(\beta)^{*}XF(\alpha),\end{split}$

where $X(J,r)$ is defined by (3.5) (and is Hermitian, invertible, and
independent of $\alpha$ and $\beta$). We are therefore back to the classical
case,

$\left[R_{\alpha}F,G\right]-\left[F,R_{\beta}G\right]-(\alpha-\overline{\beta})\left[R_{\alpha}F,R_{\beta}G\right]=iG(\beta)^{*}X(J,r)F(\alpha)$

Write $X(J,r)=Y^{*}J_{0}Y$, where $J_{0}$ is a real signature matrix. The
space $\mathfrak{P}_{1}$ of functions of the form $YF$ with inner product

(8.6) $[YF,YG]_{Y}=[F,G]$

satisfies

$\left[R_{\alpha}YF,YG\right]-\left[YF,R_{\beta}YG\right]-(\alpha-\overline{\beta})\left[R_{\alpha}YF,R_{\beta}YG\right]=iG(\beta)^{*}Y^{*}J_{0}YF(\alpha)$

and are analytic across the real line (since the functions $F$ are analytic in
a disk $|z|<\rho$ by definition of $A(r)$). This analyticity condition allows
us to apply the Pontryagin space version of de Branges’ result (see [11,
Theorem 6.10], and see [24, Theorem III, p. 447], [10, Theorem 2.3 p. 598] for
the Hilbert space version) to assert that its reproducing kernel is of the
form

(8.7) $\frac{J_{0}-\Theta_{0}(z)J_{0}\Theta_{0}(w)^{*}}{-i(z-\overline{w})}.$

from which the result follows. More precisely, we see that the index of the
reproducing kernel Pontryagin space with reproducing kernel (8.7) is equal to
$\nu_{-}(\mathfrak{P})$. ∎

######  Notation 8.2.

We will set $\mathfrak{P}=\mathfrak{P}(\Theta)$ and define

(8.8) $\nu_{-}(\Theta)=\nu_{-}(\mathfrak{P}),$

where the latter is the index of negativity of the Pontryagin space
$\mathfrak{P}$.

We now consider an inclusion and factorization result related to the above
spaces. In the statement, $\nu_{-}(\Theta_{1}^{-1}\Theta)$ is the number of
negative squares of the kernel

$\frac{X(J,r)-\Theta_{1}(r(z))^{-1}\Theta(r(z))X(J,r)\Theta(r(w))^{*}\Theta_{1}(r(z))^{-*}}{-i(r(z)-\overline{r(w)})}$

(note that the numerator involves only $X(J,r)$ and not its inverse).

######  Theorem 8.3.

In the notation of the previous theorem, let $\mathfrak{P}_{1}$ be a subspace
of $\mathfrak{P}$ which is $R_{\alpha}^{(r)}$-invariant, and non-degenerate.
Then $\mathfrak{P}_{1}$ has a reproducing kernel of the form

$(Z_{r}(z)\otimes
I_{p})\frac{X(J,r)^{-1}-\Theta_{1}(r(z))X(J,r)\Theta_{1}(r(w))^{*}}{-i(r(z)-\overline{r(w)})}(Z_{r}(w)\otimes
I_{p})^{*}$

and we have

$\nu_{-}(\Theta)=\nu_{-}(\Theta_{1})+\nu_{-}(\Theta_{1}^{-1}\Theta)$

and the direct and orthogonal decomposition

(8.9) $\mathfrak{P}(\Theta)=\mathfrak{P}(\Theta_{1})\oplus(Z_{r}\otimes
I_{p})\Theta_{1}(r)\mathfrak{P}(\Theta_{1}^{-1}\Theta).$

###### Proof.

Thanks to Corollary 2.15 the claim reduces to the decomposition

$\begin{split}\frac{X(J,r)^{-1}-\Theta(r(z))X(J,r)\Theta(r(w))^{*}}{-i(r(z)-\overline{r(w)})}&=\frac{X(J,r)^{-1}-\Theta_{1}(r(z))X(J,r)\Theta_{1}(r(w))^{*}}{-i(r(z)-\overline{r(w)})}+\\\
&+\Theta_{1}(r(z))\frac{X(J,r)-\Theta_{1}(r(z))^{-1}\Theta(r(z))X(J,r)\Theta(r(w))^{*}\Theta_{1}(r(z))^{-*}}{-i(r(z)-\overline{r(w)})}\Theta_{1}(r(z))^{*},\end{split}$

and the result follows then from classical results in reproducing kernel
Pontryagin spaces. ∎

We now turn to the circle case. The argument is similar to the one in the
previous theorem, but we need to use Corollary 4.5 at some point $w_{0}$ of
modulus $1$ not a pole of $r$. Recall that $A(r,w_{0})$ was introduced in
Definition 4.6.

######  Theorem 8.4.

Let $r$ be a rational function satisfying the hypothesis of Theorem 4.1, let
$w_{0}$ be a point on the unit circle not a pole of $r$. Let $\mathfrak{P}$ be
a reproducing kernel Pontryagin space of $\mathbb{C}^{p}$-valued functions
analytic in $r^{-1}(A(w_{0},r))$. Then, for each $\alpha,\beta\in\omega(r)\cap
A(w_{0},r)$ it holds that

(8.10)
$\left[f,g\right]_{\mathfrak{P}}+\overline{\beta}\left[f,R_{\beta}^{(r)}g\right]_{\mathfrak{P}}+\alpha\left[R_{\alpha}^{(r)}f,g\right]_{\mathfrak{P}}-(1-\alpha\overline{\beta})\left[R^{(r)}_{\alpha}f,R^{(r)}_{\beta}g\right]_{\mathfrak{P}}=(\alpha-\overline{\beta})\sum_{i,j=1}^{N}\frac{g(v_{j})^{*}}{\overline{r^{\prime}(v_{j})}}J\frac{f(u_{i})}{r^{\prime}(u_{i})}\frac{1}{u_{i}-\overline{v_{j}}}$

(with $r(u_{i})=\alpha$ and $r(v_{j})=\beta$, $i,j=1,\ldots,N$) if and only if
the reproducing kernel of $\mathfrak{P}$ is of the form

(8.11) $(Z_{r}(z)\otimes
I_{p})\frac{X(J,r)^{-1}-\Theta(r(z))X(J,r)\Theta(r(w))^{*}}{1-r(z)\overline{r(w)}}Y^{-*}(Z_{r}(w)\otimes
I_{s})^{*},$

where $\Theta$ is a $\mathbb{C}^{Np\times Np}$-valued function analytic in a
neighborhood of the origin.

