Density of disc algebra functions in de Branges-Rovnyak spaces

TL;DR

This paper proves that in de Branges-Rovnyak spaces generated by extreme points of the unit ball of H-infinity, functions that are continuous up to the boundary are dense, enhancing understanding of their structure.

Contribution

It establishes the density of boundary-continuous functions in specific de Branges-Rovnyak spaces, a novel result in the theory of these function spaces.

Findings

Boundary-continuous functions are dense in certain de Branges-Rovnyak spaces.

The result applies to spaces induced by extreme points of the unit ball of H-infinity.

This advances the understanding of the boundary behavior in these spaces.

Abstract

We prove that functions continuous up to the boundary are dense in de Branges-Rovnyak spaces induced by extreme points the unit ball of $H^\infty$.

Equations

$$\mathcal{H}(b)=(1-T_b{T}_{\overline{b}}{)}^{1/2}{H}^2.$$

$$k_b(z,\lambda )=\frac{1-\overline{b(\lambda )}b(z)}{1-\overline{\lambda }z}.$$

$$E=\{\zeta \in \mathbb{T}:|b(\zeta )|<1\}.$$

$$P_{+}\overline{b}f=-P_{+}\sqrt{1-|b{|}^2}g.$$

$$Jf=(f,g),$$

$$J(\mathcal{H}(b))^{\perp}=\Big{\{}(bh,\sqrt{1-|b|^{2}}h):h\in H^{2}\Big{\}}.$$

$$K=\Big{\{}(bh,\sqrt{1-|b|^{2}}h):h\in H^{2}\Big{\}},$$

$${\int }_{\mathbb{T}}{\overline{\zeta }}^{n}g(\zeta )\sqrt{1-|b(\zeta ){|}^2}dm(\zeta )=0,n\ge 0,$$

$$(f_{\lambda },g_{\lambda })$$

$$\displaystyle=\Big{(}\frac{1}{1-\overline{\lambda}z},0\Big{)}-\Big{(}\frac{\overline{b(\lambda)}b(z)}{1-\overline{\lambda}z},\frac{\overline{b(\lambda)}\sqrt{1-|b(z)|^{2}}}{1-\overline{\lambda}z}\Big{)}$$

$${\int }_{\mathbb{T}}{\log }^{+}(|f_n(r{e}^{it})|)dt\le C.$$

$$⟨f,C\mu ⟩=\underset{r\to 1^{-}}{\lim }{\int }_{\mathbb{T}}f(\zeta )\overline{C\mu (r\zeta )}dm(\zeta )={\int }_{\mathbb{T}}fd\overline{\mu },$$

$$C\mu (z)={\int }_{\mathbb{T}}\frac{1}{1-z\overline{\zeta }}d\mu (\zeta )$$

$$S=\{(C\mu ,h):C\mu /b\in N^{+}(\mathbb{D}),C\mu /b=h/\sqrt{1-|b{|}^2}\text{mbox}a.e.onE\}$$

$$\underset{n}{\sup }\Vert C{\mu }_n\Vert <\infty .\underset{n\to \infty }{\lim }C{\mu }_n(z)=C\mu (z),z\in \mathbb{D}.$$

$${\int }_{\mathbb{T}}{\log }^{+}(|C{\mu }_n(r{e}^{it})|)dt\lesssim {\int }_{\mathbb{T}}|C{\mu }_n(r{e}^{it}){|}^{p}dt\lesssim \Vert C{\mu }_n{\Vert }_p^p\lesssim \underset{n}{\sup }\Vert C{\mu }_n\Vert <\infty .$$

$$J(\mathcal{A}\cap \mathcal{H}(b))={\cap }_{h\in H^2}\ker l_h,$$

$$l_{h}=\Big{(}hb,h\sqrt{1-|b|^{2}}\Big{)}.$$

$$Jf=(hb,h\sqrt{1-|b{|}^2})$$

Full text

1   Introduction

Let $H^{\infty}$ be the algebra of bounded analytic functions in the unit disk $\mathbb{D}$ in the complex plane, and denote by $\mathcal{A}$ the disc algebra, i.e. the subalgebra of $H^{\infty}$ consisting of functions which extend continuously to the closed disk. The Hardy space $H^{2}$ consists of power series in $\mathbb{D}$ with square-summable coefficients. If $\mathbb{T}$ denotes the unit circle, we identify as usual $H^{2}$ with the closed subspace of $L^{2}(\mathbb{T})$ consisting of functions whose negative Fourier coefficients vanish. The orthogonal projection from $L^{2}(\mathbb{T})$ onto $H^{2}$ is denoted by $P_{+}$.

