# Generalized de Branges-Rovnyak spaces

Alexandru Aleman and Frej Dahlin

###### Abstract

Given the reproducing kernel $k$ of the Hilbert space ${{\mathcal{H}}_{k}}$ we
study spaces ${{\mathcal{H}}_{k}}(b)$ whose reproducing kernel has the form
$k(1-bb^{*})$, where $b$ is a row-contraction on ${{\mathcal{H}}_{k}}$. In
terms of reproducing kernels this it the most far-reaching generalization of
the classical de Branges-Rovnyaks spaces, as well as their very recent
generalization to several variables. This includes the so called sub-Bergman
spaces [30] in one or several variables. We study some general properties of
${{\mathcal{H}}_{k}}(b)$ e.g. when the inclusion map into
${{\mathcal{H}}_{k}}$ is compact. Our main result provides a model for
${{\mathcal{H}}_{k}}(b)$ reminiscent of the Sz.-Nagy-Foiaş model for
contractions (see also [5]). As an application we obtain sufficient conditions
for the containment and density of the linear span of
$\\{k_{w}:w\in{\mathcal{X}}\\}$ in ${{\mathcal{H}}_{k}}(b)$. In the standard
cases this reduces to containment and density of polynomials. These methods
resolve a very recent conjecture [13] regarding polynomial approximation in
spaces with kernel
$\frac{(1-b(z)b(w)^{*})^{m}}{(1-z\overline{w})^{\beta}},1\leq
m<\beta,m\in{\mathbb{N}}$.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Reproducing kernels
    2. 2.2 Some unitarily invariant kernels on the unit ball of ${\mathbb{C}}^{d}$
    3. 2.3 Weighted Bergman spaces on a domain of ${\mathbb{C}}^{d}$
  3. 3 Basic properties of generalized de Branges-Rovnyak spaces
    1. 3.1 Examples via operator inequalities
    2. 3.2 Compact embedding
    3. 3.3 A model for ${{\mathcal{H}}_{k}}(b)$
  4. 4 Approximation results
    1. 4.1 The general approximation theorem
  5. 5 Weighted sub-Bergman spaces
    1. 5.1 Approximation by polynomials in standard weighted sub-Bergman spaces on the unit ball
  6. 6 Approximation by polynomials in higher order de Branges-Rovnyak spaces

##  1 Introduction

Motivated by models for contractions, de Branges and Rovnyak initiated in the
60’s [9, 10] the study of the Hilbert spaces with a reproducing kernel of the
form

(1) $k(z,w)=\frac{1-b(z)b(w)^{*}}{1-z\overline{w}},\quad z,w\in{\mathbb{D}},$

Here, as usual, ${\mathbb{D}}$ denotes the unit disc in the complex plane and
the operator-valued function
$b:{\mathbb{D}}\to{\mathcal{B}}({\mathbb{C}},l^{2})$ is analytic with operator
norm bounded by one in ${\mathbb{D}}$. These ideas have been considerably
developed by Sarason [25] who studied extensively these spaces in the case
when $b$ is a scalar-valued analytic function, and called them _sub-Hardy
spaces_ ${\mathcal{H}}(b)$. For example, he represented them as ranges of the
defect operators $(I-T_{b}T_{b}^{*})^{1/2}$ endowed with the norm making these
operators co-isometric. Here $T_{b}$ denotes the (analytic) Toeplitz operator
with symbol $b$ on the Hardy space $H^{2}$.
${\mathcal{H}}(b)$-spaces attracted a lot of attention during the years, a
good account on those developments can be found in the book by Fricain and
Mashreghi [12] and in the references therein. An important idea of Sarason
[25] reveals a kind of dichotomy regarding the structure of these spaces.
Roughly speaking, it turns out that the properties of ${\mathcal{H}}(b)$ are
either similar to those of a shift invariant subspace of $H^{2}$, or to those
of the orthogonal complement of such a subspace (model space). The first
alternative occurs when $b$ is not an extreme point of the unit ball of
$H^{\infty}$, the algebra of bounded analytic functions in ${\mathbb{D}}$,
while the second occurs when $b$ is an extreme point of that set. In fact, it
is proved in [25] that the following are equivalent:

  1. (i)

$b\in{\mathcal{H}}(b)$,

  2. (ii)

${\mathcal{H}}(b)$ is invariant under the shift operator defined by
$Sf(z)=zf(z)$,

  3. (iii)

the polynomials are contained in ${\mathcal{H}}(b)$ and are dense in the
space.

We shall refer to the above situation as Sarason’s dichotomy.
Zhu [30], [31] developed these ideas by considering Toeplitz operators $T_{b}$
acting on the Bergman space, i.e. the range of $(I-T_{b}T_{b}^{*})^{1/2}$ with
the appropriate norm. We should point out that these ranges are quite
different. For example, it turns out that
$\operatorname{Ran}(I-T_{b}T_{b}^{*})=\operatorname{Ran}(I-T_{b}^{*}T_{b})$,
which fails dramatically in the Hardy space context.
Such spaces are called _sub-Bergman_ spaces and by analogy to (1) they have
the reproducing kernel

(2) $k(z,w)=\frac{1-b(z)\overline{b(w)}}{(1-z\overline{w})^{2}},$

i.e. it is obtained by replacing in (1) the Szegő kernel by the Bergman
kernel. All of these spaces are invariant for the forward shift and contain
the polynomials. For our purposes it is important to note that in [11] Chu
proved the density of polynomials in all sub-Bergman spaces.
This idea generated a lot of further work [31, 18, 1, 23] mainly devoted to
extensions to weighted Bergman spaces in one or several variables.

Another important and very recent development regards the generalization of
the original de Branges-Rovnyak spaces to the context of several complex
variables. From an operator-theoretic point of view, the most natural
generalization of the Szegő kernel to the unit ball in several complex
variables is the Drury-Arveson kernel

$k(z,w)=\frac{1}{1-\langle z,w\rangle},\quad z,w\in{\mathbb{B}}_{d}.$

Jury and Martin considered in [16] reproducing kernel Hilbert spaces with
kernel of the form

(3) $k_{b}(z,w)=\frac{1-b(z)b(w)^{*}}{1-\langle z,w\rangle},$

where $b$ is a row-vector pointwise multiplier of the Drury-Arveson space.
They characterized the multipliers $b$ for which the equivalent assertions
(i), or (ii) from above hold, showing in particular, that the Sarason
dichotomy does occur in this case. Shortly after, Hartz [15] completed their
work by showing that the equivalent statements (i) and (ii) above hold
precisely when $b$ is an extreme point of the unit ball of the multiplier
space. However, the density of polynomials (assertion (iii)) when (i) or (ii)
hold, remains open in this context.
Spaces with kernel of the form (3) appear also in the recent doctoral thesis
of Sautel as models for certain commuting tuples which are expansive on the
complement of their null-space. Moreover, together with Richter [26] they
provide sufficient conditions under which a kernel of the form (3) is a
complete Nevanlinna-Pick kernel.

The present paper is concerned with a very general construction of this type.
More precisely, we consider a reproducing kernel Hilbert space with
reproducing kernel $k$ on a non-void set ${\mathcal{X}}$, denoted by
${\mathcal{H}}_{k}$. In order to deal with vector-valued $b$’s as well, we
assume that $b$ is a non-constant contractive multiplier of
${\mathcal{H}}_{k}\otimes\ell^{2}$ into ${\mathcal{H}}_{k}$. This implies (see
§2.1 below) that

$k^{b}(x,y)=(1-b(x)b(y)^{*})k(x,y),\quad x,y\in{\mathcal{X}},$

is the reproducing kernel of a Hilbert function space on ${\mathcal{X}}$ which
we denote by ${\mathcal{H}}_{k}(b)$. It could be called _sub-
${\mathcal{H}}_{k}$ space_.
Adopting this general point of view reveals that a number of properties of the
particular spaces mentioned above, especially in the sub-weighted Bergman
case, only depend on general results about reproducing kernels. For example,
in §3.2 we consider the embedding of ${\mathcal{H}}_{k}(b)$ into
${\mathcal{H}}_{k}$ which is always contractive. We prove that a necessary
condition for the compactness of this embedding is that $\|b(x_{n})\|\to 1$,
whenever $k(x_{n},x_{n})\to\infty$.
In the case when ${\mathcal{H}}_{k}$ is a space of analytic functions on the
unit disc ${\mathbb{D}}$ with $k(z_{n},z_{n})\to\infty$ whenever $|z_{n}|\to
1$ and $b$ is a scalar-valued multiplier, the above condition implies that $b$
is a finite Blaschke product. In Theorem 4 we show that this condition is also
sufficient provided that it is sufficient in the simplest case $b(z)=z$. This
is a fairly far-reaching generalization of results of Zhu [30, 18] proved for
standard weighted Bergman spaces.
The results extend also to the context of several complex variables, namely
when $k$ is a power of the Drury-Arveson kernel and $b$ is an automorphism of
the unit ball. We also supply some examples showing that, in general, the
above condition fails to be sufficient.

Beyond extending these interesting results, the study of general
${\mathcal{H}}_{k}(b)$\- spaces has additional motivations. In this paper we
shall focus on the following two aspects:

  1. 1)

Reproducing kernel structure induced by operator inequalities,

  2. 2)

Reproducing kernels with a complete Nevanlinna-Pick factor.