###### Proof.

The functions $F$ are now analytic across the unit circle at $w_{0}$ and one
can use the Pontryagin space version of de Branges’ theorem for the circle
case; see [11, Theorem 6.10] for the latter. ∎

##  9\. $\mathfrak{P}(S)$ spaces

It is interesting to note that the Fock space restricted to the unit disk is a
special case of a $\mathfrak{P}(S)$ space; see [8].

######  Theorem 9.1.

Let $r$ be a real rational function satisfying the hypothesis of Theorem 4.1.
Let $\mathfrak{P}$ be a reproducing kernel Pontryagin space of
$\mathbb{C}^{p}$-valued functions analytic in
$r^{-1}(A(r))\setminus\mathbb{R}$, and let $\alpha\in\Omega(r)$. Assume that
the inequality

(9.1)
$\left[R^{(r)}_{\alpha}f,R^{(r)}_{\alpha}f\right]_{\mathfrak{P}}\leq\left[(I+\alpha
R^{(r)}_{\alpha})f,(I+\alpha
R^{(r)}_{\alpha})f\right]_{\mathfrak{P}}-(\alpha-\overline{\alpha})\sum_{i,j=1}^{N}\frac{f(u_{j})^{*}}{\overline{r^{\prime}(u_{j})}}J\frac{f(u_{i})}{r^{\prime}(u_{i})}\frac{1}{u_{i}-\overline{u_{j}}}$

holds for some preassigned signature matrix $J\in\mathbb{R}^{p\times p}$. Then
there exist a Pontryagin space $\mathfrak{C}$ such that
$\nu_{-}(\mathfrak{C})=\nu_{-}(X)$, and a
$\mathbf{L}(\mathfrak{C},\mathbb{C}^{Np}_{J_{0}})$-valued function analytic in
a neighborhood of $\alpha$ such that the reproducing kernel of $\mathfrak{P}$
is of the form

(9.2) $K(z,w)=(Z_{r}(z)\otimes
I_{p})\frac{X(J,r)^{-1}-S(r(z))S(r(w))^{*}}{1-r(z)\overline{r(w)}}(Z_{r}(w)\otimes
I_{p})^{*}.$

###### Proof.

The strategy is the same as for Theorem 8.1. We write elements of
$\mathfrak{P}$ in the form $f=(Z_{r}\otimes I_{p})F(r))$, where $F$ is
uniquely determined and analytic in $A(r)$, and with induced inner product
(8.4). Inequality (9.1) becomes

(9.3) $[R_{\alpha}F,R_{\alpha}F]\leq[(I+\alpha R_{\alpha})F,(I+\alpha
R_{\alpha})F]-F(\alpha)^{*}XF(\alpha)$

(see , [11, (6.9)]). As above, write $X=Y^{*}J_{0}Y$. The space of functions
of the form $YF$ with norm the norm of $F$ has reproducing kernel of the form

$\frac{J_{0}-S(r(z))S(r(w)^{*}}{1-r(z)\overline{r(w)}}$

where $\mathfrak{C}$ is a Pontryagin space with negative index
$\nu_{-}(J_{0})$ and where $S$ is a
$\mathbf{L}(\mathfrak{C},\mathbb{C}_{J_{0}}^{Np})$-valued function analytic in
a neighborhood of $\alpha$. This is [9, Theorem 3.1.2, p. 85] for the case
$\alpha=0$ and [18, Theorem 3.4, p. 32] for the general case. ∎

Classical $\mathcal{H}(s)$ spaces are generalizations of the orthogonal
complements of Beurling-Lax invariant subspaces. This suggests that the
counterpart of the Hardy space in the current setting is the reproducing
kernel space with reproducing kernel

(9.4) $(Z_{r}(z)\otimes
I_{p})\frac{X(J,r)^{-1}}{1-r(z)\overline{r(w)}}(Z_{r}(w)\otimes I_{p})^{*}.$

Let us set $p=1$ for simplicity. The expression

$Z_{r}(z)X(J,r)^{-1}Z_{r}(w)$

is a reproducing kernel formula.

######  Remark 9.2.

The result [18, Theorem 3.4, p. 32] will also allow to get the half-line
counterpart of the above theorem, using the inequality (see [11, Theorem 6.7
(6.10)])

$\frac{\alpha-\overline{\alpha}}{2\pi
i}[R_{\alpha}F,R_{\alpha}F]\leq\frac{[R_{\alpha}F,F]-[F,R_{\alpha}F]}{2\pi
i}-F(\alpha)^{*}F(\alpha)$

rather than (9.3), and the relation

$\frac{\alpha-\overline{\alpha}}{2\pi
i}[R^{(r)}_{\alpha}F,R^{(r)}_{\alpha}F]\leq\frac{[R^{(r)}_{\alpha}F,F]-[F,R^{(r)}_{\alpha}F]}{2\pi
i}-(\alpha-\overline{\alpha})\sum_{i,j=1}^{N}\frac{f(u_{j})^{*}}{\overline{r^{\prime}(u_{j})}}J\frac{f(u_{i})}{r^{\prime}(u_{i})}\frac{1}{u_{i}-\overline{u_{j}}}$

rather than (9.1).

##  10\. $\mathfrak{P}(E_{+},E_{-})$ spaces

We now present two structure theorems. The proof of the first one is a matrix-
valued version of [25, Théorème 23, p. 59]. The proof of the second theorem is
an adaptation to the circle case of de Branges’ proof, and appears in [3] and
[7, Theorem 6.1, p. 173]. Both proofs are repeated here for completeness.