For $\phi\in L^{\infty}(\mathbb{T})$ let $T_{\phi}$ denote the Toeplitz operator on $H^{2}$ defined by $T_{\phi}f=P_{+}\phi f$. Given $b\in H^{\infty}$ with $\|b\|_{\infty}\leq 1$ we define the corresponding de Branges-Rovnyak space $\mathcal{H}(b)$ as

[TABLE]

$\mathcal{H}(b)$ is endowed with the unique norm which makes the operator $(1-T_{b}T_{\overline{b}})^{1/2}$ a partial isometry from $H^{2}$ onto $\mathcal{H}(b)$. Alternatively, $\mathcal{H}(b)$ is defined as the reproducing kernel Hilbert space with kernel

[TABLE]

$\mathcal{H}(b)$-spaces are naturally split into two classes with fairly different structures according to whether the quantity $\int_{\mathbb{T}}\log(1-|b|)\,dm$ is finite or not. Here $m$ denotes the normalized arc-length measure on $\mathbb{T}$. The present note concerns the approximation of $\mathcal{H}(b)$-functions by functions in $\mathcal{A}\cap\mathcal{H}(b)$ and from the technical point of view there is a major difference between the two classes, which we shall briefly explain.

If $\int_{\mathbb{T}}\log(1-|b|)\,dm<\infty$, or equivalently, if $b$ is a non-extreme point of the unit ball of $H^{\infty}$ (see [8]), then $\mathcal{H}(b)$ contains all functions analytic in a neighborhood of the closed unit disk. By a theorem of Sarason, the polynomials form a norm-dense subset of the space [8]. An interesting feature of the proofs of density of polynomials in an $\mathcal{H}(b)$-space is that the usual approach of approximating a function $f$ first by its dilations $f_{r}(z)=f(rz)$, and then by their truncated Taylor series, or by their Cesàro means, does not work. Sarason’s intial proof of density of polynomials is based on a duality argument. In recent years a more involved constructive polynomial approximation scheme has been obtained in [6].

The picture changes dramatically in the case when $\int_{\mathbb{T}}\log(1-|b|)\,dm=\infty$, or equivalently when $b$ is an extreme point of the unit ball of $H^{\infty}$. Then it is in general a difficult task to identify any functions in the space other than the reproducing kernels, and it might happen that $\mathcal{H}(b)$ contains no non-zero function analytic in a neighborhood of the closed disk. A special class of extreme points are the inner functions. If $b$ is inner then $\mathcal{H}(b)=H^{2}\ominus bH^{2}$ with equality of norms, and it is a consequence of a celebrated theorem of Aleksandrov [1] that in this case the intersection $\mathcal{A}\cap\mathcal{H}(b)$ is dense in the space. The result is surprising since, as pointed out above, in most cases it is not obvious at all that $\mathcal{H}(b)$ contains any non-zero function in the disk algebra $\mathcal{A}$.

Motivated by the situation described here, E. Fricain [4], raised the natural question whether Aleksandrov’s result extends to all other $\mathcal{H}(b)$-spaces induced by extreme points $b$ of the unit ball of $H^{\infty}$. It is the purpose of this note to provide an affirmative answer to this question, contained in the main result below.

Theorem 1**.**

\thlabel

theorem If $b$ is an extreme point of the unit ball of $H^{\infty}$ then $\mathcal{A}\cap\mathcal{H}(b)$ is a dense subset of $\mathcal{H}(b)$.

Together with Sarason’s result [8] on the density of polynomials in the non-extreme case, it follows that the intersection $\mathcal{A}\cap\mathcal{H}(b)$ is dense in the space $\mathcal{H}(b)$ for any $b$ in the unit ball of $H^{\infty}$. Our proof Theorem LABEL:theorem is deferred to Section 3 and relies on a duality argument. Therefore, just as the earlier proofs of Sarason and Aleksandrov, our approach is non-constructive. Section 2 serves to a preliminary purpose.