1) When $b(0)=0$ the original de Branges-Rovnyak spaces ${\mathcal{H}}(b)$
with $b$ scalar-valued are precisely the spaces of analytic functions on
${\mathbb{D}}$ with reproducing kernel normalized at $0$ where the backward
shift $L$ is contractive and, in addition, $I-LL^{*}$ has rank one. The recent
paper [5] provides a thorough study of such spaces without the last
assumption, which yields Hilbert spaces with reproducing kernel of the general
form (1). Moreover, it turns out that the case when
$\operatorname{rank}(I-LL^{*})<\infty$ leads to a theory analogous to the one
of sub-Hardy spaces, in particular Sarason’s dichotomy holds. Extending these
ideas to the context of several variables is obviously very complicated due to
the intricate structure of left inverses to the row operator given by
multiplication of the coordinates. For this reason, the expansivity of that
row operator plays the central role in Sautel’s work [26] mentioned above.
The second hereditary inequality is more involved and was derived by Shimorin
in [28]. If $S$ denote the forward shift on a space of analytic functions
${\mathcal{H}}$ on the disc with normalized reproducing kernel at the origin,
Shimorin’s condition reads

(4) $\|Sf+g\|^{2}\leq 2(\|f\|^{2}+\|Sg\|^{2}).$

It turns out (see also §3.1 below) that it characterizes general sub-Bergman
spaces obtained by replacing the scalar function $b$ in (2) by an operator-
valued one. In §3.1 we prove a generalization of both inequalities. Assume
that ${\mathcal{H}}$ is a reproducing kernel Hilbert space of analytic
functions in ${\mathbb{D}}$ with normalized kernel at the origin. We show that
the inequality

$\left\|f_{0}+\sum_{n\geq 1}u_{n}f_{n}\right\|^{2}\leq\|\varphi
f_{0}\|^{2}+\sum_{n\geq 1}\|f_{n}\|^{2},\quad f_{n}\in{\mathcal{H}},\,n\geq
0,$

where $\varphi,u_{n},~{}n\geq 1$ are fixed multipliers, characterizes
${\mathcal{H}}_{k}(b)$-spaces, with $k$ a kernel of Bergman-type, a class
introduced by McCullough and Richter in [19].
2) By a normalized complete Nevanlinna-Pick (CNP) kernel on the non-void set
${\mathcal{X}}$, normalized at $z_{0}\in{\mathcal{X}}$ we mean a kernel of the
form

$(z,w)\mapsto\frac{1}{1-b(z)b(w)^{*}},\qquad b(z)b(w)^{*}=\langle
b(z),b(w)\rangle_{\ell^{2}},\quad z,w\in{\mathcal{X}},$

where $\|b(z)\|_{l^{2}}\leq 1,~{}z\in{\mathcal{X}},~{}b(z_{0})=0$. Such
reproducing kernels arise from solving certain interpolation problems for
multipliers [2]. These kernels have a number of remarkable properties and have
attracted a lot of attention recently. Here we are interested in a much larger
class, namely those which have a normalized CNP factor, i.e. the reproducing
kernels $k$ for which there is a normalized CNP kernel $s$ such that $k/s$ is
a reproducing kernel as well (positive definite). Such kernels play a crucial
for the very general commutant lifting theorem by S. Shimorin [27] and very
recently in [3] it has been shown that functions in spaces with such kernels
can be represented as quotients of multipliers. Given $k$ with the CNP factor
$\frac{1}{1-b(z)b(w)^{*}}$ the ”quotient” kernel

$k_{b}(z,w)=k(z,w)(1-b(z)b(w)^{*}),$

corresponds to the ${\mathcal{H}}_{k}(b)$ space which actually appears in
Shimorin’s commutant lifting theorem as well. One of the main difficulties in
dealing with such spaces is to understand which functions belong to them other
than the reproducing kernels. In §4.1 we give sufficient conditions such that
${\mathcal{H}}_{k}(b)$ contains the kernels $k_{y}\in{\mathcal{H}}_{k}$ or
even more, such that the span of these kernels is dense in
${\mathcal{H}}_{k}(b)$ (Theorem 6). The proof of this approximation theorem is
based on an analogue of the Sz.-Nagy-Foiaş model inspired by the work in [5].
The idea is interesting in its own right and is presented in §3.3.
There is a large class of examples satisfying these conditions, like the sub-
Hardy spaces ${\mathcal{H}}(b)$ with $b$ non-extremal in the unit ball of
$H^{\infty}$ and all ${\mathcal{H}}_{k}(b)$ spaces with ${\mathcal{H}}_{k}$,
essentially every weighted Bergman space on a domain in ${\mathbb{C}}^{d}$ and
$b$ a non-constant analytic row contraction on that domain. Intuitively
speaking, the kernels $k_{y}\in{\mathcal{H}}_{k},~{}y\in{\mathcal{X}}$ play
the same role as polynomials in the theory of spaces of analytic functions on
the $d$-dimensional unit ball. In particular, these results extend Chu’s
density theorem [11] to sub-Bergman spaces on very general domains in
${\mathbb{C}}^{d}$. In fact, for the standard weighted Bergman spaces on the
ball we show in §5.1 that Sarason’s dichotomy does not occur, that is
conditions (i), (ii) and (iii) hold in any of such sub- weighted Bergman
spaces.

Finally, these results and ideas apply to an interesting class of kernels
considered recently in [13], namely positive integer powers of the sub-
weighted Bergman kernels

$\frac{(1-b(z)b(w)^{*})^{m}}{(1-z\overline{w})^{\beta}}\qquad
m\in{\mathbb{N}},m<\beta\in{\mathbb{R}}.$

These kernels are naturally connected to $m$-hypercontractive operators on
standard weighted Bergman spaces. In [13] it was conjectured that polynomials
are dense in the corresponding spaces of analytic functions on ${\mathbb{D}}$.
As an application of the results mentioned above we prove that the conjecture
holds true. Our methods yield a proof in the higher dimensional case as well.

The paper is organized as follows. Section 2 serves to a preliminary purpose.
Section 3 contains general properties of ${{\mathcal{H}}_{k}}(b)$, in §3.1 we
discuss the operator inequalities mentioned before. §3.2 is devoted to the
embedding of ${{\mathcal{H}}_{k}}(b)$ in ${{\mathcal{H}}_{k}}$ and the cases
where it is compact. The analogue of the Sz.-Nagy-Foiaş model is described in
§3.3. Section 4 contains the general approximation results, while in Section 5
we apply these ideas to general weighted Bergman spaces. The proof of the
conjecture in [13] is deferred to section 6.

##  2 Preliminaries

###  2.1 Reproducing kernels

For the sake of completeness we recall some basic properties of these objects.
The material is standard, can be found in [8] and [21]. If ${\mathcal{X}}$
denotes a non-empty set, a function
$k:{\mathcal{X}}\times{\mathcal{X}}\to{\mathbb{C}}$ is called _positive_
(write $k\gg 0$) if for all finite subsets
$\\{x_{1},\ldots,x_{n}\\}\subset{\mathcal{X}}$ the matrix
$[k(x_{i},x_{j})]_{i,j=1}^{n}$ is positive semi-definite. Given $k\gg 0$ there
exists a unique Hilbert space ${{\mathcal{H}}_{k}}$ consisting of complex-
valued functions on ${\mathcal{X}}$, such that point-evaluations are bounded
linear functionals on ${{\mathcal{H}}_{k}}$ given by

$f(y)=\langle f,k_{y}\rangle_{{{\mathcal{H}}_{k}}},\quad x,y\in{\mathcal{X}},$

where we have used the common notation $k_{y}=k(\cdot,y)$. For this reason,
$k$ is called a _reproducing kernel_.

It is easy to verify that if $k,h$ are reproducing kernels on ${\mathcal{X}}$
with $k-h\gg 0$, then ${\mathcal{H}}_{h}$ is contractively contained in
${{\mathcal{H}}_{k}}$. In fact, the map $k_{y}\to h_{y}$ extends by linearity
to a contraction, whose adjoint is the inclusion map. This yields the
following result.

######  Proposition 1.

Let $k$ be a reproducing kernel on ${\mathcal{X}}$, then
$f:{\mathcal{X}}\to{\mathbb{C}}$ is in ${\mathcal{H}}_{k}$ if and only if

$c^{2}k(x,y)-f(x)\overline{f(y)}\gg 0$

for some $c>0$ and the least such $c$ is equal to $\|f\|_{{\mathcal{H}}_{k}}$.

The classical Schur’s product theorem for positive semi-definite matrices
applies to reproducing kernels and gives the following result.

######  Theorem 1 (Schur’s product theorem).

Let $k$ and $l$ be reproducing kernels on ${\mathcal{X}}$. Then $kl$ is also a
reproducing kernel.

######  Definition 1.

Let $k$ be a reproducing kernel on ${\mathcal{X}}$. A function
$\varphi:{\mathcal{X}}\to{\mathbb{C}}$ is a _multiplier_ of
${\mathcal{H}}_{k}$ if $\varphi f\in{\mathcal{H}}_{k}$ for every
$f\in{\mathcal{H}}_{k}$. The algebra of these operators is denoted
$\operatorname{Mult}({{\mathcal{H}}_{k}})$.

Note that if $\varphi\in\operatorname{Mult}({{\mathcal{H}}_{k}})$, then the
induced linear map $M_{\varphi}$ on ${\mathcal{H}}_{k},~{}M_{\varphi}f=\varphi
f$ is bounded by the closed graph theorem. Moreover, we have

(5) $M_{\varphi}^{*}k_{y}=\overline{\varphi(y)}k_{y}.$

Consequently, $\sup_{x\in{\mathcal{X}}}|\varphi(x)|\leq\|M_{\varphi}\|$. Using
also the fact that the linear span of $\\{k_{y}:y\in{\mathcal{X}}\\}$ is dense
in ${{\mathcal{H}}_{k}}$ one can derive the following result.

######  Proposition 2.

Let $k$ be a reproducing kernel on ${\mathcal{X}}$, then $\varphi$ is a
multiplier of ${\mathcal{H}}_{k}$ if and only if

$k(x,y)(c^{2}-\varphi(x)\overline{\varphi(y)})\gg 0$

for some $c>0$ and the least such $c$ is equal to $\|M_{\varphi}\|$.

Given two sets ${\mathcal{F}},{\mathcal{G}}$ of complex-valued functions on
${\mathcal{X}}$ shall consider the linear span of their pointwise products,
that is,

$\operatorname{span}{\mathcal{F}}\cdot{\mathcal{G}}=\operatorname{span}\\{fg:f\in{\mathcal{F}},g\in{\mathcal{G}}\\}.$

The next result is essentially a consequence of the above. We include a short
proof for the convenience of the reader.

######  Proposition 3.

Let $s,t$ be reproducing kernels on ${\mathcal{X}}$ and $k=st$. Then:

  1. (i)

$\operatorname{span}\operatorname{Mult}({\mathcal{H}}_{s})\cdot\operatorname{Mult}({\mathcal{H}}_{t})\subset\operatorname{Mult}({\mathcal{H}}_{k})$,
in particular
$\operatorname{Mult}({\mathcal{H}}_{s})\subset\operatorname{Mult}({\mathcal{H}}_{k})$
contractively,

  2. (ii)

If $f\in{\mathcal{H}}_{s}$ and $g\in{\mathcal{H}}_{t}$, then
$fg\in{\mathcal{H}}_{k}$ and
$\|fg\|_{{\mathcal{H}}_{k}}\leq\|f\|_{{\mathcal{H}}_{s}}\|g\|_{{\mathcal{H}}_{t}}$,
in particular if the constant function $1\in{\mathcal{H}}_{t}$, then
${\mathcal{H}}_{s}$ is boundedly contained in ${\mathcal{H}}_{k}$,

  3. (iii)

If ${\mathcal{F}}$ and ${\mathcal{G}}$ are dense sets in ${\mathcal{H}}_{s}$
and ${\mathcal{H}}_{t}$, respectively, then
$\operatorname{span}{\mathcal{F}}\cdot{\mathcal{G}}$ is dense in
${\mathcal{H}}_{k}$.