######  Theorem 10.1.

(The line case) Let $\mathfrak{P}$ be a reproducing kernel Pontryagin space of
$\mathbb{C}^{p}$ functions analytic in an open set $\Omega$, symmetric with
respect to the real line and intersecting it, and assume that there is a point
$\alpha\not=0\in\Omega$ such that both $K(\alpha,\alpha)$ and
$K(\overline{\alpha},\overline{\alpha})$ are invertible. Assume that the
following condition holds for every $F,G\in\mathfrak{P}$: if $F(v)=G(v)=0$ for
$v\not=\overline{v}\in\Omega$ then the functions
$z\mapsto\frac{z-\overline{v}}{z-v}F(z)$ and
$z\mapsto\frac{z-\overline{v}}{z-v}G(z)$ belongs to $\mathfrak{P}$ and

(10.1)
$\left[F(z),G(z)\right]=\left[\frac{z-\overline{v}}{z-v}F(z),\frac{z-\overline{v}}{z-v}G(z)\right].$

Then, there exist $\mathbb{C}^{p\times p}$-valued functions $E_{+}$ and
$E_{-}$ and a signature matrix $J\in\mathbb{c}^{p\times p}$ such that the
reproducing kernel of $\mathfrak{P}$ is of the form

(10.2)
$K_{\mathfrak{P}}(z,w)=\frac{E_{+}(z)JE_{+}(w)^{*}-E_{-}(z)JE_{-}(w)^{*}}{-i(z-\overline{w})},$

where $J$ is a signature matrix with same signature as $K(\alpha,\alpha)$.

###### Proof.

Without loss of generality, and by maybe replacing $\alpha$ by its conjugate,
we will assume ${\rm Im}\,\alpha>0$. Let $c\in\mathbb{C}^{p}$. The function

(10.3)
$z\mapsto\left(K(z,w)-K(z,\alpha)K(\alpha,\alpha)^{-1}K(\alpha,w)\right)c$

belongs to $\mathfrak{P}$ and vanishes at the point $\alpha$. So the function

$z\mapsto\frac{z-\overline{\alpha}}{z-\alpha}\left(K(z,w)-K(z,\alpha)K(\alpha,\alpha)^{-1}K(\alpha,w)\right)c$

belongs to $\mathfrak{P}$. Let $F\in\mathfrak{P}$ which vanishes at
$\overline{\alpha}$. Using (10.1) with $v=\overline{\alpha}$ we can write

$\begin{split}\left[F(z),\frac{z-\overline{\alpha}}{z-\alpha}\left(K(z,w)-K(z,\alpha)K(\alpha,\alpha)^{-1}K(\alpha,w)\right)c\right]&=\\\
&=\left[F(z)\frac{z-\alpha}{z-\overline{\alpha}},\frac{z-\alpha}{z-\overline{\alpha}}\frac{z-\overline{\alpha}}{z-\alpha}\left(K(z,w)-K(z,\alpha)K(\alpha,\alpha)^{-1}K(\alpha,w)\right)c\right]\\\
&=\left[F(z)\frac{z-\alpha}{z-\overline{\alpha}},\left(K(z,w)-K(z,\alpha)K(\alpha,\alpha)^{-1}K(\alpha,w)\right)c\right]\\\
&=\left[F(z)\frac{z-\alpha}{z-\overline{\alpha}},\left(K(z,w)-K(z,\alpha)K(\alpha,\alpha)^{-1}K(\alpha,w)\right)c\right]\\\
&=c^{*}F(w)\frac{w-\alpha}{w-\overline{\alpha}}\\\
&=\left[F(z),\left(K(z,w)-K(z,\overline{\alpha})K(\overline{\alpha},\overline{\alpha})^{-1}K(\overline{\alpha},w)\right)c\frac{\overline{w}-\overline{\alpha}}{\overline{w}-\alpha}\right].\end{split}$

This chain of equalities holds for all $F$ vanishing at $\overline{\alpha}$
and hence we get

$\begin{split}\frac{z-\overline{\alpha}}{z-\alpha}\left(K(z,w)-K(z,\alpha)K(\alpha,\alpha)^{-1}K(\alpha,w)\right)&=\\\
&=\left(K(z,w)-K(z,\overline{\alpha})K(\overline{\alpha},\overline{\alpha})^{-1}K(\overline{\alpha},w)\right)\frac{\overline{w}-\overline{\alpha}}{\overline{w}-\alpha}.\end{split}$

We can thus write:

$\begin{split}\left(\frac{z-\overline{\alpha}}{z-\alpha}-\frac{\overline{w}-\overline{\alpha}}{\overline{w}-\alpha}\right)K(z,w)&=\\\
&=\frac{z-\overline{\alpha}}{z-\alpha}K(z,\alpha)K(\alpha,\alpha)^{-1}K(\alpha,w)-\frac{\overline{w}-\overline{\alpha}}{\overline{w}-\alpha}K(z,\overline{\alpha})K(\overline{\alpha},\overline{\alpha})^{-1}K(\overline{\alpha},w).\end{split}$

Therefore:

(10.4)
$K(z,w)=\frac{F_{+}(z)K(\alpha,\alpha)^{-1}F_{+}(w)^{*}-F_{-}(z)K(\overline{\alpha},\overline{\alpha})^{-1}F_{-}(w)^{*}}{(-i(\alpha-\overline{\alpha}))(-i(z-\overline{w}))},$

with

(10.5) $\displaystyle F_{+}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{-i(\alpha-\overline{\alpha})}}(z-\overline{\alpha})K(z,\alpha)$
(10.6) $\displaystyle F_{-}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{-i(\alpha-\overline{\alpha})}}(z-{\alpha})K(z,\overline{\alpha}).$

By analyticity, $\det K(z,\alpha)\not\equiv 0$ and so taking a point on the
unit circle where $\det K(z,\alpha)\not=0$ and $\det
K(z,\overline{\alpha})\not=0$ we see that $K(\alpha,\alpha)$ and
$K(\overline{\alpha},\overline{\alpha})$ have the same signature, and we can
write with a common signature matrix $J$

$K(\alpha,\alpha)^{-1}=MJM^{*}\quad{\rm and}\quad
K(1/\overline{\alpha},1/\overline{\alpha})^{-1}=NJN^{*}.$

The result follows with $E_{+}=F_{+}M$ and $E_{-}=F_{-}N$.

