Toeplitz operators on backward shift invariant subspaces of H^p

TL;DR

This paper extends the theory of compressed Toeplitz operators from the Hilbert space setting to more general Hardy spaces, broadening understanding of their structure on backward shift invariant subspaces.

Contribution

It introduces new results on compressed Toeplitz operators on $H^p$ spaces, generalizing previous work from $H^2$ to a wider class of Hardy spaces.

Findings

Extended Toeplitz operator results from $H^2$ to $H^p$ spaces.

Characterized properties of compressed Toeplitz operators on backward shift invariant subspaces.

Provided new insights into operator behavior in non-Hilbert Hardy spaces.

Abstract

We extend results on compressed Toeplitz operators on the backward shift invariant subspaces of $H^2 $ to the context of the spaces $H^p$, $1<p<\infty.$

Equations

$$\underset{0<r<1}{\sup }\frac{1}{2\pi }{\int }_0^{2\pi }|f(r{e}^{i\theta }){|}^{p}d\theta <\infty .$$

$$(S^{\ast }f)(z)=\frac{f(z)-f(0)}{z}.$$

$$\mathcal{M}=K_I^p=H^p\cap I\overline{H_0^p}=(I{H}^q{)}^{\perp }$$

$$(P_{+}f)(z)=\frac{1}{2\pi }{\int }_0^{2\pi }\frac{f(e^{i\theta })}{1-e^{-i\theta }z}d\theta$$

$$\bar{I}f=P_{+}(\bar{I}f)+P_{-}(\bar{I}f).$$

$$f=I\bar{I}f=I{P}_{+}(\bar{I}f)+I{P}_{-}(\bar{I}f)$$

$$P_{+}(\bar{I}f)\cap P_{-}(\bar{I}f)=\{0\},$$

$$P_{I}f=I{P}_{-}(\bar{I}f).$$

$$⟨f,g⟩=\frac{1}{2\pi }{\int }_0^{2\pi }f(e^{i\theta })\overline{g(e^{i\theta })}d\theta (f\in H^p,g\in H^q).$$

$$H^p=I{H}^p\oplus K_I^p.$$

$$⟨Ih,I{P}_{-}(\bar{I}f)⟩=⟨h,P_{-}(\bar{I}f)⟩=0.$$

$$T_{\phi }(f)=P_{+}(\phi f).$$

$$⟨T_{\bar{a}}f,Ig⟩=⟨\bar{a}f,Ig⟩=⟨f,aIg⟩=0,$$

$$\inf \{|f(z)|+|g(z)|:z\in \mathbb{D}\}=\delta >0.$$

$$a(z)u(z)+I(z)v(z)=1,z\in \mathbb{D}.$$

$$f=T_{1}f=T_{\bar{a}\bar{u}+\bar{I}\bar{v}}f=T_{\bar{a}}{T}_{\bar{u}}f+T_{\bar{I}}{T}_{\bar{v}}f.$$

$$T_{\bar{I}}{T}_{\bar{v}}f=T_{\bar{v}}{T}_{\bar{I}}f=T_{\bar{v}}{T}_{\bar{I}}(I{P}_{-}(\bar{I}f))=0.$$

$$f=T_{\bar{u}}{T}_{\bar{a}}f=T_{\bar{a}}{T}_{\bar{u}}f,$$

$$\underset{n\to \infty }{\lim }(|I(z_n)|+|a(z_n)|)=0.$$

$$b_n(z)=\frac{z-z_n}{1-{\overline{z}}_{n}z}$$

$$b_n{I}_n=I-I(z_n),b_n{a}_n=a-a(z_n).$$

$$k_n(z)=\frac{(1-|z_n{|}^2{)}^{1/q}}{1-{\bar{z}}_{n}z},\text{where}\frac{1}{p}+\frac{1}{q}=1.$$

$${\left({\int }_0^{2\pi }\frac{d\theta }{|1-z{e}^{-i\theta }{|}^p}\right)}^{1/p}\sim \frac{1}{(1-|z{|}^2{)}^{1/q}}\text{as }|z|\to 1^{-}.$$