###### Proof.

(i) Let $\varphi\in\operatorname{Mult}({\mathcal{H}}_{s})$ and
$\psi\in\operatorname{Mult}({\mathcal{H}}_{t})$, then by Proposition 2

$\displaystyle s(x,y)\|M_{\varphi}\|^{2}\gg
s(x,y)\varphi(x)\overline{\varphi(y)}$ $\displaystyle\gg 0,$ $\displaystyle
t(x,y)\|M_{\psi}\|^{2}\gg t(x,y)\psi(x)\overline{\psi(y)}$ $\displaystyle\gg
0.$

By Schur’s product theorem, the order is preserved when multiplying the left-
hand side and the right-hand side, respectively. That is

$k(x,y)\|M_{\varphi}\|^{2}\|M_{\psi}\|^{2}\gg\varphi(x)\psi(x)\overline{\varphi(y)\psi(y)}$

and it follows that $\varphi\psi\in\operatorname{Mult}({\mathcal{H}}_{k})$.
Clearly, any element in
$\operatorname{span}\operatorname{Mult}({\mathcal{H}}_{s})\cdot\operatorname{Mult}({\mathcal{H}}_{t})$
is a sum of products as above. The contractive containment follows by letting
above $\psi=1$ which gives

$k(x,y)(\|M_{\varphi}\|^{2}-\varphi(x)\overline{\varphi(y)})\gg 0.$

(ii) is very similar and will be omitted.
(iii) By (ii) we have that
${\mathcal{F}}\cdot{\mathcal{G}}\subset{\mathcal{H}}_{k}$. If
$y\in{\mathcal{X}},~{}\varepsilon>0$, and
$f\in{\mathcal{F}},~{}g\in{\mathcal{G}}$ with

$\|f-s_{y}\|_{{\mathcal{H}}_{s}}<\varepsilon,\quad\|g-t_{y}\|_{{\mathcal{H}}_{t}}<\varepsilon,$

by the first part of (ii) we have

$\displaystyle\|fg-k_{y}\|_{{\mathcal{H}}_{k}}$
$\displaystyle\leq\|g\|_{{\mathcal{H}}_{t}}\|f-s_{y}\|_{{\mathcal{H}}_{s}}$
$\displaystyle\leq(\varepsilon+\|t_{y}\|_{{\mathcal{H}}_{t}})\varepsilon+(\varepsilon+\|s_{y}\|_{{\mathcal{H}}_{s}})\varepsilon$

which gives the desired result by the density of the linear span of the kernel
functions $k_{y},\,y\in{\mathcal{X}}$ in ${\mathcal{H}}_{K}$. ∎

A repeated direct application of the last part of the proposition leads to the
following result.

######  Corollary 1.

Let ${\mathcal{F}}$ be an algebra of functions on ${\mathcal{X}}$ and let
$s^{j},~{}j=1,\ldots,n$ be reproducing kernels on ${\mathcal{X}}$ such that
${\mathcal{F}}$ is dense in ${\mathcal{H}}_{s^{j}},j=1,\ldots,n$. If
$k=\prod_{j=1}^{n}s^{j}$ then ${\mathcal{F}}$ is dense in ${\mathcal{H}}_{k}$.

We close the paragraph with a few words about the vector-valued case. The
reproducing kernel of a Hilbert spaces of vector-valued functions is an
operator valued function of two variables with similar properties to the
scalar-valued version.
In this paper we shall be concerned with a very special case. Given a scalar
reproducing kernel $k$ on ${\mathcal{X}}$, we shall consider the Hilbert space
${{\mathcal{H}}_{k}}\otimes\ell^{2}$, that is, the space of functions
$f:{\mathcal{X}}\to\ell^{2}$ such that

$f=\begin{pmatrix}f_{0}\\\ f_{1}\\\
\vdots\end{pmatrix},\,f_{j}\in{{\mathcal{H}}_{k}},j\geq
0,\qquad\|f\|_{{{\mathcal{H}}_{k}}\otimes\ell^{2}}^{2}=\sum_{j\geq
0}\|f_{j}\|_{{\mathcal{H}}_{k}}^{2}<\infty.$

This Hilbert space has the operator-valued reproducing kernel $(x,y)\mapsto
I_{\ell^{2}}k(x,y)$. In other words, given
$f\in{{\mathcal{H}}_{k}}\otimes\ell^{2},~{}y\in{\mathcal{X}},~{}e\in\ell^{2}$,
we have

$\langle f(y),e\rangle_{\ell^{2}}=\langle
f,k_{y}e\rangle_{{{\mathcal{H}}_{k}}\otimes\ell^{2}}.$

A multiplier $\varphi$ from ${{\mathcal{H}}_{k}}\otimes\ell^{2}$ to
${{\mathcal{H}}_{k}}$ is row-vector valued function such that $\varphi
f\in{{\mathcal{H}}_{k}}$ for every $f\in{{\mathcal{H}}_{k}}\otimes\ell^{2}$.
We write
$\varphi\in\operatorname{Mult}({{\mathcal{H}}_{k}}\otimes\ell^{2},{{\mathcal{H}}_{k}})$.
In this case (5) becomes

(6) $M_{\varphi}^{*}k_{y}=\varphi(y)^{*}k_{y}.$

Note that $\varphi(y)^{*}$ can be interpreted as a column-vector. Finally, let
us also mention that Proposition 2 extends verbatim to this context by simply
replacing $\overline{\varphi(y)}$ by $\varphi(y)^{*}$.

###  2.2 Some unitarily invariant kernels on the unit ball of
${\mathbb{C}}^{d}$

Let $d$ be a positive integer and

${\mathbb{B}}_{d}=\\{z\in{\mathbb{C}}^{d}:\|z\|<1\\},$

be open the unit ball in ${\mathbb{C}}^{d}$. A kernel $k$ on
${\mathbb{B}}_{d}$ is invariant under the unitary group on ${\mathbb{C}}^{d}$
if and only if $k(z,w)$ is a function of $\langle z,w\rangle.$

An important class of examples, see [7], are reproducing kernels of radially
weighted Besov spaces

$B_{\omega}^{s}=\left\\{f\in
Hol({\mathbb{B}}_{d}):~{}\int_{{\mathbb{B}}_{d}}|R^{s}f|^{2}\omega
dV<\infty\right\\}.$

Here $R^{s}$ is a power of the radial derivative operator
$R=\sum_{j=1}^{d}z_{j}\frac{\partial}{\partial z_{j}}$, $V$ is the Lebesgue
measure on the ball, and $\omega$ is a radial integrable function on
${\mathbb{B}}_{d}$. It is easy to verify that monomials form an orthogonal
basis in these spaces. A particular case is the Drury-Arveson kernel on
${{\mathbb{B}}_{d}}$ is defined by

$s(z,w)\coloneq\frac{1}{1-\langle z,w\rangle},\qquad
z,w\in{{\mathbb{B}}_{d}}.$

When $d=1$ it is the reproducing kernel for the Hardy space $H^{2}$ on the
unit disk ${\mathbb{D}}$ and $s$ is actually called the Szegő kernel. For
$d>1$, the corresponding space is called the Drury-Arveson space and plays a
key role in the theory of commuting operator tuples. The recent survey [14]
contains a lot of interesting material in this direction.
For any $\beta>0$ observe that

$s(z,w)^{\beta}=\sum_{n=0}^{\infty}a_{n}\langle z,w\rangle^{n},$

where $a_{n}\geq 0$ therefore by Schur’s product theorem $s^{\beta}$ is a
reproducing kernel.

######  Definition 2.

Let $\beta>0$ we write ${{\mathcal{H}}_{\beta}}$ for the space of holomorphic
functions on ${{\mathbb{B}}_{d}}$ that corresponds to the reproducing kernel

$s^{\beta}(z,w)\coloneq\frac{1}{(1-\langle z,w\rangle)^{\beta}}$

These spaces are extensively studied in the literature. If $\beta>d$, then
${\mathcal{H}}_{s^{\beta}}$ is a (standard) weighted Bergman space on
${{\mathbb{B}}_{d}}$ as noted in the next paragraph.

###  2.3 Weighted Bergman spaces on a domain of ${\mathbb{C}}^{d}$

Let $\Omega$ be a domain in ${\mathbb{C}}^{d}$ and let $\mu$ be a regular
Borel positive measure on $\Omega$ such that the analytic functions in
$L^{2}(\mu)$ form a closed subspace where point-evaluations are continuous
linear functionals. These spaces are usually called weighted Bergman spaces
and we denote them by $L^{2}_{a}(\mu)$. By assumption the space
$L^{2}_{a}(\mu)$ has a reproducing kernel which we denote by $k^{\mu}$, and,
as pointed out before, the norm is given by

$\|f\|^{2}=\int_{\Omega}|f(z)|^{2}d\mu(z),\quad f\in L_{a}^{2}(\mu).$

If $\beta>d$, the spaces ${{\mathcal{H}}_{\beta}}$ from the previous paragraph
satisfy ${{\mathcal{H}}_{\beta}}=L^{2}_{a}(\mu)$ with
$\Omega={\mathbb{B}}_{d}$, $d\mu=c_{\alpha}(1-|z|^{2})^{\alpha}dV$, where
$\alpha=\beta-(d+1)$, and $c_{\alpha}$ is a normalizing constant.

The vector valued version $L^{2}_{a}(\mu)\otimes\ell^{2}$ is defined with help
of the norm

$\|f\|^{2}=\int_{\Omega}\|f(z)\|^{2}d\mu(z)<\infty.$

The following simple observation plays an important role for the sequel. We
denote throughout by $V$ the Lebesgue measure on ${\mathbb{C}}^{d}$.

######  Proposition 4.

If $L_{a}^{2}(\mu)$ is a reproducing kernel closed subspace of $L^{2}(\mu)$
then there exists a measure $\tilde{\mu}$ such that
$L_{a}^{2}(\tilde{\mu})\otimes\ell^{2}=L_{a}^{2}(\mu)\otimes\ell^{2}$ and for
each compact subset $K$ of $\Omega$ there exists $c(K)>0$ with

(7) $\tilde{\mu}|K\geq c(K)V|K,$

###### Proof.

Choose $v\in L^{1}(V),~{}v>0$ a.e. with $\text{essinf }v|K>0$ for any compact
subset $K$ of $\Omega$ and set

$d\tilde{\mu}=d\mu+\frac{v(z)}{1+\|k^{\mu}_{z}\|^{2}}dV.$

Since for every $f\in L_{a}^{2}(\mu)\otimes\ell^{2}$ and $z\in\Omega$ we have

$\frac{\|f(z)\|_{\ell^{2}}^{2}}{1+\|k^{\mu}_{z}\|^{2}}\leq 1,$

we obtain
$L_{a}^{2}(\tilde{\mu})\otimes\ell^{2}=L_{a}^{2}(\mu)\otimes\ell^{2}$ and that
the norms are comparable. ∎

One verifies that $\operatorname{Mult}(L^{2}_{a}(\mu))$ equals the algebra of
bounded analytic functions on $\Omega$ with equality of norms. Similarly,
$\operatorname{Mult}(L^{2}_{a}(\mu)\otimes\ell^{2},L^{2}_{a}(\mu))$ consists
of bounded analytic row-vectors and the multiplier norm coincides with the
supremum norm. For our purposes it is important to note that these multipliers
never attain their norm unless the symbols are constant. Indeed, if
$\|bf\|=\|b\|_{\infty}\|f\|$ for some non-zero $f\in
L^{2}_{a}(\mu)\otimes\ell^{2}$, then $\|b\|_{\infty}=\|b(z)\|_{\ell^{2}}$ for
all $z\in\Omega$, which easily implies that $b$ is constant. In the
terminology of [6] these Bergman spaces have no non-trivial _sub-inner_
multipliers. This is in strong contrast to smaller reproducing kernel spaces
on the ball. In fact, more than that is true for measures $\mu$ as above.