∎

######  Theorem 10.2.

(The circle case) Let $\mathfrak{P}$ be a reproducing kernel Pontryagin space
of $\mathbb{C}^{p}$ functions analytic in an open set $\Omega$, symmetric with
respect to the unit circle and intersecting it, and assume that there is a
point $\alpha\not=0\in\Omega$ such that both $K(\alpha,\alpha)$ and
$K(1/\overline{\alpha},1/\overline{\alpha})$ are invertible. Assume that the
following condition holds for every $F,G\in\mathfrak{P}$: if $F(v)=G(v)=0$ for
$v\in\Omega\setminus\mathbb{T}$ then the functions
$z\mapsto\frac{1-z\overline{v}}{z-v}F(z)$
$z\mapsto\frac{1-z\overline{v}}{z-v}G(z)$ belongs to belongs to $\mathfrak{P}$
and

(10.7)
$\left[F(z),G(z)\right]=[\frac{1-z\overline{v}}{z-v}F(z),\frac{1-z\overline{v}}{z-v}G(z)].$

Then, there exist $\mathbb{C}^{p\times p}$-valued functions $E_{+}$ and
$E_{-}$ and a signature matrix $J\in\mathbb{C}^{p\times p}$ such that the
reproducing kernel of $\mathfrak{P}$ is of the form

(10.8)
$K_{\mathfrak{P}}(z,w)=\frac{E_{+}(z)JE_{+}(w)^{*}-E_{-}(z)JE_{-}(w)^{*}}{1-z\overline{w}},$

where $J$ is a signature matrix with same signature as $K(\alpha,\alpha)$.

###### Proof.

Let $c\in\mathbb{C}^{p}$. The function (10.3) and vanishes at the point
$\alpha$. So the function

$z\mapsto\frac{1-z\overline{\alpha}}{z-\alpha}\left(K(z,w)-K(z,\alpha)K(\alpha,\alpha)^{-1}K(\alpha,w)\right)c\in\mathfrak{P}.$

Let $F\in\mathfrak{P}$ which vanishes at $1/\overline{\alpha}$. Using (10.7)
with $v=1/\overline{\alpha}$ we can write

$\begin{split}\left[F(z),\frac{1-z\overline{\alpha}}{z-\alpha}\left(K(z,w)-K(z,\alpha)K(\alpha,\alpha)^{-1}K(\alpha,w)\right)c\right]&=\\\
&=\left[F(z)\frac{1-z/\alpha}{z-1/\overline{\alpha}},\frac{1-z/\alpha}{z-1/\overline{\alpha}}\frac{1-z\overline{\alpha}}{z-\alpha}\left(K(z,w)-K(z,\alpha)K(\alpha,\alpha)^{-1}K(\alpha,w)\right)c\right]\\\
&=\left[F(z)\frac{1-z/\alpha}{z-1/\overline{\alpha}},\frac{\overline{\alpha}}{\alpha}\left(K(z,w)-K(z,\alpha)K(\alpha,\alpha)^{-1}K(\alpha,w)\right)c\right]\\\
&=\left[F(z)\frac{z-\alpha}{1-z\overline{\alpha}},\left(K(z,w)-K(z,\alpha)K(\alpha,\alpha)^{-1}K(\alpha,w)\right)c\right]\\\
&=c^{*}F(w)\frac{w-\alpha}{1-w\overline{\alpha}}\\\
&=\left[F(z),\left(K(z,w)-K(z,1/\overline{\alpha})K(1/\overline{\alpha},1/\overline{\alpha})^{-1}K(1/\overline{\alpha},w)\right)c\frac{\overline{w}-\overline{\alpha}}{1-\overline{w}\alpha}\right].\end{split}$

This chain of equalities holds for all $F$ vanishing at $1/\overline{\alpha}$
and hence we get

$\begin{split}\frac{1-z\overline{\alpha}}{z-\alpha}\left(K(z,w)-K(z,\alpha)K(\alpha,\alpha)^{-1}K(\alpha,w)\right)&=\\\
&=\left(K(z,w)-K(z,1/\overline{\alpha})K(1/\overline{\alpha},1/\overline{\alpha})^{-1}K(1/\overline{\alpha},w)\right)\frac{\overline{w}-\overline{\alpha}}{1-\overline{w}\alpha}.\end{split}$

Hence

$\begin{split}\left(\frac{1-z\overline{\alpha}}{z-\alpha}-\frac{\overline{w}-\overline{\alpha}}{1-\overline{w}\alpha}\right)K(z,w)&=\\\
&=\frac{1-z\overline{\alpha}}{z-\alpha}K(z,\alpha)K(\alpha,\alpha)^{-1}K(\alpha,w)-\frac{\overline{w}-\overline{\alpha}}{1-\overline{w}\alpha}K(z,1/\overline{\alpha})K(1/\overline{\alpha},1/\overline{\alpha})^{-1}K(1/\overline{\alpha},w),\end{split}$

so that

(10.9)
$K(z,w)=\frac{F_{+}(z)K(\alpha,\alpha)^{-1}F_{+}(w)^{*}-F_{-}(z)K(1/\overline{\alpha},1/\overline{\alpha})^{-1}F_{-}(w)^{*}}{(1-|\alpha|^{2})(1-z\overline{w})},$

with

(10.10) $\displaystyle F_{+}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1-|\alpha|^{2}}}(1-z\overline{\alpha})K(z,\alpha)$
(10.11) $\displaystyle F_{-}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1-|\alpha|^{2}}}(z-{\alpha})K(z,1/\overline{\alpha}).$

By analyticity, $\det K(z,\alpha)\not\equiv 0$ and so taking a point on the
unit circle where $\det K(z,\alpha)\not=0$ and $\det
K(z,1/\overline{\alpha})\not=0$ we see that $K(\alpha,\alpha)$ and
$K(1/\overline{\alpha},1/\overline{\alpha})$ have the same signature, and we
can write with a common signature matrix $J$

$K(\alpha,\alpha)^{-1}=MJM^{*}\quad{\rm and}\quad
K(1/\overline{\alpha},1/\overline{\alpha})^{-1}=NJN^{*}.$

The result follows with $E_{+}=F_{+}M$ and $E_{-}=F_{-}N$. ∎

In the case of a Hilbert space we have $J=I_{p}$ and the following proposition
holds:

######  Proposition 10.3.

$K(\alpha,\alpha)$ is invertible if and only if linear span of the functions
$f(\alpha)$ when $f$ runs through $\Omega$ is equal to $\mathbb{C}^{p}$.