$$f_n=P_I{I}_n{k}_n.$$

$$\Vert f_n{\Vert }_{H^p}$$

$$P_{-}({\bar{b}}_n{k}_n)(z)=(1-|z_n{|}^2{)}^{1/q}\frac{1}{z-z_n}=(1-|z_n{|}^2{)}^{1/q}\frac{\bar{z}}{1-z_n\bar{z}},$$

$$\Vert T_{\bar{a}}{f}_n{\Vert }_{H^p}$$

$$\Vert T_{\bar{a}}{f}_n{\Vert }_{H^p}=\underset{\begin{array}{c}g\in L^q,\\ \Vert g\Vert =1\end{array}}{\sup }|⟨\bar{a}(P_{-}(k_n{\bar{b}}_n)-I(z_n)P_{-}(\bar{I}{k}_n{\bar{b}}_n)),P_{-}(\bar{I}{P}_{+}g)⟩|.$$

$$\underset{\begin{array}{c}g\in L^q,\\ \Vert g\Vert =1\end{array}}{\sup }|⟨\bar{a}I(z_n)P_{-}(\bar{I}{k}_n{\bar{b}}_n),P_{-}(\bar{I}{P}_{+}g⟩|\to 0\text{as}n\to \infty .$$

$$\underset{\begin{array}{c}g\in L^q,\\ \Vert g\Vert =1\end{array}}{\sup }|⟨\bar{a}{P}_{-}(k_n{\bar{b}}_n),P_{-}(\bar{I}{P}_{+}g)⟩|$$

Full text

1. Introduction

Let $\mathbb{D}$ denote the unit disc in the complex plane and let $\mathbb{T}$ be its boundary. For $1\leqslant p<\infty$ let $H^{p}$ be the Hardy space of functions in $L^{p}({\mathbb{T}})$ with vanishing negative Fourier coefficients. $H^{p}$ consists of the boundary values of functions holomorphic in $\mathbb{D}$ and satisfying

[TABLE]

Let $H^{\infty}$ denote the Banach space of all bounded holomorphic functions in $\mathbb{D}$ equipped with the usual supremum norm. We will write $H_{0}^{p}$ for the subspace of $H^{p}$ consisting of functions vanishing at zero and $\overline{H^{p}}$ for the subspace of the complex conjugates of functions in $H^{p}$.

Let $S^{\ast}$ denote the backward shift operator on $H^{p}$ defined in the unit disc $\mathbb{D}$ by

[TABLE]

A closed subspace $\mathcal{M}$ of $H^{p}$ is called backward shift invariant, or $S^{\ast}$- invariant, if $f\in{\mathcal{M}}$ implies $S^{\ast}f\in{\mathcal{M}}$. It is well known that for $1<p<\infty$, all $S^{\ast}$-invariant subspaces are of the form

[TABLE]

for some inner function $I$, where $(IH^{q})^{\bot}$ is the annihilator of the space $IH^{q}$ and $\frac{1}{p}+\frac{1}{q}=1$ (see [CR, pp. 81–83]).

For $p=2$ the space $K_{I}=K_{I}^{2}$ is the orthogonal complement of the shift invariant subspace $IH^{2}$ and is called a model space. The model spaces play an important role in the model theory for Hilbert space contractions ([N3],[SzNB]). In this paper we deal mainly with $K^{p}_{I}$, $1<p<\infty$, and Toeplitz operators on these spaces. Recently these spaces have been studied e.g. in [Dy1], [Dy2], [HS]. The theory of model spaces is based on Hilbert space methods that are not always easily transferred to the context of general $H^{p}$ spaces. Often, however, results in $H^{p}$ spaces are suggested by those in $H^{2}$.

In Section 3 we prove that the Toeplitz operator $T_{\bar{a}}$ with the bounded co-analytic symbol $\bar{a}$ restricted to $K^{p}_{I}$ is invertible if and only if the functions $a$ and $I$ form a corona pair. This extends the result obtained by P. A. Fuhrmann for $p=2$ ([F1], [F2] , see also [P, pp. 17–19]) to $1<p<\infty.$

In Section 4 we refer to the commutant of the compressed shift, or, equivalently, the commutant of the restricted backward shift, investigated by Sarason [S1] in 1967. The description of the commutant of $S^{\ast}$ restricted to $K^{p}_{I}$ was conjectured in [CR, p.111]. Here we give its proof.

