Truncated Toeplitz operators and complex symmetries

TL;DR

This paper characterizes truncated Toeplitz operators using complex symmetries, confirming a conjecture and extending previous partial results.

Contribution

It provides a complete characterization of truncated Toeplitz operators through complex symmetries, resolving a conjecture in the field.

Findings

Truncated Toeplitz operators are characterized by complex symmetries.

The conjecture by Klis-Garlicka, Lanucha, and Ptak is proved in full generality.

The work extends previous partial results to a comprehensive proof.

Abstract

We show that truncated Toeplitz operators are characterized by a collection of complex symmetries. This was conjectured by Klis-Garlicka, Lanucha, and Ptak, and proved by them in some special cases.

Equations

$$C_u(C_{u/v}(f))=vf.$$

$$Af=P_{K_u}(\phi f)$$

$$K_u^0=\{g\in K_u:Sg\in K_u\}.$$

$$Q_A(f)=Q_A(Sf)$$

$${\omega }_a(f)=\frac{\sqrt{1-|a{|}^2}}{1-\bar{a}\chi }f\circ b_a,f\in K_u.$$

$$Q_A(Sf)=Q_A(C_u{C}_{u/z}f)=Q_A(C_{u/\chi }f))=Q_A(f).$$

$$e_{\nu }(z)=\exp \left(-{\int }_{\mathbb{T}}\frac{\zeta +z}{\zeta -z}d\nu (\zeta )\right),z\in \mathbb{D}.$$

$$\frac{e_{{\mu }_n}-1}{{\mu }_n(\mathbb{T})}(\chi -\eta )g,n\in \mathbb{N},$$

$$d{\mu }_n=\sqrt{\frac{|I_n|}{\nu (I_n)}}{\chi }_{I_n}d\nu ,n\in \mathbb{N}.$$

$$\underset{n\to \infty }{\lim }\frac{e_{{\mu }_n}(z)-1}{{\mu }_n(\mathbb{T})}=\frac{z+\eta }{z-\eta },z\in \mathbb{D}.$$

$$\underset{z\in \mathbb{D},n\in \mathbb{N}}{\sup }|z-\eta |\frac{|e_{{\mu }_n}(z)-1|}{{\mu }_n(\mathbb{T})}<\infty .$$

$$|z-\eta |\frac{|e_{{\mu }_n}(z)-1|}{{\mu }_n(\mathbb{T})}\le \frac{20|I_n|}{\sqrt{|I_n|\nu (I_n)}}=20\sqrt{\frac{|I_n|}{\nu (I_n)}},$$

$$|e^{\lambda }-1|\le |\lambda |e^{|\lambda |},\lambda \in \mathbb{C},$$

$$\left|\frac{\zeta -z}{\zeta +z}\right|<3,\zeta \in I_n,z\in \mathbb{D},|z-\zeta |>\frac{10}{n},$$

$$|e_{{\mu }_n}(z)-1|\le 3{\mu }_n(\mathbb{T})e^{3{\mu }_n(\mathbb{T})}.$$

$$|z-\eta |\frac{|e_{{\mu }_n}(z)-1|}{{\mu }_n(\mathbb{T})}\le 6e^{3{\mu }_n(\mathbb{T})}<6e^{3\nu (\mathbb{T})}.$$

$$C_{e_{\nu }}(C_{e_{\nu }/e_{{\mu }_m}}((\chi -\eta )f)=e_{{\mu }_m}(\chi -\eta )f.$$

$$Q_A(e_{{\mu }_m}(\chi -\eta )f)=Q_A((\chi -\eta )f),m\ge n$$

$$\begin{split}0&=\frac{1}{\mu_{m}(\mathbb{T})}\big{(}\langle A(e_{\mu_{m}}(\chi-\eta)f),e_{\mu_{m}}(\chi-\eta)f\rangle-\langle A((\chi-\eta)f),(\chi-\eta)f\rangle\big{)}\\ &=\left\langle\frac{e_{\mu_{m}}-1}{\mu_{m}(\mathbb{T})}(\chi-\eta)f,A^{*}(e_{\mu_{m}}(\chi-\eta)f)\right\rangle\\ &\hskip 85.35826pt+\left\langle A((\chi-\eta)f),\frac{e_{\mu_{m}}-1}{\mu_{m}(\mathbb{T})}(\chi-\eta)f\right\rangle\end{split}$$

$$⟨A((\chi +\eta )f),(\chi -\eta )f⟩+⟨A((\chi -\eta )f),(\chi +\eta )f⟩=0.$$

Full text

HB was supported in part by a grant of the National Science Foundation.