###### Proof.

Assume that there is $c\in\mathbb{C}^{p}$ such that $K(\alpha,\alpha)c=0$
Then, the Cauchy-Schwarz inequality implies that

$\begin{split}|c^{*}f(\alpha)|^{2}&=|\langle f,K(\cdot,\alpha)\rangle|^{2}\\\
&\leq\|f\|^{2}\cdot(c^{*}K(\alpha,\alpha)c)\\\ &=0\end{split}$

and so $c$ is orthogonal to the said linear span.

Conversely, if there is such a $c$, the map $f\mapsto c^{*}f(\alpha)$ is
identically equal to $0$, and hence its Riesz representation corresponds to
the zero vector, i.e. $K(\cdot,\alpha)c\equiv 0$, and so in particular
$K(\alpha,\alpha)c=0$. ∎

In the case of a Pontryagin space, the second part of the proof is still
valid, since Riesz representation theorem holds in Pontryagin space, and we
have:

######  Corollary 10.4.

Assume that the linear span is not dense. Then $K(\alpha,\alpha)$ is not
invertible.

Next theorem further generalizes results in Section 6:

######  Theorem 10.5.

Let $r$ be a rational function satisfying the hypothesis of Theorem 4.1, and
let $A(r)$ be defined by (4.5). Let $\mathfrak{P}$ be a reproducing kernel
Pontryagin space of $\mathbb{C}^{p}$-valued functions analytic in
$r^{-1}(A(r))$ and having full rank, in the sense that there exist
$\alpha\not=\overline{\alpha}\in\Omega_{0}$ such that the matrices
$K(\alpha,\alpha)$ and $K(\overline{\alpha},\overline{\alpha})$ are
invertible. Assume that for $f,g\in\mathfrak{P}$ and $v\in\Omega$,

$f(w_{n})=g(w_{n})=0,\quad n=1,\ldots,N\,\,\Longrightarrow
f(z)\frac{r(z)+\overline{v}}{r(z)-v}\,\,and\,\,\,g(z)\frac{r(z)+\overline{v}}{r(z)-v}\,\,\in\mathfrak{P},$

where $w_{1},\ldots,w_{n}$ are the roots, assumed pairwise different, of
$r(w)=v$, and

(10.12)
$\left[f(z),g(z)\right]_{\mathfrak{P}}=\left[f(z)\frac{r(z)+\overline{\alpha}}{r(z)-\alpha},g(z)\frac{r(z)+\overline{\alpha}}{r(z)-\alpha}\right]_{\mathfrak{P}}.$

Then the reproducing kernel of $\mathfrak{P}$ is of the form

$K(z,w)=(Z_{r}(z)\otimes
I_{p})\frac{E_{+}(r(z))JE_{+}(r(w))^{*}-E_{-}(r(z))JE_{-}(r(w))^{*}}{-i(r(z)-\overline{r(w)})}(Z_{r}(w)\otimes
I_{p})^{*},$

where $J$ is a signature matrix with same signature as $K(\alpha,\alpha)$.

###### Proof.

Using the vector-valued version of Lemma 4.3 we write $f(z)=Z_{r}(z)F(r(z))$,
where $F$ is $\mathbb{C}^{Np}$-valued and analytic in $\Omega_{0}$. We
consider the space of functions

(10.13) $\mathfrak{M}=\left\\{F(z)\,:\,Z_{r}(z)F(r(z))\in\mathfrak{H},\,\,{\rm
with\,\,inner\,\,product}\,\,\ [F,G]=[f,g]_{\mathfrak{P}}\right\\}.$

By Corollary 2.13 the condition $f(w_{n})=0$, $n=1,\ldots,N$ implies that
$F(\alpha)=0$.
Equation (10.14) implies that

$\left[F(z),G(z)\right]=\left[F(z)\frac{z+\overline{v}}{z-v},G(z)\frac{z+\overline{v}}{z-v}\right].$

By Theorem 10.2 the reproducing kernel of the form (10.2) and hence the
result.

∎

In the case of the circle we have the following result; the proof is similar
and will be omitted.

######  Theorem 10.6.

Let $r$ be a rational function satisfying the hypothesis of Theorem 4.1, and
let $A(r)$ be defined by (4.5). Let $\mathfrak{P}$ be a reproducing kernel
Pontryagin space of $\mathbb{C}^{p}$-valued functions analytic in
$r^{-1}(A(r))$ and having full rank, in the sense that there exist
$\alpha\not=\overline{\alpha}\in\Omega_{0}$ such that the matrices
$K(\alpha,\alpha)$ and $K(1/\overline{\alpha},1/\overline{\alpha})$ are
invertible. Assume that for $f,g\in\mathfrak{P}$ and $v\in\Omega$

$f(w_{n})=g(w_{n})=0,\quad n=1,\ldots,N\,\,\Longrightarrow
f(z)\frac{1-r(z)\overline{\alpha}}{r(z)-\overline{\alpha}}\,\,and\,\,\,g(z)\frac{1-r(z)\overline{\alpha}}{r(z)-\alpha}\,\,\in\mathfrak{P},$

where $w_{1},\ldots,w_{n}$ are the roots, assumed pairwise different, of
$r(w)=v$, and

(10.14)
$\left[f(z),g(z)\right]_{\mathfrak{P}}=\left[f(z)\frac{1-r(z)\overline{v}}{r(z)-v},g(z)\frac{1-r(z)\overline{v}}{r(z)-v}\right]_{\mathfrak{P}}.$

Then the reproducing kernel of $\mathfrak{P}$ is of the form

$K(z,w)=(Z_{r}(z)\otimes
I_{p})\frac{E_{+}(r(z))JE_{+}(r(w))^{*}-E_{-}(r(z))JE_{-}(r(w))^{*}}{-i(r(z)-\overline{r(w)})}(Z_{r}(w)\otimes
I_{p})^{*},$

where $J$ is a signature matrix with same signature as $K(\alpha,\alpha)$.