######  Proposition 5.

Let $b:\Omega\to\ell^{2}$ be analytic non-constant and such that $b(z)$ is a
row-contraction for every $z\in\Omega$. If $\mu$ satisfies (7), then for every
$z\in\Omega$ there exists $c_{z}>0$ such that

$\|f\|^{2}-\|bf\|^{2}\geq c_{z}\|f(z)\|_{\ell^{2}}^{2},$

for all $f\in L^{2}_{a}(\mu)\otimes\ell^{2}$.

###### Proof.

Let $z\in\Omega$ and $B_{z}$ be an open ball in ${\mathbb{C}}^{d}$ centered at
$z$ with $\overline{B_{z}}\subset\Omega$. By the assumption on $b$ there is
$a_{z}>0$ such that $1-\|b(\zeta)\|_{\ell^{2}}^{2}\geq a_{z}$, $\zeta\in
B_{z}$. This implies for each $e\in\ell^{2}$,

$\|e\|_{\ell_{2}}^{2}-|b(\zeta)e|^{2}\geq(1-\|b(\zeta)\|_{\ell^{2}}^{2})\|e\|_{\ell^{2}}^{2}\geq
a_{z}\|e\|_{\ell^{2}}^{2}.$

Using also (7) we have for $f\in L^{2}_{a}(\mu)\otimes\ell^{2}$

$\displaystyle\|f(z)\|_{\ell^{2}}^{2}$
$\displaystyle\leq\frac{1}{V(B_{z})}\int_{B_{z}}\|f\|_{\ell^{2}}dV$
$\displaystyle\leq\frac{1}{V(B_{z})c(\overline{B_{z}})}\int_{B_{z}}\|f\|_{\ell^{2}}^{2}d\mu$
$\displaystyle\leq\frac{1}{V(B_{z})c(\overline{B_{z}})a_{z}}\int_{B_{z}}(\|f\|_{\ell^{2}}^{2}-|bf|^{2})d\mu$
$\displaystyle\leq\frac{\|f\|^{2}-\|bf\|^{2}}{V(B_{z})c(\overline{B_{z}})a_{z}},$

which completes the proof. ∎

##  3 Basic properties of generalized de Branges-Rovnyak spaces

Let us recall the general notion of a de Branges-Rovnyak space pointed out in
the Introduction. Given be a scalar-valued reproducing kernel $k$ on the non-
empty set ${\mathcal{X}}$ let ${\mathcal{H}}_{k}$ denote the corresponding
Hilbert function space.

######  Definition 3.

Let
$b\in\operatorname{Mult}({\mathcal{H}}_{k}\otimes\ell^{2},{\mathcal{H}}_{k})$
with $\|M_{b}\|\leq 1$. The sub-${\mathcal{H}}_{k}$ space, or the de Branges-
Rovnyak space with respect to ${\mathcal{H}}_{k}$ and $b$ is the Hilbert space
corresponding to the reproducing kernel

$k^{b}(x,y)\coloneq k(x,y)(1-b(x)b(y)^{*}),\qquad b(x)b(y)^{*}=\langle
b(x),b(y)\rangle_{\ell^{2}},$

which is positive due to the multiplier criterion. We denote this space by
${\mathcal{H}}_{k}(b)$.

One can verify that as a set, ${\mathcal{H}}_{k}(b)$ is equal to
$\operatorname{Ran}(I-M_{b}M_{b}^{*})^{\frac{1}{2}}$ and its norm satisfies

$\langle(I-M_{b}M_{b}^{*})^{\frac{1}{2}}f,(I-M_{b}M_{b}^{*})^{\frac{1}{2}}g\rangle_{{\mathcal{H}}_{k}(b)}=\langle
f,g\rangle_{{\mathcal{H}}_{k}},$

whenever
$f,g\in\operatorname{Ran}(I-M_{b}M_{b}^{*})^{\frac{1}{2}}={\mathcal{H}}_{k}\ominus\ker(I-M_{b}M_{b}^{*})^{\frac{1}{2}}.$
Furthermore, ${\mathcal{H}}_{k}(b)$ is contractively contained in
${\mathcal{H}}_{k}$, this can be seen either from the above formula, or the
fact that

$k(z,w)-k^{b}(z,w)=b(x)b(y)^{*}k(x,y),$

which is positive by Schur’s product theorem.

###  3.1 Examples via operator inequalities

It is well known that the backward shift

$Lf(z)=\frac{f(z)-z}{z},$

is a contraction on any de Branges-Rovnyak space
${\mathcal{H}}(b)={\mathcal{H}}_{s}(b)$, with
$s(z,w)=\frac{1}{1-\overline{w}z}$ and $b(0)=0$. The starting point of the
work in [5] is the observation that the converse holds true as well, as the
following result shows.

######  Proposition 6 ([5]).

Let ${\mathcal{H}}$ be a Hilbert space of analytic functions on ${\mathbb{D}}$
with a reproducing kernel $k$ normalized at $0$. $L$ is contractive on
${\mathcal{H}}$ if and only if there exists an analytic row-contraction $b$
with $b(0)=0$ such that

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

For reproducing kernel Hilbert spaces of analytic functions in ${\mathbb{D}}$
having the identity function $z$ as a multiplier and a reproducing kernel
normalized at the origin, the contractivity of $L$ is equivalent to the fact
that the forward shift $M_{z}$ is expansive, that is, $\|zf\|\geq\|f\|$. In
particular, such spaces have a general de Branges-Rovnyak kernel as above.

There is a very recent extension of this idea to the context of several
complex variables due to S. Richter and J. Sautel [26]. It uses the Drury-
Arveson kernel $s(z,w)=\frac{1}{1-\langle z,w\rangle}$ instead of the Szegö
kernel. The proof is much more involved in this case.

######  Theorem 2.

Let $T=(T_{1},\ldots,T_{d}):{\mathcal{H}}^{d}\to{\mathcal{H}}$ be a bounded
row operator on an infinite dimensional Hilbert space ${\mathcal{H}}$, and let
$m\in{\mathbb{N}}$. T is unitarily equivalent to $(M_{z},{\mathcal{H}}_{s}(b)$
for some
$b\in\operatorname{Mult}({\mathcal{H}}_{s}\otimes\mathcal{D},{\mathcal{H}}_{s}\otimes\mathcal{E})$
and Hilbert spaces $\mathcal{D}$ and $\mathcal{E}$ such that $b(0)=0$,
$\dim\mathcal{E}=m$, and that $M_{z}$ is a bounded row operator if and only if
$T$ satisfies the following four conditions:

  1. (i)

$T_{i}T_{j}=T_{j}T_{i}$ for all $1\leq i,j\leq d$,

  2. (ii)

$\dim([\operatorname{Ran}(T-\lambda)]^{\perp})=m$ for every
$\lambda\in{{\mathbb{B}}_{d}}$,

  3. (iii)

$\|Tx\|_{\mathcal{H}}\geq\|x\|_{{\mathcal{H}}^{d}}$ for all
$x\in\operatorname{Ker}(T)^{\perp}$,

  4. (iv)

$\bigcap\limits_{n\geq 0}\sum\limits_{|w|=n}T_{w}{\mathcal{H}}=\\{0\\}$.

Sub-Bergman spaces are also related to an operator inequality which is
considerably more subtle. It is due to S. Shimorin [28] and it asserts that on
the (unweighted) Bergman space on ${\mathbb{D}}$, the forward shift
$M_{z}f=zf$ satisfies

(8) $\|zf+g\|^{2}\leq 2(\|f\|^{2}+\|zg\|^{2}).$

As the previous inequalities, (8) is _hereditary_ in the sense that it
continues to hold for restrictions of the operator involved to its invariant
subspaces. This estimate has far-reaching consequences regarding invariant
subspaces of the Bergman shift. A crucial fact (see also [4]) is that (8) is
closely related to sub-Bergman kernels in the following way.

######  Proposition 7 (Shimorin’s inequality ).

Let ${\mathcal{H}}={{\mathcal{H}}_{k}}$ be a Hilbert space of analytic
functions on ${\mathbb{D}}$ such that the reproducing kernel $k$ is normalized
at $0$, $M_{z}$ and $L$ are bounded on ${\mathcal{H}}$. Then the inequality
(8) holds on ${\mathcal{H}}$ if and only if there exists an analytic row-
contraction $b$ with $b(0)=0$ such that

(9) $k(z,w)=\frac{1-b(z)b(w)^{*}}{(1-z\overline{w})^{2}},$

In order to study $M_{z}$-invariant subspaces of the Bergman space one needs
to drop both assumptions in the above proposition, that is, to consider non-
normalized kernels, as well as left inverses of the forward shift with a null
space of arbitrary dimension.

Several interesting extensions of Shimorin’s inequality have been found by
Olofsson and Wennman (see [20] and the references therein).
A very general class of kernels containing all the above examples are the so-
called Bergman-type kernels introduced by McCullough and Richter in [19] These
are reproducing kernels in ${\mathbb{D}}$ of the form

(10) $k(z,w)=\frac{1}{1-\varphi(z)\overline{\varphi(w)}(1-u(z)u(w)^{*})},\quad
z,w\in{\mathbb{D}},$

with $\varphi(0)=0$ and $k(z,z)\to\infty$ when $|z|\to 1^{-}$. It is shown in
[19] that with $\rho(z)=\frac{\varphi(z)}{z}$ can be chosen to be outer and
bounded from above and below. Clearly, the Szegö kernel is obtained for
$\varphi(z)=z,u=0$, and the unweighted Bergman kernel for
$\varphi(z)=z\sqrt{2},u(z)=\frac{z}{\sqrt{2}}$.
Also, if $k$ is of Bergman-type we have $Mult({\mathcal{H}}(k))=H^{\infty}$
with equality of norms (see again [19]). The following theorem extends all
previous results in one complex variable to the general context of Bergman-
type kernels.