1. Introduction

The systematic study of truncated Toeplitz operators was initiated by Sarason [6]. Given an inner function $u$ on the unit disc, this class, denoted by $\mathcal{T}_{u}$ consists of those bounded operators on $K_{u}=H^{2}\ominus uH^{2}$ that are compressions of multiplication operators operators. A recent survey of results in this area is contained in [4].

Sarason observed that, while every operator in $\mathcal{T}_{u}$ is complex symmetric (relative to the natural conjugation on $K_{u}$; see [3]), not every complex symmetric operator on $K_{u}$ belongs to $\mathcal{T}_{u}$. Operators in $\mathcal{T}_{u}$ satisfy additional complex symmetry conditions and the authors of [5] conjectured that every operator on $K_{u}$ that satisfies these additional symmetries necessarily belongs to $\mathcal{T}_{u}$. This conjecture is proved in [5] in many cases in which $u$ is a Blaschke product. The purpose of this note is to provide a proof of this conjecture for arbitrary inner functions $u$. In the case in which $u$ has at least one zero, it turns out that the operators in $\mathcal{T}_{u}$ are characterized by the fact that they satisfy just two complex symmetries. In case $u$ is singular, one needs to require a countable collection of complex symmetries.

2. Notation and preliminaries

We denote by $\mathbb{C}$ the complex plane, by $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ the unit disc, and by $\mathbb{T}=\{z\in\mathbb{C}:|z|=1\}$ the unit circle. As usual, we view the Hardy space $H^{2}$ on $\mathbb{D}$ as a subspace of $L^{2}=L^{2}(\mathbb{T})$ (relative to the normalized arclength measure on $\mathbb{T}$) by identifying functions analytic in $\mathbb{D}$ with their radial limits (which exist almost everywhere). Similarly, the algebra $H^{\infty}$ of bounded analytic functions in $\mathbb{D}$ can be viewed as a closed subalgebra of $L^{\infty}=L^{\infty}(\mathbb{T})$. We denote by $S$ the shift operator in $H^{2}$, defined by $(Sf)(z)=zf(z)$, $f\in H^{2}$, $z\in\mathbb{D}$.

A function $u\in H^{\infty}$ is said to be inner if $|u|=1$ almost everywhere on $\mathbb{T}$. For instance, the function $\chi\in H^{\infty}$ defined by $\chi(z)=z$, $z\in\mathbb{D}$, is inner. If $u$ is an inner function, the model space ${K_{u}}$ (often denoted $\mathcal{H}(u)$ in the literature) is defined by ${K_{u}}=H^{2}\ominus uH^{2}$ and $P_{{K_{u}}}:L^{2}\to K_{u}$ denotes the orthogonal projection onto ${K_{u}}$.

Given an arbitrary bounded operator $A$ on a Hilbert space ${\mathcal{H}}$, we denote by $Q_{A}$ the quadratic form on ${\mathcal{H}}$ defined by $Q_{A}(f)=\langle Af,f\rangle$, $f\in\mathcal{H}$. A conjugation on a Hilbert space ${\mathcal{H}}$ is an isometric, conjugate linear involution, that is, $C\circ C=I_{\mathcal{H}}$ and $\langle Ch,Ck\rangle=\langle k,h\rangle$ for $h,k\in\mathcal{H}$. An operator $A$ is then said to be $C$-symmetric [3] or simply complex symmetric when $C$ is understood, if $A^{*}=CAC$. This condition is easily seen to be equivalent to $Q_{A}(f)=Q_{A}(Cf)$, $f\in{\mathcal{H}}$.

Given an arbitrary inner function $u$, there is a conjugation $C_{u}$ on $L^{2}$ defined by $C_{u}f=u\overline{\chi f}$. This conjugation maps $K_{u}$ bijectively onto itself and therefore it also defines a conjugation on this space. We record for further use the following result whose proof is a simple calculation.