The previous theorems are of special interest in the Hilbert space setting,
when the space $\mathfrak{M}$ (see (10.13)) consists of entire functions.

##  11\. $\mathfrak{L}({\mathsf{N}})$ spaces

We first recall:

######  Definition 11.1.

The function ${\mathsf{N}}$ analytic in the open set
$\Omega\subset\mathbb{C}$, symmetric with respect to the real line is called a
generalized Nevanlinna function if the kernel

$K_{\mathsf{N}}(z,w)=\frac{{\mathsf{N}}(z)-{\mathsf{N}}(w)^{*}}{z-\overline{w}}$

has a finite number of negative squares in $\Omega$.

The following result appears, for the scalar and positive setting, in [25,
Théorème 6, p. 13]; a first Pontryagin space version, with more restrictive
conditions and different proof, appears in [1]. Here we are treating the case
of negative squares, which corresponds to generalized Nevanlinna functions.
The latter class of functions has been introduced and thoroughly studied by
Krein and Langer; see e.g. [40, 41]. These functions are closely related to
the positive real functions and generalized positive real functions from
linear system theory; see e.g. [31].

######  Theorem 11.2.

Let $\mathfrak{P}$ be a reproducing kernel Pontryagin space of
$\mathbb{C}^{p}$-valued functions analytic in a set $\Omega$ symmetric with
respect to the real line and assume that $\mathfrak{P}$ is resolvent
invariant. Assume moreover that

(11.1)
$[F(z),(R_{\beta}G)(z)]_{\mathfrak{P}}-[(R_{\alpha}F)(z),F(z)]_{\mathfrak{P}}+(\alpha-\overline{\beta})[(R_{\alpha}F)(z),(R_{\beta}G)(z)]_{\mathfrak{P}}=0$

for every $\alpha,\beta\in\Omega$. Then there is a $\mathbb{C}^{p\times
p}$-valued function ${\mathsf{N}}(z)$ analytic in $\Omega$ and such that the
reproducing kernel of $\mathfrak{P}$ is

(11.2)
$K_{\mathsf{N}}(z,w)=\frac{{\mathsf{N}}(z)-{\mathsf{N}}(w)^{*}}{z-\overline{w}},\quad
z,w\in\Omega.$

###### Proof.

The core of the proof is as in [25, p. 13], and we adapt the arguments to the
case of negative squares:

STEP 1: The resolvent operators are bounded in $\mathfrak{P}$:

Indeed, they are closed in view of the reproducing kernel property, and
everywhere defined. Hence they are bounded since we are in a Pontryagin space.

STEP 2: The reproducing kernel $K(z,w)$ satisfies:

(11.3)
$(\overline{w}-z)K(z,w)=-(z-\overline{\alpha})K(z,\alpha)+(\overline{w}-\overline{\alpha})K(\overline{\alpha},w),\quad\alpha,z,w\in\Omega.$

As in [25, p. 13] this follows from setting $G(z)=K(z,w)d$ and
$\alpha=\overline{\beta}$ in (11.5). We then get for any $F\in\mathfrak{P}$

$\begin{split}\left[F(z),\frac{K(z,w)-K(\overline{\alpha},w)}{z-\overline{\alpha}}d\right]&=\left[\frac{F(z)-F({\alpha})}{z-{\alpha}},K(z,w)d\right]\\\
&=d^{*}\frac{F(w)-F(\alpha)}{w-{\alpha}}\\\
&=\left[F(z),\frac{K(z,w)-K(z,\alpha)}{\overline{w}-\overline{\alpha}}d\right]\end{split}$

and so

$\frac{K(z,w)-K(\overline{\alpha},w)}{z-\overline{\alpha}}=\frac{K(z,w)-K(z,\alpha)}{\overline{w}-\overline{\alpha}},$

from which (11.3) follows.

It follows from the previous step that

(11.4)
$K(z,w)=\frac{{\mathsf{N}}_{1}(z)-{\mathsf{N}}_{2}(w)^{*}}{z-\overline{w}}$

with ${\mathsf{N}}_{1}(z)=(z-\overline{\alpha})K(z,\alpha)$ and
${\mathsf{N}}_{2}(w)=(w-\alpha)K(w,\overline{\alpha})$.

STEP 3: It holds that
${\mathsf{N}}_{1}(z)-{\mathsf{N}}_{2}(w)=(\alpha-\overline{\alpha})K(\alpha,\alpha)$.

Indeed, from the equality $K(z,w)=K(w,z)^{*}$ we have

$\frac{{\mathsf{N}}_{1}(z)-{\mathsf{N}}_{2}(w)^{*}}{z-\overline{w}}=\left(\frac{{\mathsf{N}}_{1}(w)-{\mathsf{N}}_{2}(z)^{*}}{w-\overline{z}}\right)^{*},$

from which we get

${\mathsf{N}}_{1}(z)-{\mathsf{N}}_{2}(w)^{*}={\mathsf{N}}_{2}(z)-{\mathsf{N}}_{1}(w)^{*}$

and hence the result.

We conclude by taking (see [25, p. 14]),

${\mathsf{N}}(z)=\frac{(\alpha-\overline{\alpha})}{2}K(\alpha,\alpha)+(z-\overline{\alpha})K(z,\alpha).$

Then, $K(z,w)=\frac{{\mathsf{N}}(z)-{\mathsf{N}}(w)^{*}}{z-\overline{w}}$. ∎

######  Theorem 11.3.

Let $r$ be a rational function satisfying the hypothesis of Theorem 4.1, and
let $A(r)$ be defined by (4.5). Let $\mathfrak{P}$ be a reproducing kernel
Pontryagin space of $\mathbb{C}^{p}$-valued functions analytic in
$r^{-1}(A(r))$, and assume that $\mathfrak{P}$ is $R_{\alpha}^{(r)}$ resolvent
invariant. Assume moreover that