######  Theorem 3.

Let $k$ be a Bergman-type kernel on ${\mathbb{D}}$ of the form (10). Let
${\mathcal{H}}$ be a Hilbert space of analytic functions on ${\mathbb{D}}$
whose reproducing kernel is normalized at $0$ and assume that $\varphi\in
Mult({\mathcal{H}})$, the range of $M_{\varphi}$ is closed of co-dimension
$1$, and that $u\in Mult({\mathcal{H}}\otimes\ell^{2})$. Then
${\mathcal{H}}={\mathcal{H}}_{k}(b)$ for an analytic row-contraction $b$ in
${\mathbb{D}}$ with $b(0)=0$, if and only

(11) $\left\|f_{0}+\sum_{n\geq 1}u_{n}f_{n}\right\|^{2}\leq\|\varphi
f_{0}\|^{2}+\sum_{n\geq 1}\|f_{n}\|^{2},$

whenever $f_{0},f_{1}\ldots,f_{n},\ldots\in{\mathcal{H}}$.

###### Proof.

The assumptions on $M_{\varphi}$ imply that its range equals the space of
functions in ${\mathcal{H}}$ vanishing at the origin. Consequently, the
operator $L_{\varphi}$ defined by

$L_{\varphi}f=\frac{f-f(0)}{\varphi},\quad f\in{\mathcal{H}},$

is a bounded left-inverse of $M_{\varphi}$. Then the inequality (11) is
equivalent to

(12) $\left\|L_{\varphi}f_{0}+\sum_{n\geq
1}u_{n}f_{n}\right\|^{2}\leq\|f_{0}\|^{2}+\sum_{n\geq 1}\|f_{n}\|^{2}.$

Indeed, if (11) holds, we apply it to $L_{\varphi}f_{0},f_{1},\ldots$. From
the fact that the kernel of ${\mathcal{H}}$ is normalized we obtain

$\|\varphi L_{\varphi}f_{0}\|^{2}=\|f_{0}-f_{0}(0)\|^{2}\|\leq\|f_{0}\|,$

and (12) follows.

Conversely, if (12) holds we apply it to $\varphi f_{0},f_{1},\ldots$ and
obtain (11). Now (12) is further equivalent to the fact that the operator
$T:\bigoplus_{\mathbb{N}}{\mathcal{H}}\to{\mathcal{H}}$ defined by

$T\sum_{n\geq 0}f_{n}=L_{\varphi}f_{0}+\sum_{n\geq 1}u_{n}f_{n},$

is contractive. This can be re-written as

$I-TT^{*}\geq 0\,\Leftrightarrow(I-TT^{*})h\gg
0\,\Leftrightarrow(I-TT^{*})h=bb^{*},$

where $h$ is the reproducing kernel in ${\mathcal{H}}$ and $b$ is an analytic
row-contraction in ${\mathbb{D}}$. Since

$L_{\varphi}L_{\varphi}^{*}h(z,w)=\frac{h(z,w)-1}{z\overline{w}},\quad
M_{u}M_{u}^{*}h(z,w)=u(z)\overline{u(w)}h(z,w),\,u\in Mult({\mathcal{H}}),$

we can solve for $h$ in $(I-TT^{*})h=bb^{*}$ and obtain after a
straightforward calculation

$h(z,w)=(1-b(z)b^{*}(w))k(z,w),$

i.e. $h$ is the reproducing kernel of ${\mathcal{H}}_{k}(b)$. ∎

Besides the Szegő, or the unweighted Bergman kernel, some standard examples of
Bergman-type kernels are

$k_{\beta}(z,w)=\frac{1}{(1-z\overline{w})^{\beta}},\quad
z,w\in{\mathbb{D}},\,1<\beta<2.$

These examples are discussed in [20]. The corresponding Hilbert spaces are
standard weighted Bergman spaces. Of course, there are many other examples in
this class, in particular in [19] there are produced examples such that the
forward shift on the corresponding space is not subnormal.

###  3.2 Compact embedding

As mentioned in the beginning of this section ${\mathcal{H}}_{k}(b)$ is always
contained contractively in ${\mathcal{H}}_{k}$. It is natural to ask when the
contractive inclusion map is compact. As pointed out in the Introduction this
question attracted attention in the context of sub-weighted Bergman spaces.

If the original space ${\mathcal{H}}_{k}$ is smaller, then the question is
less interesting as we shall see below. We start with a simple necessary
condition for the compactness of the inclusion map which holds under mild
assumptions in the most general context of a kernel $k$ on the non-void set
${\mathcal{X}}$.

######  Proposition 8.

Let $k$ be a reproducing kernel on a non-empty set ${\mathcal{X}}$ with the
property that $k_{y}$ is a bounded function for every $y\in{\mathcal{X}}$. Let
$b\in\operatorname{Mult}({\mathcal{H}}_{k}\otimes\ell^{2})$ and suppose that
the inclusion map from ${\mathcal{H}}_{k}(b)$ to ${\mathcal{H}}_{k}$ is
compact. Then for every sequence $(x_{n})$ in ${\mathcal{X}}$ with
$\lim_{n\to\infty}k(x_{n},x:n)=\infty$, we have

$\lim_{n\to\infty}\|b(x_{n})\|_{\ell^{2}}=1.$

###### Proof.

Let us begin with the simple observation that if
$J:{\mathcal{H}}_{k}(b)\to{\mathcal{H}}_{k}$ is the inclusion map,then its
adjoint $J^{*}$ satisfies

$j^{*}k_{y}(x)=(1-b(x)b^{*}(y))k_{y}(x),\quad x,y\in{\mathcal{X}}.$

Now let $(x_{n})$ be a sequence as in the statement and set
$f_{n}=\frac{k_{x_{n}}}{\|k_{x_{n}}\|}$, so that $\|f_{n}\|=1$. Furthermore,
by the assumption on $k$ we have

$\langle
f_{n},k_{y}\rangle=\frac{k(x_{n},y)}{\|k_{x_{n}}\|}=\frac{\overline{k_{y}(x_{n})}}{\|k_{x_{n}}\|}\to
0\text{as $n\to\infty$}.$

This proves that $(f_{n})$ tends weakly to $0$ in ${\mathcal{H}}_{k}$, hence
if $J$ is compact, so is $J^{*}$, that is, $J^{*}f_{n}$ must tend in norm to
0. Thus,

$\|J^{*}f_{n}\|^{2}=1-\|b(x_{n})\|^{2}\to 0.$

and the result follows. ∎

The result is of particular interest for most common reproducing kernel
Hilbert spaces of analytic functions in the unit disc, in the case when $b$ is
scalar-valued. Indeed, if ${\mathcal{H}}_{k}$ is such a space with

$k(z,z)\to\infty,\quad|z|\to 1^{-},$

and $b\in Mult({\mathcal{H}}_{k})$ is contractive, by Proposition 8 the
compactness of the inclusion $J:{\mathcal{H}}_{k}(b)\to{\mathcal{H}}_{k}$
implies that the bounded analytic function $b$ satisfies for all
$\lambda\in\mathbb{T}$

$\lim_{z\to\lambda}|b(z)|=1,$

hence $b$ is a finite Blaschke product. However, in this generality, finite
Blaschke products may fail to be contractive multipliers so the corresponding
generalized de Branges-Rovnyak spaces might not exist.

######  Example 1.

  1. a)

In the Dirichlet space $D$ with kernel

$-\frac{1}{z\overline{w}}\log(1-z\overline{w}),\quad z,w\in{\mathbb{D}},$

every finite Blaschke product $b$ is a multiplier, but (see [22])

$\|bf\|>\|f\|,\quad f\in D\setminus\\{0\\}.$

  2. b)

Let ${\mathcal{H}}_{2}$ be the unweighted Bergman space and
${\mathcal{H}}_{1}=H^{2}$ be the Hardy space on the unit disc. Consider the
reproducing kernel Hilbert space ${\mathcal{H}}$ consisting of analytic
functions $f$ in ${\mathbb{D}}$ which can be written (uniquely) as

$f(z)=u_{1}(z^{2})+zu_{2}(z^{2}),\quad
u_{1}\in{\mathcal{H}}_{1},\,u_{2}\in{\mathcal{H}}_{2},$

with norm

$\|f\|^{2}=\|u_{1}\|^{2}_{{\mathcal{H}}_{1}}+\|u_{2}\|^{2}_{{\mathcal{H}}_{2}}.$

Then obviously, $M_{z}$ is unbounded on ${\mathcal{H}}$ while if $\varphi\in
H^{\infty}$, the function $\psi(z)=\varphi(z^{2})$ is a multiplier with
$\|M_{\psi}\|=\|\psi\|_{\infty}.$

######  Proposition 9.

Given a reproducing kernel Hilbert space ${\mathcal{H}}_{k}$ of analytic
functions in ${\mathbb{D}}$, the finite Blaschke products are contractive
multipliers if and only if the identity function $\zeta(z)=z$ is a contractive
multiplier.

###### Proof.

One direction is obvious since the identity is a finite Blaschke product,
while the converse follows by standard functional calculus for contractions
from the equality $b(M_{z})=M_{b}$ valid for any finite Blaschke product. ∎

A similar result can be derived for the compactness of the embedding of
${\mathcal{H}}_{k}(b)$ into ${\mathcal{H}}_{k}$, but the proof is somewhat
more involved. Our next result provides a far-reaching generalization of the
result in [18].

######  Theorem 4.

Assume that ${\mathcal{H}}_{k}$ is a reproducing kernel Hilbert space of
analytic functions on ${\mathbb{D}}$ such that $M_{z}$ is contractive on it.
Then if $b\in Mult({\mathcal{H}}_{k})$ is contractive, the inclusion map From
${\mathcal{H}}_{k}(b)$ into ${\mathcal{H}}_{k}$ is compact if and only if
$I-M_{z}M_{z}^{*}$ is compact.