Our proofs are based upon ideas similar to those for the case $p=2$. However, we apply neither the functional calculus, nor the commutant lifting theorem. The main tools we use in our reasoning are the bounded projection $P_{I}$ of $H^{p}$ onto $K^{p}_{I}$ and the representation of linear functionals on $H^{p}$.

2. Backward shift invariant spaces

Let us recall that a closed subspace $\mathcal{M}$ of a Banach space $X$ is complemented if there exists a closed subspace $\mathcal{N}$ of $X$ such that $X=\mathcal{M}\oplus\mathcal{N}$, i.e. $X=\mathcal{M}+\mathcal{N}$ and $\mathcal{M}\cap\mathcal{N}=\{0\}$.

For $1<p<\infty$, the M. Riesz theorem implies that $H^{p}$ is a complemented subspace of $L^{p}({\mathbb{T}})$ and $H^{p}\oplus\overline{H^{p}_{0}}=L^{p}({\mathbb{T}})$. Moreover, the Riesz projection $P_{+}$ defined by

[TABLE]

maps $L^{p}({\mathbb{T}})$ onto $H^{p}$. We also set $P_{-}=\operatorname{id}-P_{+}$, where $\operatorname{id}$ denotes the identity operator on $L^{p}(\mathbb{T})$.

Now we show the following

Proposition 2.1**.**

For $1<p<\infty$ and any inner function $I$, the space $K_{I}^{p}$ is a complemented subspace of $H^{p}$.

Proof.

In view of the known characterization of complemented subspaces (see [R], p. 126) it is enough to show that there exists a continuous projection $P_{I}$ from $H^{p}$ onto $K^{p}_{I}$. To this end note that for $f\in H^{p}$,

[TABLE]

Since

[TABLE]

and

[TABLE]

we get

[TABLE]

∎

Remark 2.2*.*

It was proved in [St, Thm. 2.2] that if $I$ is an inner function, then the operator $T_{\bar{I}}$ is bounded on $H^{1}$ if and only if $I$ is a finite Blaschke product. Consequently, the projection $P_{I}$ is bounded only for such inner functions $I$. Hence Proposition 2.1 is not generally true for $p=1$.

It is well known that for $1<p<\infty$ the dual space $(H^{p})^{\ast}$ can be identified with $H^{q}$, $\frac{1}{p}+\frac{1}{q}=1$, via the pairing

[TABLE]

Clearly, the definition of $\langle f,g\rangle$ can be extended for $f\in L^{p}({\mathbb{T}})$ and $g\in L^{q}({\mathbb{T}})$. In the sequel we often use this symbol for such functions.

Remark 2.3*.*

It is worth-while to notice that for $1<p<\infty$ the following equality holds true

[TABLE]

In view of (1) it is enough to observe that if $f\in H^{p}$, then $IP_{-}(\bar{I}f)\in(IH^{q})^{\bot}$. Indeed, for any $h\in H^{q}$ we have

[TABLE]

3. Restricted Toeplitz operators

For $\varphi\in L^{\infty}({\mathbb{T}})$ the Toeplitz operator $T_{\varphi}$ on $H^{p}$, $1<p<\infty$, is defined by

[TABLE]

It is easy to check that if $\varphi\in H^{\infty}$ is outer, then both the operators $T_{\varphi}$ and $T_{\bar{\varphi}}$ are injective. Moreover, we have

Proposition 3.1**.**

For $\varphi\in H^{\infty}$, the operator $T_{\bar{\varphi}}$ is surjective if and only if $T_{\varphi}$ is left invertible.

Proof.

If $\varphi=I\varphi_{0}$ with $I$ the inner factor and $\varphi_{0}$ the outer factor of $\varphi$, then $\ker T_{\bar{\varphi}}=K^{p}_{I}$. Since $K_{I}^{p}$ is a complemented subspace of $H^{p}$, surjectivity of $T_{\bar{\varphi}}$ is equivalent to right invertibility of $T_{\bar{\varphi}}$ (see [M, p. 92, Thm. 16]). ∎

For any $a\in H^{\infty}$ we have $T_{\bar{a}}(K^{p}_{I})\subset K^{p}_{I}$. To see this inclusion it is enough to observe that for $f\in K^{p}_{I}$ and $g\in H^{q}$,

[TABLE]

since $K^{p}_{I}=(IH^{q})^{\bot}$.