(11.5)
$[F(z),(R_{\beta}^{(r)}G)(z)]_{\mathfrak{P}}-[(R_{\alpha}^{(r)}F)(z),F(z)]_{\mathfrak{P}}+(\alpha-\overline{\beta})[(R_{\alpha}^{(r)}F)(z),(R_{\beta}^{(r)}G)(z)]_{\mathfrak{P}}=0.$

Then there is a $\mathbb{C}^{p\times p}$-valued function ${\mathsf{N}}(z)$
analytic in $\Omega$ and such that the reproducing kernel of $\mathfrak{P}$ is

(11.6)
$K(z,w)=Z_{r}(z)\frac{{\mathsf{N}}(r(z))-{\mathsf{N}}(r(w))^{*}}{r(z)-\overline{r(w)}}Z_{r}(w)^{*},\quad
z,w\in r^{-1}(A(r)).$

###### Proof.

As in earlier proofs in the paper, we using (2.26) and $f=Z_{r}F(r)$ and
$g=Z_{r}G(r)$; we consider the space of functions corresponding space of
functions $F$ with the inner product

$[F,G]=[f,g].$

We obtain by Theorem 11.2 a space $\mathfrak{L}({\mathsf{N}})$ and the result
follows. ∎

When ${\mathsf{N}}(z)\equiv i/2$ in the open upper half-plane, we get the
counterpart of the Hardy space

$\frac{Z_{r}(z)Z_{r}(w)^{*}}{-i(r(z)-\overline{r(w)})}=\sum_{n=1}^{N}\frac{e_{n}(z)\overline{e_{n}(w)}}{-i(r(z)-\overline{r(w)})},$

which is positive definite on the set of $w$ such that ${\rm Im}\,r(w)\geq 0$
and with associated reproducing kernel Hilbert space

$f(z)=\sum_{n=1}^{N}e_{n}(z)f_{n}(r(z)),\quad
f_{1},\ldots,f_{N}\in\mathbf{H}^{2}(\mathbb{C}_{+}).$

The new Hardy space reduces to the classical one when

$\sum_{n=1}^{N}e_{n}(z)\overline{e_{n}(w)}=\frac{-i(r(z)-\overline{r(w)})}{-i(z-\overline{w})}.$

This will happen if and only if $r$ satisfies the symmetry
$r(z)=\overline{r(\overline{z})}$. See [15] for the corresponding Cuntz type
decomposition.

Acknowledgements: It is a pleasure to thank Prof. Dan Volok, Kansas State
University, for help with Theorem 4.1.