The proof is essentially based on the identity below. A similar formula
appears in [13]. Given $a\in{\mathbb{D}}$ we denote by $\varphi_{a}$ the disc-
automorphism

$\varphi_{a}(z)=\frac{z-a}{1-\overline{a}z},\quad z\in{\mathbb{D}}.$

######  Lemma 1.

Let $s(z,w)=\frac{1}{1-z\overline{w}}$ and let $b$ be a finite Blaschke
product with zeros $a_{1},\ldots,a_{n}\in{\mathbb{D}}$. Then

(13)
$(1-b(z)\overline{b(w)})s(z,w)=\sum_{l=1}^{n}(\varphi_{a_{l}}^{\prime}(z)\overline{\varphi_{a_{l}}^{\prime}(w)})^{\frac{1}{2}}\prod_{j=1}^{l-1}\varphi_{a_{j}}(z)\overline{\varphi_{a_{j}}(w)}.$

###### Proof.

We proceed by induction. For $a\in{\mathbb{D}}$ a direct computation gives

$(1-\varphi_{a}(z)\overline{\varphi_{a}(w)})s(z,w)=(\varphi_{a}^{\prime}(z)\overline{\varphi_{a}^{\prime}(w)})^{\frac{1}{2}},$

which is the case $n=1$. Suppose that (13) holds for any Blaschke product with
$n$ zeros, and let $b$ be the finite Blaschke product with zeros
$a_{1},\ldots,a_{n+1}\in{\mathbb{D}}$. Let $b_{0}=1$ and let $b_{k},~{}k\geq
1,$ denote the Blaschke product having only the first $k$ zeros,
$a_{1},\ldots,a_{k}$. Write

$1-b(z)\overline{b(w)}=1-b_{n}(z)\overline{b_{n}(w)}+b_{n}(z)\overline{b_{n}(w)}(1-\varphi_{a_{n+1}}(z)\overline{\varphi_{a_{n+1}}(w)}).$

By the induction hypotheses

$(1-b_{n}(z)\overline{b_{n}(w)})s(z,w)=\sum_{j=1}^{n}(\varphi_{a_{l}}^{\prime}(z)\overline{\varphi_{a_{l}}^{\prime}(w)})^{\frac{1}{2}}b_{l-1}(z)\overline{b_{l-1}(w)},$

and by the case $n=1$,

$b_{n}(z)\overline{b_{n}(w)}(1-\varphi_{a_{n+1}}(z)\overline{\varphi_{a_{n+1}}(w)})s(z,w)=(\varphi_{a_{n+1}}^{\prime}(z)\overline{\varphi_{a_{n+1}}^{\prime}(w)})^{\frac{1}{2}}b_{n}(z)\overline{b_{n}(w)},$

which proves the induction step and the result follows. ∎

Proof of Theorem 4. The assumption $\|M_{z}\|\leq 1$ implies that for all
$a\in{\mathbb{D}}$, the functions
$u_{a}(z)=(1-\overline{a}z)^{-1},~{}z\in{\mathbb{D}}$ are multipliers, in fact
$M_{u_{a}}=(I-\overline{a}M_{z})^{-1}$. Assume that $I-M_{z}M_{z}^{*}$ is
compact and let $b$ be a finite Blaschke product with zeros
$a_{1},\ldots,a_{n}\in{\mathbb{D}}$. Multiply both sides of (13) by
$k(z,w)/s(z,w)$, use the identity
$u(z)\overline{u(w)}k(z,w)=M_{u}M_{u}^{*}k_{w}(z)$ together with the density
of the linear span of $\\{k_{w}:~{}w\in{\mathbb{D}}\\}$ to obtain an equality
of the form

(14)
$I-M_{b}M_{b}^{*}=\sum_{l=1}^{n}M_{v_{l}}(I-M_{z}M_{z}^{*})M_{v_{l}}^{*},$

with $v_{1},\ldots,v_{n}\in Mult({\mathcal{H}}_{k})$. Thus, by assumption,
each operator in the sum is compact, hence, so is $I-M_{b}M_{b}^{*}$. Now if
$J:{\mathcal{H}}_{k}(b)\to{\mathcal{H}}_{k}$ is the inclusion map, as above it
follows that

$J^{*}k_{w}(z)=(1-b(z)\overline{b(w)})k(z,w),$

hence $JJ^{*}=I-M_{b}M_{b}^{*}$ is compact. Conversely, if
$JJ^{*}=I-M_{b}M_{b}^{*}$ is compact, by Proposition 8, $b$ must be a Blaschke
product, so that (14) holds. Note that all operators on the right hand side
are positive, and in the first term we have

$v_{1}(z)=(\varphi_{a_{1}}^{\prime}(z))^{\frac{1}{2}}=\frac{(1-|a_{1}|^{2})^{\frac{1}{2}}}{1-\overline{a_{1}}z},$

is an invertible multiplier. This leads to

$M_{v_{1}}^{-1}(I-M_{b}M_{b}^{*})(M_{v_{1}}^{-1})^{*}\geq I-M_{z}M_{z}^{*}.$

Since the left hand side is compact and the right hand side is positive it
follows that $I-M_{z}M_{z}^{*}$ is compact and the proof is complete. ∎

The theorem applies not only to arbitrary weighted Bergman spaces on
${\mathbb{D}}$, or the Hardy space. The result holds in many other reproducing
kernel spaces, for example those corresponding to radial kernels

$k(z,w)=\sum_{n\geq 0}k_{n}(z\overline{w})^{n},\quad z,w\in{\mathbb{D}},$

with $0<k_{n}\leq k_{n+1},~{}\lim_{n\to\infty}\frac{k_{n}}{k_{n+1}}=1.$
It is an interesting question to characterize the compact embedding of
${\mathcal{H}}_{k}(b)$ into ${\mathcal{H}}_{k}$ for a a contractive row
multiplier $b$.

The situation is much more complicated for analytic reproducing kernels on the
unit ball ${\mathbb{B}}_{d},~{}d>1$. A particular case that can be treated
with the method above is when $b$ is an automorphism of ${{\mathbb{B}}_{d}}$
and ${\mathcal{H}}_{k}={\mathcal{H}}_{\beta}$ for some $\beta>1$. In this case
we have the following identity similar to (13) (see for example [24]),

$1-\langle\psi(z),\psi(w)\rangle=\frac{(1-\langle a,a\rangle)(1-\langle
z,w\rangle)}{(1-\langle z,a\rangle)(1-\langle a,w\rangle)}$

where $a=b^{-1}(0)$. This leads to

$s^{\beta}(z,w)(1-\psi(z)\psi(w)^{*})=s^{\beta-1}(z,w)\varphi(z)\overline{\varphi(w)},$

where $\varphi(z)=\frac{(1-\langle a,a\rangle)^{\frac{1}{2}}}{1-\langle
z,a\rangle}$ is an invertible multiplier of ${\mathcal{H}}_{\beta-1}$. In
other words, ${\mathcal{H}}_{\beta}(b)={\mathcal{H}}_{\beta-1}$ with
equivalent norms. This proves the following result.

######  Proposition 10.

If $\beta>1$ and $b$ is an automorphism of ${{\mathbb{B}}_{d}}$, then
${{\mathcal{H}}_{\beta}}(b)$ is compactly embedded in
${{\mathcal{H}}_{\beta}}$.

###  3.3 A model for ${{\mathcal{H}}_{k}}(b)$

One of the main difficulties that occur when dealing with de-Branges-Rovnyak
spaces and their generalizations is finding functions in the space other than
finite linear combinations of reproducing kernels. In some cases this led to a
number of recent non-trivial results. For example, the main result in [5]
shows that in one variable, ${\mathcal{H}}_{s}(b)$ contains a dense subset of
functions that extend continuously to $\overline{{\mathbb{D}}}$. Further
refinements of that result have been recently obtained by Limani and Malman
[17]. the method in [5] relies on a special formula for the norm in
${\mathcal{H}}_{s}(b)$ which turns out to be essentially related to the famous
Nagy-Foiaş model for the backward shift on these spaces (see [29]). Some of
these ideas do extend to the general context of ${\mathcal{H}}_{k}(b)$-spaces
and provide useful information about their structure. To illustrate this,
consider a scalar-valued reproducing kernel $k$ on the non-empty set
${\mathcal{X}}$ with corresponding Hilbert space ${\mathcal{H}}_{k}$ and fix a
contractive row-multiplier
$b\in\operatorname{Mult}({\mathcal{H}}_{k}\otimes\ell^{2},{\mathcal{H}}_{k})$.
The considerations below apply to a scalar, or finite dimensional vector $b$
is a trivial way. Therefore for the rest of the subsection we shall consider
the case when
$b\in\operatorname{Mult}({\mathcal{H}}_{k}\otimes\ell^{2},{\mathcal{H}}_{k})$.

Denote by
$\Delta=(I-M_{b}^{*}M_{b})^{\frac{1}{2}}:{\mathcal{H}}_{k}\otimes\ell^{2}\to{\mathcal{H}}_{k}\otimes\ell^{2}$.
If $\Delta$ (or $I-M^{*}_{b}M_{b}$) is injective, so that

$\|f\|_{\Delta}=\|\Delta f\|,$

is a norm on ${\mathcal{H}}_{k}\otimes\ell^{2}$. In this case we let
${\mathcal{L}}_{\Delta}$ be the completion of
${\mathcal{H}}_{k}\otimes\ell^{2}$ with respect to $\|\cdot\|_{\Delta}$. We
shall be interested in the closed subspace
${\mathcal{M}}\subset{\mathcal{H}}_{k}\oplus{\mathcal{L}}_{\Delta}$ defined by

(15) ${\mathcal{M}}=\\{(bf,f):f\in{\mathcal{H}}_{k}\otimes\ell^{2}\\}$

Let us record two simple facts about this subspace.

######  Proposition 11.

Suppose that $\Delta$ is injective and let
${\mathcal{M}}\subset{\mathcal{H}}_{k}\oplus{\mathcal{L}}_{\Delta}$ be the
subspace given by (15). Then:

  1. (i)

${\mathcal{M}}$ is closed,

  2. (ii)

For $(u,v)\in{\mathcal{H}}_{k}\oplus{\mathcal{H}}_{k}\otimes\ell^{2}$ its
orthogonal projection onto ${\mathcal{M}}$, $P_{\mathcal{M}}(u,v)$ is given by

(16) $P_{\mathcal{M}}(u,v)=(bw,w),\qquad w=M_{b}^{*}u+\Delta^{2}v.$

###### Proof.

(i) If $((bf_{n},f_{n}))$ converges in
${\mathcal{H}}_{k}\oplus{\mathcal{L}}_{\Delta}$ then from

$\|(bf_{n},f_{n})-(bf_{m},f_{m})\|^{2}=\|b(f_{n}-f_{m})\|^{2}_{{\mathcal{H}}_{k}}+\|\Delta(f_{n}-f_{m})\|^{2}_{{\mathcal{H}}_{k}\otimes\ell^{2}}=\|f_{n}-f_{m}\|^{2}_{{\mathcal{H}}_{k}\otimes\ell^{2}},$

we see that $(f_{n})$ is Cauchy in ${\mathcal{H}}_{k}\otimes\ell^{2}$, hence
convergent.

(ii) With $f\in{\mathcal{H}}_{k}\otimes\ell^{2}$ and $w$ as in the statement
we have

$\displaystyle\langle(bf,f),(bw,w)\rangle_{{\mathcal{H}}_{k}\oplus{\mathcal{L}}_{\Delta}}$
$\displaystyle=\langle f,w\rangle_{{\mathcal{H}}_{k}\otimes\ell^{2}}$
$\displaystyle=\langle
f,M_{b}^{*}u+\Delta^{2}v\rangle_{{\mathcal{H}}_{k}\otimes\ell^{2}}$
$\displaystyle=\langle(bf,f),(u,v)\rangle{{\mathcal{H}}_{k}\oplus{\mathcal{L}}_{\Delta}}.$

∎

The main result of this subsection is given below.