2   Preliminaries

2.1  The norm on $\mathcal{H}(b)$.

An essential step is the following useful representation of the norm in $\mathcal{H}(b)$. The authors have originally deduced the result using the techniques in [3] (see also [2, Chapter 3]), but once the goal is identified, several available techniques provide simpler proofs. For example, the proposition below can be deduced from results in [8]. For the sake completeness, we include a new shorter proof.

Proposition 2**.**

\thlabel

normformula Let $b$ be an extreme point of the unit ball of $H^{\infty}$ and let

[TABLE]

Then for $f\in\mathcal{H}(b)$ the equation

[TABLE]

has a unique solution $g\in L^{2}(E)$, and the map $J:\mathcal{H}(b)\to H^{2}\oplus L^{2}(E)$ defined by

[TABLE]

is an isometry. Moreover,

[TABLE]

Proof.

Let

[TABLE]

and let $P_{1}$ be the projection from $H^{2}\oplus L^{2}(E)$ onto the first coordinate $H^{2}$, i.e., $P_{1}(f,g)=f$. We observe first that $P_{1}|K^{\perp}$ is injective. Indeed, if $K^{\perp}$ contains a tuple of the form $(0,g)\in H^{2}\oplus L^{2}(E)$, it follows that

[TABLE]

and consequently the function $g\sqrt{1-|b|^{2}}$ coincides a.e. with the boundary values of the complex conjugate of a function $f\in H^{2}_{0}$. But the assumption that $b$ is an extreme point then implies that $\int_{\mathbb{T}}\log|f|\,dm=-\infty$, and since $f\in H^{2}$, we conclude that $f=0$, i.e, $g=0$. Thus, the space $\mathcal{H}=P_{1}K^{\perp}$ with the norm $\|f\|_{\mathcal{H}}=\|P_{1}^{-1}f\|_{H^{2}\oplus L^{2}(E)}$ is a Hilbert space of analytic functions on $\mathbb{D}$, contractively contained in $H^{2}$, in particular, it is a reproducing kernel Hilbert space. We now show that $\mathcal{H}$ equals $\mathcal{H}(b)$ by verifying that the reproducing kernels of the two spaces coincide. This follows from a simple computation. For $\lambda\in\mathbb{D}$, the tuple

[TABLE]

is obviously orthogonal to $K$, while the last tuple on the right hand side is in $K$, so that $f_{\lambda}$ is the reproducing kernel in $\mathcal{H}$, which obviously equals the reproducing kernel in $\mathcal{H}(b)$. The first assertion in the statement is now self-explanatory. ∎

2.2  The Khintchin-Ostrowski theorem.

Recall that analytic functions $f$ in $\mathbb{D}$ satisfy $\sup_{0<r<1}\int_{\mathbb{T}}\log^{+}|f_{r}|dm<\infty$ if and only if they are quotients of $H^{\infty}$-functions, in particular they have finite nontangential limits a.e. on $\mathbb{T}$ which define a boundary function denoted also by $f$. The class $N^{+}(\mathbb{D})$ consists of quotients of $H^{\infty}$-functions such that the denominator can be chosen to be outer, it contains all Hardy spaces $H^{p},~{}p>0$. The Khintchin-Ostrowski theorem reads as follows. A proof can be found in [7].

Theorem 3**.**

\thlabel

ostrowski Let $\{f_{n}\}$ be a sequence of functions analytic in the unit disk satisfying the following conditions:

1. (i)

There exists a constant $C>0$ such that

[TABLE] 2. (ii)

On some set $E$ of positive measure, the sequence $f_{n}$ converges in measure to a function $\phi$.

Then the sequence $f_{n}$ converges uniformly on compact subsets of the unit disk to a function $f$ which satisfies $f=\phi$ a.e. on $E$.

3   Proof of the main result

Due to Proposition LABEL:normformula we can now implement Aleksandrov’s strategy from [1] which will then be combined with the Khintchin-Ostrowski theorem.