Two functions $f,g\in H^{\infty}$ are said to form a corona pair if

[TABLE]

The next theorem is a generalization of a result due to Fuhrmann for $p=2$ ([F1], [P, Thm. 2.7]).

Theorem 3.2**.**

Assume that $1<p<\infty$, $a\in H^{\infty}$, and $I$ is an inner function. Then the restriction of the operator $T_{\bar{a}}$ to $K^{p}_{I}$ is invertible if and only if the functions $a$ and $I$ form a corona pair.

Proof.

Suppose first that functions $a$ and $I$ form a corona pair. We show that the operator $T_{\bar{a}}$ is invertible on $K^{p}_{I}$. By the corona theorem there exist functions $u,v\in H^{\infty}$ such that

[TABLE]

Hence for $f\in H^{p}$

[TABLE]

Note that if $f\in K_{I}^{p}$, then by (2)

[TABLE]

In view of (4) we get

[TABLE]

which proves invertibility of $T_{\bar{a}}$ on $K^{p}_{I}$.

Assume now that the functions $a$ and $I$ do not form a corona pair. Then there exists a sequence $\{z_{n}\}$ of points from $\mathbb{D}$ such that

[TABLE]

Put

[TABLE]

and define functions $I_{n}$ and $a_{n}$ by

[TABLE]

Finally, let

[TABLE]

We will also need the following asymptotic estimates for the integral means (see, e.g. [Z, Thm. 1.12]). For $z\in\mathbb{D}$,

[TABLE]

Let $f_{n}$ be defined by

[TABLE]

First we show that $\|f_{n}\|_{H^{p}}\geqslant\gamma>0$ for $n$ large enough and some constant $\gamma>0$. We have

[TABLE]

Since for $|z|=1$,

[TABLE]

estimates (5) imply that there exists $c>0$ such that $\|P_{-}(\bar{b}_{n}k_{n})\|_{L^{p}}\geqslant c$.

Furthermore, $\|P_{-}((|I_{n}|^{2}-1)\bar{b}_{n}k_{n})\|_{L^{p}}\to 0$ as $n\to\infty$ (see [P, p. 18]).

We now claim that $\|T_{\bar{a}}f_{n}\|_{H^{p}}\rightarrow 0$ as $n\rightarrow\infty$.

It follows from the above that $T_{\bar{a}}(K^{p}_{I})\subset K^{p}_{I}$. Moreover, since $IH^{q}$ is the annihilator of $K_{I}^{p}$, we get

[TABLE]

Since $b_{n}I_{n}=I-I(z_{n})$, we have $I_{n}=\overline{b}_{n}(I-I(z_{n}))$. Therefore

[TABLE]

It is clear that

[TABLE]

Moreover,

[TABLE]

where $g_{0}=P_{-}(\bar{I}P_{+}g)\in H^{q}_{0}$.

Since

[TABLE]

and

[TABLE]

our claim follows. ∎

4. The commutant of the restricted backward shift

Let us recall that the commutant $\{S^{\ast}\}^{\prime}$ of the backward shift consists of all bounded operators $A$ on $H^{p}$, $1<p<\infty$, commuting with $S^{\ast}$, i.e.

[TABLE]

It is well-known that (see e.g. [CR, pp. 109–110])

[TABLE]

Here we describe $\{S^{\ast}|K^{p}_{I}\}^{\prime}$ the commutant of the restricted backward shift operator $S^{\ast}$ to the subspace $K^{p}_{I}$, $1<p<\infty$. For $p=2$ the commutant of this operator was characterized by Sarason [S1]. The result says

[TABLE]

We will extend this result to the subspaces $K^{p}_{I}$, $1<p<\infty$, using an approach analogous to that suggested by N. K. Nikolskii for $p=2$ [N1, pp. 179–182] (see also [P, pp.13–15 ]). To this end, we first define Hankel operators on $H^{p}$, $1<p<\infty$.