######  Theorem 5.

Suppose that $\Delta$ is injective and let
${\mathcal{M}}\subset{\mathcal{H}}_{k}\oplus{\mathcal{L}}_{\Delta}$ be the
subspace given by (15). Then the map
$J:{\mathcal{M}}^{\perp}\to{\mathcal{H}}_{k}(b)$, $J(u,v)=u$ is unitary.
Equivalently, for every $f\in{\mathcal{H}}_{k}(b)$ there exists a unique
$g\in{\mathcal{L}}_{\Delta}$ such that $(f,g)\in{\mathcal{M}}^{\perp}$ and

$\|f\|_{{\mathcal{H}}_{k}(b)}^{2}=\|f\|_{{\mathcal{H}}_{k}}^{2}+\|g\|_{{\mathcal{L}}_{\Delta}}^{2}.$

###### Proof.

Let us note first that if
$(0,g)\in{\mathcal{M}}^{\perp},~{}g\in{\mathcal{L}}_{\Delta}$, then $g=0$.
Indeed, in this case $g$ is orthogonal in ${\mathcal{L}}\Delta$ to the dense
subspace ${\mathcal{H}}_{k}\otimes\ell^{2}$, hence $g=0$. In particular, if
$P_{{\mathcal{M}}^{\perp}}=I-P_{\mathcal{M}}$, this implies that the linear
span of $\\{P_{{\mathcal{M}}^{\perp}}(k_{y},0):~{}y\in{\mathcal{X}}\\}$ is
dense in ${\mathcal{M}}^{\perp}$. By Proposition 11 (ii) we can easily
calculate the elements of this set. Indeed, according to (16) we have

$P_{\mathcal{M}}(k_{y},0)=(bw,w),\qquad w=M_{b}^{*}k_{y},$

so that

$(I-P_{\mathcal{M}})(k_{y},0)=P_{{\mathcal{M}}^{\perp}}(k_{y},0)=(k_{y}-b(x)b^{*}(y)k_{y},-b^{*}(y)k_{y})=(k_{y}^{b},-b^{*}(y)k_{y}),$

where $k^{b}$ denotes the reproducing kernel in ${\mathcal{H}}_{k}(b)$. But
then

$\displaystyle\langle
P_{{\mathcal{M}}^{\perp}}(k_{y},0),P_{{\mathcal{M}}^{\perp}}(k_{z},0)\rangle_{{\mathcal{H}}_{k}\oplus{\mathcal{L}}_{\delta}}$
$\displaystyle=\langle
k_{y}-bb^{*}(y)k_{y},k_{z}-bb^{*}(z)k_{z}\rangle_{{\mathcal{H}}_{k}}+\langle\Delta
b^{*}(y)k_{y},b^{*}(z)k_{z}\rangle_{{\mathcal{H}}_{k}\otimes\ell^{2}}$
$\displaystyle=k_{y}(z)-b(z)b^{*}(y)k_{y}(z)-\overline{b(y)b^{*}(z)k_{z}(y)}+\langle
bb^{*}(y)k_{y},bb^{*}(z)k_{z}\rangle_{{\mathcal{H}}_{k}}$
$\displaystyle\quad+b(z)b^{*}(y)k_{y}(z)-\langle
bb^{*}(y)k_{y},bb^{*}(z)k_{z}\rangle_{{\mathcal{H}}_{k}}$
$\displaystyle=k_{y}(z)-b(z)b^{*}(y)k_{y}(z)=k_{y}^{b}(z).$

Thus the map $J$ in the statement preserves the norm on a dense subset of
${\mathcal{M}}^{\perp}$ and maps it onto a dense subset of
${\mathcal{H}}_{k}(b)$. The result follows.

∎

The hypotheses that $\Delta$ is injective might seem restrictive. It does not
apply to inner functions $b$ in the case of the Hardy space on ${\mathbb{D}}$,
or, more generally to sub-inner multipliers of ${\mathcal{H}}_{k}$. It turns
out that smaller spaces than $H^{2}$ on ${\mathbb{D}}$, like for example
standard weighted Dirichlet spaces possess a large set of sub-inner
multipliers (see [6], Theorem 14.9). On the other hand, recall that weighted
Bergman spaces do not have such multipliers.

##  4 Approximation results

###  4.1 The general approximation theorem

Given a space ${\mathcal{H}}_{k}(b)$, one of the main difficulties in
understanding its structure is the lack of knowledge about its elements other
than the reproducing kernels. Therefore it is natural to relate this problem
to elements in the larger (”known”) space ${\mathcal{H}}_{k}$, in particular
to the original kernels $k_{y},~{}y\in X$. We shall consider the map $J$ from
Theorem 5, more precisely, its inverse
$J^{-1}:{\mathcal{H}}_{k}(b)\to{\mathcal{H}}_{k}\oplus{\mathcal{L}}_{\Delta}$
and denote, as in the previous subsection, by ${\mathcal{M}}$ the closed
subspace of ${\mathcal{H}}_{k}\oplus{\mathcal{L}}_{\Delta}$ with

${\mathcal{M}}=\\{(bg,g):~{}g\in{\mathcal{H}}_{k}\otimes\ell^{2}\\}.$

######  Proposition 12.

Assume that $\Delta$ is injective. If $y\in{\mathcal{X}}$ then
$k_{y}\in{\mathcal{H}}_{k}(b)$ if and only if there exists $c_{y}>0$ such that

(17) $|b(y)g(y)|\leq c_{y}\|g\|_{\Delta},\quad
g\in{\mathcal{H}}_{k}\otimes\ell^{2}.$

In this case, there exists a unique $l_{y}\in{\mathcal{L}}_{\Delta}$ with

(18) $\langle g,l_{y}\rangle_{{\mathcal{L}}_{\Delta}}=-b(y)g(y),\quad
g\in{\mathcal{H}}_{k}\otimes\ell^{2},$

and $J^{-1}k_{y}=(k_{y},l_{y})$.

###### Proof.

If $k_{y}\in{\mathcal{H}}_{k}(b)$ and
$J^{-1}k_{y}=(k_{y},l_{y})\in{\mathcal{M}}^{\perp}$, then $l_{y}$ must satisfy
(18) which implies (17). Conversely, if (17) holds, the existence and
uniqueness of $l_{y}$ follows by the Riesz representation theorem and Theorem
5 gives $k_{y}\in{\mathcal{H}}(b)$. ∎

Under the assumption $k_{y}\in{\mathcal{H}}(b),~{}y\in{\mathcal{X}}$, the
natural question regarding approximation in this space is whether the linear
span of these functions is dense. We provide a sufficient condition for this
conclusion.

######  Theorem 6.

Suppose that $\Delta$ is injective and that:

  1. (i)

If $y\in{\mathcal{X}},~{}e\in\ell^{2}$ there exists $d_{y}>0$ such that

(19) $|\langle g(y),e\rangle_{\ell^{2}}|\leq
d_{y}\|g\|_{\Delta}\|e\|_{2},\quad g\in{\mathcal{H}}_{k}\otimes\ell^{2}.$

Equivalently, point evaluations are continuous on ${\mathcal{L}}_{\Delta}$.

  1. (ii)

If $f\in{\mathcal{L}}_{\Delta}$ and $bf\in{\mathcal{H}}_{k}$ then
$f\in{\mathcal{H}}_{k}\otimes\ell^{2}.$

Then the linear span of $\\{k_{y}:y\in{\mathcal{X}}\\}$ is contained and dense
in ${\mathcal{H}}_{k}(b)$.

###### Proof.

The hypothesis (i) together with Proposition 12 imply that
$k_{y}\in{\mathcal{H}}_{k}(b),\,y\in{\mathcal{X}}$. Now suppose that
$f\in{\mathcal{H}}_{k}(b)$ annihilates $k_{y},~{}y\in{\mathcal{X}}$. Let

$J^{-1}f=(f,g),\,\,J^{-1}k_{y}=(k_{y},l_{y})\in{\mathcal{M}}^{\perp},\quad
y\in{\mathcal{X}},$

and apply Proposition 12 to conclude that (18) holds for all
$y\in{\mathcal{X}}$. Thus

$0=\langle f,k_{y}\rangle_{{\mathcal{H}}_{k}(b)}=\langle
J^{-1}f,J^{-1}k_{y}\rangle_{{\mathcal{H}}_{k}\oplus{\mathcal{L}}_{\Delta}}=f(y)-b(y)g(y),$

for all $y\in{\mathcal{X}}$, i.e. $f=bg\in{\mathcal{H}}_{k}$. But then by
assumption (ii), we have $g\in{\mathcal{H}}_{k}\otimes\ell^{2}$ and
$J^{-1}f=(f,g)=(bg,g)\in{\mathcal{M}}$ which implies $(f,g)=0$ and the result
follows. ∎

######  Remark 1.

  1. 1)

In the classical case when ${\mathcal{H}}_{k}$ equals the Hardy space $H^{2}$
and $b$ is a scalar analytic function bounded by $1$ in ${\mathbb{D}}$,
assumption (i) is equivalent to the fact that $b$ is not extremal in
$H^{\infty}$. This follows immediately by the standard Szegö theorem.
Moreover, in this case there exists an outer function $a$ with
$|a|^{2}+|b|^{2}=1$ a.e. on the unit circle, which easily implies that (ii)
holds automatically.

  2. 2)

In the general case, an analogue of the above condition is to assume that
there exists a multiplier $a$ such that $I-M_{b}^{*}M_{b}\geq M_{a}^{*}M_{a}$
(see for example [16] and the discussion in the Introduction). If $a$ is zero-
free then this assumption implies assumption (i) in the theorem above.
However, we do not know whether it implies (ii) as well.

##  5 Weighted sub-Bergman spaces

Let $\Omega$ be a domain of ${\mathbb{C}}^{d}$, $\mu$ be a positive measure on
$\Omega$ such that the Bergman space $L^{2}_{a}(\mu)$ is a closed subspace of
$L^{2}(\mu)$ with reproducing kernel $k^{\mu}$ (see section 2.3. For an
analytic row-contraction $b$ on $\Omega\subset{\mathbb{C}}^{d}$ we consider
the sub- Bergman space ${\mathcal{H}}_{k^{\mu}}(b)$. The aim of this
subsection is to show that the general result given in Theorem 6 does always
apply to these spaces.

######  Theorem 7.

Assume that $\mu$ satisfies (7) and let $b$ be any analytic row-contraction on
$\Omega$ with $\|b(z_{0})\|<1$ for some $z_{0}\in\Omega$. Then the linear span
of $\\{k^{\mu}_{z}:~{}z\in\Omega\\}$ is dense in ${\mathcal{H}}_{k^{\mu}}(b)$.