Recall that the dual $\mathcal{A}^{\prime}$ of the disk algebra $\mathcal{A}$ can be identified with the space $\mathcal{C}$ of Cauchy transforms of finite measures on $\mathbb{T}$ ([5]) via the pairing

[TABLE]

where

[TABLE]

is the Cauchy transform of $\mu$. The space $\mathcal{C}$ is endowed with the obvious quotient norm and is continuously contained in all $H^{p}$ spaces for $0<p<1$. The following result extends Alexandrov’s approach to the context of $\mathcal{H}(b)$-spaces, when $b$ is extremal in the unit ball of $H^{\infty}$.

Lemma 4**.**

\thlabel

wsclosed Let $E=\{\zeta\in\mathbb{T}:|b(\zeta)|<1\}$, $B=\mathcal{A}\oplus L^{2}(E)$ and $B^{\prime}=C\oplus L^{2}(E)$. Then the set

[TABLE]

is weak- closed in $B^{\prime}$.*

Proof.

Since $\mathcal{A}\oplus L^{2}(E)$ is separable, it will be sufficient to show that $S$ is weak-* sequentially closed. Let $(C\mu_{n},h_{n})$ converge weak-* to $(C\mu,h)$, where $(C\mu_{n},h_{n})\in S$ for $n\geq 1$. Equivalently, $h_{n}\to h$ weakly in $L^{2}(E)$, and

[TABLE]

Now by passing to a subsequence and the Cesàro means of that subsequence we can assume that $h_{n}\to h$ in the $L^{2}$-norm. Finally using another subsequence we may also assume that $h_{n}\to h$ pointwise a.e. on $E$. Let $I_{b}$ be the inner factor of $b$. Since $C\mu_{n}/I_{b}\in N^{+}(\mathbb{D})$, it follows by Vinogradov’s theorem ([5, Theorem 6.5.1]) that $\{C\mu_{n}/I_{b}\}_{n}$ is a bounded sequence in $\mathcal{C}$ converging pointwise on $\mathbb{D}$ to $C\mu/I_{b}$. This implies weak-* convergence in $\mathcal{C}$, in particular, $C\mu/I_{b}\in C\subset N^{+}(\mathbb{D})$, and consequently, $C\mu/b\in N^{+}(\mathbb{D})$. Moreover, we have a.e. on $E$ that $C\mu_{n}/b=h_{n}/\sqrt{1-|b|^{2}}$ which converges pointwise to $h/\sqrt{1-|b|^{2}}$, hence we conclude that the sequence $C\mu_{n}$ converges in measure to some function $\phi$ on $E$. Finally, if $p\in(0,1)$ and $\|\cdot\|_{p}$ denotes the $H^{p}$-norm, then

[TABLE]

Thus the assumptions of \threfostrowski are satisfied, and so (a subsequence of) $C\mu_{n}$ converges a.e. on $E$ to $C\mu$. This clearly implies $C\mu/b=h/\sqrt{1-|b|^{2}}$ a.e. on $E$, i.e. $(C\mu,h)\in S$. ∎

We are now ready to complete the proof of the main theorem.

Proof of \threftheorem.

Let $J$ denote the embedding in \threfnormformula. Based on the pairing described at the beginning of this section, a direct application of \threfnormformula gives

[TABLE]

where the functionals $l_{h}$ are identified with elements of $\mathcal{C}\oplus L^{2}(E)$ as

[TABLE]

It is a consequence of the Hahn-Banach theorem that the annihilator $J(\mathcal{A}\cap\mathcal{H}(b))^{\perp}$ is the weak-* closure of the set of the functionals $l_{h}$. Since for all $h\in H^{2}$ we have $l_{h}\in S$, the set considered in \threfwsclosed, by the lemma we conclude that $J(\mathcal{A}\cap\mathcal{H}(b))^{\perp}\subset S$. Thus if $f\in\mathcal{H}(b)$ is orthogonal to $\mathcal{A}\cap\mathcal{H}(b)$, we must have $Jf\in S$, that is

[TABLE]

for some $h\in H^{2}$. But then by \threfnormformula, $Jf\in J(\mathcal{H}(b))^{\perp}$, which gives $Jf=0$ and the proof is complete. ∎