For $\psi\in L^{p}(\mathbb{T})$, $1\leqslant p<\infty$, we define the Hankel operator $H_{\psi}$ on the dense subset of $H^{p}$ (e.g. $H^{\infty}$ or analytic polynomials) by

[TABLE]

It is easy to see that $H_{\psi}$ is bounded on $H^{p}$ if the function $\psi\in L^{\infty}(\mathbb{T})$.

For $n\in\mathbb{Z}$ let $\chi_{n}(z)=z^{n}$, $z\in{\mathbb{T}}$. The functions $\chi_{n}$, $n\in\mathbb{Z}$, form a Schauder basis for the space $L^{p}(\mathbb{T})$. Let $\mathcal{S}$ denote the bilateral shift on $L^{p}(\mathbb{T})$, i.e. ${\mathcal{S}}\chi_{n}=\chi_{n+1}$ for $n\in\mathbb{Z}$ and $S={\mathcal{S}}|H^{p}$ be the unilateral shift on $H^{p}$.

The next theorem contains the known characterizations of Hankel operators on $H^{p}$, $1<p<\infty$. We include its proof for the convenience of the reader. We note that a version of this theorem can be found in [BS, pp. 54–55, Thm. 2.11].

Theorem 4.1**.**

The following statements are equivalent:**

1. (i)

$A\colon\,H^{p}\to\overline{H^{p}_{0}}$* is a bounded linear operator such that $\langle A\chi_{k},\bar{\chi}_{j}\rangle=a_{j+k}$ for *$j\geqslant 1,k\geqslant 0$; 2. (ii)

*there exists $\psi\in L^{\infty}(\mathbb{T})$ such that *$A=H_{\psi}$; 3. (iii)

$A$* is a bounded linear operator from $H^{p}$ into $\overline{H^{p}_{0}}$ such that $P_{-}(\mathcal{S}A)=AS$.*

Proof.

(i) $\Rightarrow$ (ii): By assumption the equality

[TABLE]

holds for any polynomials $g$ and $h$, $h(0)=0$. Since polynomials are dense in $H^{p}$, the above equality is true for all $g\in H^{p}$ and $h\in{H^{q}_{0}}$. Consequently,

[TABLE]

and

[TABLE]

This means that $\Phi\colon\,H^{1}_{0}\rightarrow{\mathbb{C}}$ defined by

[TABLE]

is a bounded linear functional on $H^{1}_{0}$. By the Hahn-Banach theorem this functional can be extended to a bounded functional on $L^{1}(\mathbb{T})$. Hence there exists $\psi\in L^{\infty}(\mathbb{T})$ such that

[TABLE]

for any $f\in L^{1}(\mathbb{T})$.

Therefore if $f=gh$, where $g\in H^{p}$ and $h\in H^{q}_{0}$, then

[TABLE]

or equivalently by (6),

[TABLE]

Taking $h=\chi_{n}$, $n\geqslant 1$, in (7) yields

[TABLE]

Since $Ag\in\overline{H^{p}_{0}}$, the last equality implies that for $g\in H^{p}$,

[TABLE]

or, in other words, $A=H_{\psi}$.

(ii) $\Rightarrow$ (iii): If $A=H_{\psi}$ with $\psi\in L^{\infty}(\mathbb{T})$, then for $f\in H^{p}$,

[TABLE]

Thus

[TABLE]

or

[TABLE]

(iii) $\Rightarrow$ (i): Assume now that a bounded operator $A\colon\,H^{p}\rightarrow\overline{H^{p}_{0}}$ satisfies $P_{-}(\mathcal{S}A)=AS$. Then for $j\geqslant 1$ and $k\geqslant 1$,

[TABLE]

since $\langle P_{+}(\mathcal{S}A)\chi_{k-1},\bar{\chi}_{j}\rangle=0$. This implies that the operator $A$ is represented by the Hankel matrix with respect to the basis $\{\chi_{k}\}_{k\geqslant 0}$ in $H^{p}$ and $\{\bar{\chi}_{j}\}_{j\geqslant 1}$ in $\overline{H^{p}_{0}}$.

∎

Let $S^{I}$ be the compression of the shift $S$ on $H^{p}$ to the subspace $K^{p}_{I}$, i.e.