###### Proof.

The statement is obvious when $b$ is a constant with $\|b\|<1$, so we may
assume $b$ is not constant. We want to apply Theorem 6. To this end we need to
verify the assumptions (i) and (ii) in that theorem. Assumption (i) follows by
an application of Proposition 5. To verify (ii), let
$g\in{\mathcal{L}}_{\Delta}$ with $bg\in L_{a}^{2}(\mu)$ and let $(g_{n})$ be
a sequence in $L_{a}^{2}(\mu)\otimes\ell^{2}$ with $\|g_{n}-g\|_{\Delta}\to
0$. By assumption (i) we have for each $z\in\Omega$,
$\lim_{n\to\infty}bg_{n}(z)=bg(z)$, and that $(g_{n}(z))$ converges weakly to
$g(z)$ in $\ell^{2}$, in particular,
$\|g(z)\|_{2}\leq\liminf_{n\to\infty}\|g_{n}(z)\|_{2}$. Then since $b(z)$ is a
contractive row contraction we can apply Fatou’s lemma to obtain

$\displaystyle\|g\|_{\Delta}^{2}=\lim_{n\to\infty}\|g_{n}\|_{\Delta}^{2}=\lim_{n\to\infty}\int_{\Omega}(\|g_{n}(z)\|_{2}^{2}-|bg_{n}(z)|^{2})d\mu(z)$
$\displaystyle\qquad\geq\int_{\Omega}\liminf_{n\to\infty}(\|g_{n}(z)\|_{2}^{2}-|bg_{n}(z)|^{2})d\mu(z)\geq\int_{\Omega}(\|g(z)\|_{2}^{2}-|bg(z)|^{2})d\mu(z).$

This gives

$\int_{\Omega}\|g(z)\|^{2}d\mu(z)\leq\|g\|_{\Delta}^{2}+\int_{\Omega}|bg(z)|^{2}d\mu(z),$

that is, assumption (ii) is verified as well. Then the result follows by a
direct application of Theorem 6. ∎

###  5.1 Approximation by polynomials in standard weighted sub-Bergman spaces
on the unit ball

In this subsection we shall specialize to unitarily invariant weighted Bergman
spaces on ${\mathbb{B}}_{d}$, the unit ball in ${\mathbb{C}}^{d}$. These are
the spaces ${\mathcal{H}}_{\beta}$ defined in §2.2, where $\beta>d$
corresponding to measures that satisfy (7). Recall that their reproducing
kernel is given by

$s^{\beta}(z,w)=\frac{1}{(1-\langle z,w\rangle)^{\beta}},\quad
z,w\in{\mathbb{B}}_{d}.$

For such $\beta$ we let $b$ be an analytic row-contraction with $\|b(0)\|<1$.
The sub-Bergman space ${{\mathcal{H}}_{\beta}}(b)$ is the Hilbert space
${\mathcal{H}}_{s^{\beta}}(b)$, that is, the Hilbert space with reproducing
kernel

$s^{\beta}(z,w)(1-b(z)b(w)^{*})=\frac{1-b(z)b(w)^{*}}{(1-\langle
z,w\rangle)^{\beta}}.$

To simplify notations we write
${\mathcal{H}}_{s^{\beta}}(b)={\mathcal{H}}_{\beta}(b)$. Let us note first the
following property of the multipliers of these spaces.

######  Proposition 13.

Let $\beta>d$ and let $b$ an analytic row-contraction with $\|b(0)\|<1$. Then
for every $0<\gamma<\beta-d$, $\operatorname{Mult}({\mathcal{H}}_{\gamma})$ is
contractively contained in $\operatorname{Mult}({\mathcal{H}}_{\beta}(b))$. In
particular, if $u$ is analytic in a neighborhood of
$\overline{{\mathbb{B}}_{d}}$ then
$u\in\operatorname{Mult}({\mathcal{H}}_{\beta}(b))$.

###### Proof.

The first assertion is a direct application of Proposition 3 (i). The fact
that analytic functions in a neighborhood of $\overline{{\mathbb{B}}_{d}}$ are
multipliers of ${\mathcal{H}}_{\gamma},~{}\gamma>0$, is well known and easy to
prove. We omit the details. ∎

With this observation in hand we can turn to the approximation result for
these spaces. It actually shows that within this context Sarason’s dichotomy
does not occur.

######  Corollary 2.

Let $\beta>d$ and let $b$ an analytic row-contraction with $\|b(0)\|<1$. Then
polynomials form a dense subset of ${\mathcal{H}}_{\beta}(b)$. Moreover, for
any $e\in\ell^{2}$, the function $z\mapsto b(z)e$ belongs to
${{\mathcal{H}}_{\beta}}(b)$.

###### Proof.

By Theorem 7, ${\mathcal{H}}_{\beta}(b)$ contains all kernels
$s_{w}^{\beta},~{}w\in{\mathbb{B}}_{d}$, of ${\mathcal{H}}_{\beta}$. By
Proposition 13 we have
$s_{w}^{-\beta}\in\operatorname{Mult}({\mathcal{H}}_{\beta}(b))$, hence
$1=s_{w}^{-\beta}s_{w}^{\beta}\in{\mathcal{H}}_{\beta}(b)$. The same argument
shows that $1\in{\mathcal{H}}_{\beta-\gamma}(b)$ whenever
$\gamma>0,~{}\beta-\gamma>d$. But then Proposition 3 (ii) implies that for
such $\gamma$, ${\mathcal{H}}_{\gamma}$ is continuously contained in
${\mathcal{H}}_{\beta}(b)$. Since the kernels $s_{w}^{\beta}$ can be
approximated by polynomials in ${\mathcal{H}}_{\gamma}$ the same holds in
${\mathcal{H}}_{\beta}(b)$ and another application of Theorem 7 shows that the
closure of polynomials in ${\mathcal{H}}_{\beta}(b)$ contains the dense set
$\text{span}\\{s_{w}^{\beta}:~{}w\in{\mathbb{B}}_{d}\\}$ which concludes the
proof.

The previous paragraph shows that $z\mapsto b(z)b(w)^{*}$ belongs to
${{\mathcal{H}}_{\beta}}$ for any $w\in{{\mathbb{B}}_{d}}$. Note that $e\perp
b(w)^{*}$ is equivalent to $b(w)e=0$, this proves the last statement of the
Corollary. ∎

Using a completely different method, Chu [11] proved the density of
polynomials for scalar-valued $b$ in the case when $d=1$ and $\beta=2$.

##  6 Approximation by polynomials in higher order de Branges-Rovnyak spaces

Let $k$ be a reproducing kernel on a non-empty set ${\mathcal{X}}$, and let
$b$ be a contractive multiplier of ${\mathcal{H}}_{k}$. In this section we
shall consider kernels of the form

(20) $k^{b,m}(x,y)=k(x,y)(1-b(x)\overline{b(y)})^{m},\quad
x,y\in{\mathcal{X}},$

where $m$ is a fixed positive integer. Such kernels have been recently
introduced and thoroughly studied in [13], in connection with
$m-$hypercontractive operators.

######  Definition 4.

A bounded operator $A$ on the Hilbert space ${\mathcal{H}}$ is called an _$m$
-hypercontraction_ if

(21) $\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}A^{*j}A^{j}\geq 0,$

for all $0\leq n\leq m$, where $m,n\in{\mathbb{N}}$.

The result below is a slight simplification of [13, Corollary 3.6].

######  Proposition 14.

Let $b$ be a contractive multiplier of ${\mathcal{H}}_{k}$ such that
$|b(x)|<1$ for every $x\in{\mathcal{X}}$. Then $M_{b}^{*}$ is a
$m$-hypercontraction if and only if

(22) $k^{b,m}(x,y)=k(x,y)(1-b(x)\overline{b(y)})^{m}\gg 0.$

In this case,

$k(x,y)(1-b(x)b(y))^{n}\gg 0,$

for every integer $n$ with $0\leq n\leq m$.

###### Proof.

We use repeatedly that the bounded operator $T$ on the reproducing kernel
Hilbert space ${\mathcal{H}}_{k}$ is positive if and only if $Tk(x,y)\gg 0$.
Thus if $M_{b}^{*}$ is $m-$hypercontractive then (21) holds for $n=m$, hence

$\displaystyle\sum_{j=0}^{m}(-1)^{j}\binom{n}{j}M_{b}^{j}M_{b}^{*j}k(x,y)$
$\displaystyle=\sum_{j=0}^{m}((-1)^{j}\binom{n}{j}b(x)^{j}\overline{b(y)}^{j})k(x,y)$
$\displaystyle=(1-b(x)\overline{b(y)})^{m}k(x,y)\gg 0.$

Conversely, if (22) holds then (21) holds for $n=m$. Moreover, by Schur’s
product theorem any positive power of a complete Nevanlinna-Pick kernel is
positive, therefore

$\frac{1}{(1-b(x)\overline{b(y)})^{m-n}}\gg 0,$

whenever $n<m$. Then another application of Schur’s theorem gives that the
product of the last two positive definite functions is positive definite, i.e.

$k(x,y)(1-b(x)b(y))^{n}\gg 0,$

whenever $n<m$. Equivalently, (21) holds for $0\leq n<m$. Thus $M_{b}^{*}$ is
$m-$hypercontractive and the result follows. ∎

We denote by ${\mathcal{H}}_{k}(b,m)$ the Hilbert space with reproducing
kernel $k^{b,m}$. We shall investigate polynomial approximation in
$k=s^{\beta}$. To simplify notations we denote these spaces by
${\mathcal{H}}_{\beta}(b,m)$. The following result completely solves a
conjecture stated in [13] in one complex variable ($d=1$).

######  Corollary 3.

Let $d,m\geq 1$ be integers with $md<\beta$, let $k=s^{\beta}$ and $b\in
H^{\infty}({\mathbb{B}}_{d})$ with $\|b\|_{\infty}\leq 1$ and $|b(0)|<1$. Then
polynomials form a dense subset of ${\mathcal{H}}_{\beta}(b,m)$.

###### Proof.

By corollary 2, polynomials form a dense subset in
${\mathcal{H}}_{\beta/m}(b)$, hence by corollary 1 they form a dense subset of
${\mathcal{H}}_{\beta}(b,m)$. ∎

######  Remark 2.

The result holds when $d=1$ and $m=\beta$, provided that $b$ is a non-
extremal point in the unit ball of $H^{\infty}$. Instead of corollary 2 just
apply Sarason’s dichotomy [25] mentioned in the Introduction.

These methods generalize to cover other situations. For example, letting
$b_{1},\ldots,b_{m}$ be non-constant in the unit ball of $H^{\infty}$ one can
prove similar results about the space with kernel

$\frac{\prod_{j=1}^{m}(1-b_{j}(z)\overline{b_{j}(w)})}{(1-z\overline{w})^{\beta}},$

where $\beta\geq m$.

Furthermore, we can also consider the case when $b$ is vector-valued. However,
in this case, when $\beta=m$ Sarason’s dichotomy is only known to hold when
$b$ is a finite dimensional vector.

When $\beta>m$ all results mentioned in this remark extend to the context of
several complex variables.

######  Question 1.

If $d=1$, $\beta=m$ and $b(z)=z$, then clearly
${\mathcal{H}}_{\beta}(b,m)={\mathbb{C}}$. Does there exist $m>1$ and $b$
extremal in the unit ball of $H^{\infty}$ such that
${\mathcal{H}}_{\beta}(b,m)$ contains all polynomials? Do they form a dense
subset?

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