## References

  * [1] D. Alpay.  Reproducing kernel Krein spaces of analytic functions and inverse scattering.  PhD thesis, Feinberg Graduate School, Weizmann Institute of Science, Israel, 1986.
  * [2] D. Alpay.  Algorithme de Schur, espaces à noyau reproduisant et théorie des systèmes, volume 6 of Panoramas et Synthèses [Panoramas and Syntheses].  Société Mathématique de France, Paris, 1998.
  * [3] D. Alpay.  A structure theorem for reproducing kernel Pontryagin spaces.  Journal of Computational and Applied Mathematics, 99:413–422, 1998.
  * [4] D. Alpay.  A complex analysis problem book.  Birkhäuser/Springer, Cham, 2016.  Second edition.
  * [5] D. Alpay, J. Ball, I. Gohberg, and L. Rodman.  Realization and factorization of rational matrix functions with symmetries, volume 47 of  Operator Theory: Advances and Applications , pages 1–60.  Birkhäuser Verlag, Basel, 1990.
  * [6] D. Alpay, V. Bolotnikov, A. Dijksma, and H.S.V. de Snoo.  On some operator colligations and associated reproducing kernel Pontryagin spaces.  J. Funct. Anal., 136:39–80, 1996.
  * [7] D. Alpay, V. Bolotnikov, and Ph. Loubaton.  On tangential ${H}_{2}$ interpolation with second order norm constraints.  Integral Equations and Operator Theory, 24:156–178, 1996.
  * [8] D. Alpay, F. Colombo, and I. Sabadini.  The Fock space as a De Branges-Rovnyak space.  Integral Equations Operator Theory, 91(6):Paper No. 51, 12, 2019\.
  * [9] D. Alpay, A. Dijksma, J. Rovnyak, and H. de Snoo.  Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, volume 96 of Operator theory: Advances and Applications.  Birkhäuser Verlag, Basel, 1997.
  * [10] D. Alpay and H. Dym.  Hilbert spaces of analytic functions, inverse scattering and operator models, I.  Integral Equation and Operator Theory, 7:589–641, 1984.
  * [11] D. Alpay and H. Dym.  On applications of reproducing kernel spaces to the Schur algorithm and rational $J$-unitary factorization.  In I. Gohberg, editor, I. Schur methods in operator theory and signal processing, volume 18 of Operator Theory: Advances and Applications, pages 89–159. Birkhäuser Verlag, Basel, 1986.
  * [12] D. Alpay and H. Dym.  On reproducing kernel spaces, the Schur algorithm, and interpolation in a general class of domains.  In Operator theory and complex analysis (Sapporo, 1991), volume 59 of Oper. Theory Adv. Appl., pages 30–77. Birkhäuser, Basel, 1992.
  * [13] D. Alpay and H. Dym.  On a new class of structured reproducing kernel Hilbert spaces.  J. Funct. Anal., 111:1–28, 1993.
  * [14] D. Alpay and H. Dym.  On a new class of realization formulas and their applications.  Linear Algebra Appl., 241/243:3–84, 1996.
  * [15] D. Alpay, P. Jorgensen, I. Lewkowicz, and I. Martziano.  Infinite product representations for kernels and iterations of functions.  In Recent advances in inverse scattering, Schur analysis and stochastic processes, volume 244 of Oper. Theory Adv. Appl., pages 67–87. Birkhäuser/Springer, Cham, 2015.
  * [16] D. Alpay, P. Jorgensen, I. Lewkowicz, and I. Marziano.  Representation formulas for Hardy space functions through the Cuntz relations and new interpolation problems.  In Xiaoping Shen and Ahmed Zayed, editors, Multiscale signal analysis and modeling, pages 161–182. Springer, 2013.
  * [17] D. Alpay, P. Jorgensen, I. Lewkowicz, and D. Volok.  A new realization of rational functions, with applications to linear combination interpolation, the Cuntz relations and kernel decompositions.  Complex Var. Elliptic Equ., 61(1):42–54, 2016.
  * [18] D. Alpay and I. Sabadini.  Beurling-Lax type theorems in the complex and quaternionic setting.  Linear Algebra Appl., 530:15–46, 2017.
  * [19] D. Alpay and V. Vinnikov.  Finite dimensional de Branges spaces on Riemann surfaces.  J. Funct. Anal., 189(2):283–324, 2002.
  * [20] D. Alpay and A. Yger.  About a non-standard interpolation problem.  Comput. Methods Funct. Theory, 19(1):97–115, 2019.
  * [21] J. Ball.  Models for noncontractions.  J. Math. Anal. Appl., 52:235–259, 1975.
  * [22] H. Bart, I. Gohberg, and M.A. Kaashoek.  Minimal factorization of matrix and operator functions, volume 1 of Operator Theory: Advances and Applications.  Birkhäuser Verlag, Basel, 1979.
  * [23] H. Bart, I. Gohberg, M.A. Kaashoek, and A.C.M. Ran.  A state space approach to canonical factorization with applications, volume 200 of Operator Theory: Advances and Applications.  Birkhäuser Verlag, Basel; Birkhäuser Verlag, Basel, 2010.  Linear Operators and Linear Systems.
  * [24] L. de Branges.  Some Hilbert spaces of analytic functions I.  Trans. Amer. Math. Soc., 106:445–468, 1963.
  * [25] L. de Branges.  Espaces Hilbertiens de fonctions entières.  Masson, Paris, 1972.
  * [26] L. de Branges and J. Rovnyak.  Canonical models in quantum scattering theory.  In C. Wilcox, editor, Perturbation theory and its applications in quantum mechanics, pages 295–392. Wiley, New York, 1966.
  * [27] L. de Branges and J. Rovnyak.  Square summable power series.  Holt, Rinehart and Winston, New York, 1966.
  * [28] Ola Bratteli and Palle E. T. Jorgensen.  Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale $N$.  Integral Equations Operator Theory, 28(4):382–443, 1997.
  * [29] M. S. Brodskiĭ.  Triangular and Jordan representations of linear operators.  American Mathematical Society, Providence, R.I., 1971.  Translated from the Russian by J. M. Danskin, Translations of Mathematical Monographs, Vol. 32.
  * [30] J. Cuntz.  Simple $C^{*}$-algebras generated by isometries.  Comm. Math. Phys., 57(2):173–185, 1977.
  * [31] B. Dickinson, Ph. Delsarte, Y. Genin, and Y. Kamp.  Minimal realizations of pseudo–positive and pseudo–bounded rational matrices.  IEEE Transactions on Circuits and Systems, 32:603–605, 1985.
  * [32] H. Dym.  ${J}$–contractive matrix functions, reproducing kernel Hilbert spaces and interpolation.  Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1989.
  * [33] H. Dym and H.P. McKean.  Gaussian processes, function theory and the inverse spectral problem.  Academic Press, 1976.
  * [34] E. Fricain and J. Mashreghi.  The theory of $\mathcal{H}$($b$) spaces. Vol. 1, volume 20 of New Mathematical Monographs.  Cambridge University Press, Cambridge, 2016.
  * [35] E. Fricain and J. Mashreghi.  The theory of $\mathcal{H}(b)$ spaces. Vol. 2, volume 21 of New Mathematical Monographs.  Cambridge University Press, Cambridge, 2016.
  * [36] P.A. Fuhrmann.  A polynomial approach to linear algebra.  Universitext. Springer-Verlag, New York, 1996.
  * [37] R.A. Horn and C.R. Johnson.  Topics in matrix analysis.  Cambridge University Press, Cambridge, 1994.  Corrected reprint of the 1991 original.
  * [38] Palle E. T. Jorgensen.  Analysis and probability: wavelets, signals, fractals, volume 234 of Graduate Texts in Mathematics.  Springer, New York, 2006.
  * [39] Palle E. T. Jorgensen.  Certain representations of the Cuntz relations, and a question on wavelets decompositions.  In Operator theory, operator algebras, and applications, volume 414 of Contemp. Math., pages 165–188. Amer. Math. Soc., Providence, RI, 2006.
  * [40] M.G. Kreĭn and H. Langer.  Über die verallgemeinerten Resolventen und die charakteristische Funktion eines isometrischen Operators im Raume ${\Pi_{k}}$.  In Hilbert space operators and operator algebras (Proc. Int. Conf. Tihany, 1970), pages 353–399. North–Holland, Amsterdam, 1972.  Colloquia Math. Soc. János Bolyai.
  * [41] M.G. Kreĭn and H. Langer.  Über die ${Q}$-Funktion eines $\pi$-hermiteschen Operators im Raume $\pi_{\kappa}$.  Acta Sci. Math. (Szeged), 34:191–230, 1973.
  * [42] H. Lev-Ari and T. Kailath.  Triangular factorisation of structured hermitian matrices.  In I. Gohberg, editor, I. Schur methods in operator theory and signal processing, volume 18 of Operator theory: Advances and Applications , pages 301–324. Birkhäuser Verlag, Basel, 1986.
  * [43] M.S. Livs̆ic.  Operator colligations, waves, open systems.  Transl. Math. Monog. AMS. AMS, Providence, R.I., 1973.
  * [44] Adolf A. Nudelman.  Some generalizations of classical interpolation problems.  In Operator extensions, interpolation of functions and related topics (Timişoara, 1992), volume 61 of Oper. Theory Adv. Appl., pages 171–188. Birkhäuser, Basel, 1993.
  * [45] T. Poggio and F. Girosi.  Networks for approximation and learning.  Proceedings of the IEEE, 78:1481–1497, 1990.
  * [46] J. Rovnyak.  Characterization of spaces ${H(M)}$.  Unpublished paper, $1968$. Available at the URL http://www.people.virginia.edu/ jlr5m/home.html.
  * [47] M. Stone.  Linear transformations in Hilbert space.  American Mathematical Society, Providence, Rhode Island, 1932\.