[TABLE]

for $f\in K_{I}^{p}$. Also let for $\varphi\in H^{\infty}$ the operator $T^{I}_{\varphi}f=P_{I}(T_{\varphi}f)$ be the compression of the Toeplitz operator $T_{\varphi}$ on $H^{p}$ to $K_{I}^{p}$.

Now our aim is to prove the following

Theorem 4.2**.**

If $T$ is a bounded linear operator on $K_{I}^{p}$, $1<p<\infty,$ that commutes with the operator $S^{I}$, then there exists $\varphi\in H^{\infty}$ such that $T=T_{\varphi}^{I}$.

We first prove the following lemma (cf. [N1, p. 181], [P, Lemma 2.2, p. 14]).

Lemma 4.3**.**

Let $T$ be a bounded linear operator on $K_{I}^{p}$, $1<p<\infty,$ and let an operator $A\colon\,H^{p}\rightarrow\overline{H^{p}_{0}}$ be given by

[TABLE]

Then $T$ commutes with $S^{I}$ if and only if $A$ is a Hankel operator.

Proof.

In view of Theorem 4.1, $A$ is a Hankel operator if and only if

[TABLE]

thus

[TABLE]

or equivalently

[TABLE]

By formula (2) we have

[TABLE]

Now observe that for $f\in IH^{p}$ the left-hand side of the last equality equals to zero since $P_{I}f=0$. Hence it follows from (10) that $TP_{I}zf=0$ for $f\in IH^{p}$. Consequently,

[TABLE]

for $f\in K^{p}_{I}$.

Finally, note that for $f\in K^{p}_{I}$,

[TABLE]

since $P_{I}f=f$. Therefore equality (8) (or (9)) is equivalent to

[TABLE]

∎

Proof of Theorem 4.2.  If an operator $T$ satisfies assumptions of Theorem 4.2, then the operator $A$ from Lemma 4.3 is a bounded Hankel operator. In view of Theorem 4.1 there exists $\psi\in L^{\infty}$ such that $Af=H_{\psi}$. Thus

[TABLE]

for $f\in H^{p}$. As in the proof of Lemma 4.3, this equality implies $P_{-}\psi f=0$ for $f\in IH^{p}$, and so

[TABLE]

Put $\varphi=\psi I$. Then $\varphi\in H^{\infty}$, $\psi=\varphi\bar{I}$ and for $f\in K^{p}_{I}$ (i. e. for $f$ such that $P_{I}f=f$)

[TABLE]

and so

[TABLE]

$\square$

We know that under pairing (3), the dual space of $K^{p}_{I}$ can be identified with $K^{q}_{I}$ (see e.g. [CR, p. 109]). Moreover, the adjoint of the compression of the shift operator $S$ to $K_{I}^{p}$, i.e. $\left(P_{I}S|K_{I}^{p}\right)^{\ast}$ is equal to $T_{\bar{z}}|K_{I}^{q}$. Similarly, for $\varphi\in H^{\infty}$, the adjoint of the compression of $T_{\varphi}$ to $K^{p}_{I}$, i.e. $\left(P_{I}T_{\varphi}|K_{I}^{p}\right)^{\ast}$ is equal to $T_{\bar{\varphi}}|K_{I}^{q}$.

Theorem 4.2 implies the following

Corollary 4.4**.**

For $1<p<\infty$,

[TABLE]

Final Remark. After this paper was submitted for publication we learned from A. Hartmann that Theorem 4.2 had been proved in his paper [H1, Théorème 1.14]. He observed that the $H^{\infty}$-functional calculus of B. Sz.-Nagy and Foiaş for the compressed shift operator to $K^{p}_{I}$, $1<p<\infty$, can be defined exactly in the same way as in the case $p=2$. Therefore, using the same reasoning, Sarason’s theorem on the commutant of the compressed shift can be extended to the non-Hilbertian case.

Furthermore, the proof of [FHR, Lemma 4.9, p. 1270–1271] can be carried out in the case $1<p<\infty$ by means of this generalized $H^{\infty}$-functional calculus for the compressed shift operator.

The advantage of our presentation is that the proofs of Theorems 3.2 and 4.2 are direct. We use neither the functional calculus, nor the commutant lifting theorem.

Acknowledgement. The second named author would like to thank the Institute of Mathematics of the Maria Curie-Skłodowska University for supporting his visit to Lublin where part of this paper was written.