Lemma 2.1**.**

Suppose that $u$ and $v$ are inner functions in $H^{\infty}$ and $v$ divides $u$. Then for every $f\in L^{2}$ we have

[TABLE]

The space ${K_{u}}$ is a reproducing kernel space of analytic functions on $\mathbb{D}$. The following well-known lemma is the $H^{2}$ version of a results that holds in arbitrary reproducing kernel Hilbert spaces.

Lemma 2.2**.**

Suppose that $\{f_{n}\}_{n\in\mathbb{N}}\subset H^{2}$. Then:

• (i)

The sequence $\{f_{n}\}_{n\in\mathbb{N}}$ converges weakly to a function $f\in H^{2}$ if and only if $\sup_{n\in\mathbb{N}}\|f_{n}\|<+\infty$ and $\lim_{n\to\infty}f_{n}(z)=f(z)$ for every $z\in\mathbb{D}$.

• (ii)

The sequence $\{f_{n}\}_{n\in\mathbb{N}}$ converges in norm to a function $f\in H^{2}$ if and only if $\lim_{n\to\infty}\|f_{n}\|=\|f\|$ and $\lim_{n\to\infty}f_{n}(z)=f(z)$ for every $z\in\mathbb{D}$.

We recall [6] that a bounded linear operator $A$ on $K_{u}$ is called a truncated Toeplitz operator if there exists a function $\varphi\in L^{2}$ (called a symbol of $A$) such that

[TABLE]

for every bounded function $f\in K_{u}$. The truncated Toeplitz operators on ${K_{u}}$ form a weakly closed subspace $\mathcal{T}_{u}$ of ${\mathcal{L}}({K_{u}})$. There is a simple characterization of the operators in $\mathcal{T}_{u}$ that does not require a symbol. The space

[TABLE]

is closed in $K_{u}$ and it has codimension 1. Its orthogonal complement is generated the vector by $S^{*}u=\overline{\chi}(u-u(0))$. The following result is [6, Theorem 8.1].

Lemma 2.3**.**

A bounded linear operator $A$ on ${K_{u}}$ belongs to ${\mathcal{T}}_{u}$ if and only if

[TABLE]

for every $f\in K_{u}^{0}$.

Fix $a\in\mathbb{D}$ and denote by $b_{a}(z)=({z-a})/({1-\bar{a}z})$, $z\in\mathbb{D}$, the corresponding Blaschke factor. The following result is used in [1, Section 4] as well as [5].

Lemma 2.4**.**

There is a unitary operator $\omega_{a}:K_{u}\to K_{u\circ b_{a}}$ defined by

[TABLE]

We have $\omega_{a}C_{u}=C_{u\circ b_{a}}\omega_{a},$ and $\omega_{a}{\mathcal{T}}_{u}\omega_{a}^{*}={\mathcal{T}}_{u\circ b_{a}}$.

3. Truncated Toeplitz operators and conjugations

Suppose that $u$ is an inner function and $A\in\mathcal{T}_{u}$. Then $A$ is $C_{u}$ symmetric, that is, $Q_{A}(f)=Q_{A}(C_{u}f)$ for every $f\in K_{u}$. Let $v$ be an inner divisor of the function $u$. Then $K_{v}\subset K_{u}$ and it was observed in [5] that $P_{v}A|K_{v}$ is also $C_{v}$-symmetric. The authors of [5] formulate the following conjecture.

Conjecture 3.1**.**

A bounded linear operator $A$ on $K_{u}$ belongs to ${\mathcal{T}}_{u}$ if and only if, for every inner divisor $v$ of $u$, the compression $P_{v}A|K_{v}$ is $C_{v}$-symmetric.

This conjecture is proved in [5] for certain Blaschke products $u$, namely, Blaschke products with a single zero, finite Blaschke products with simple zeros, and interpolating Blaschke products. The arguments rely on a characterization [2] of the class $\mathcal{T}_{u}$ in terms of its matrix entries in a particular orthonormal basis for $K_{u}$. In this section, we prove the conjecture for those inner functions $u$ that have at least one zero. The case of singular inner functions is treated in the following section.

Theorem 3.2**.**

Suppose that $u\in H^{\infty}$ is an inner function and $u(a)=0$ for some $a\in\mathbb{D}$. Then an operator $A\in{\mathcal{L}}(K_{u})$ belongs to ${\mathcal{T}}_{u}$ if and only if it is $C_{u}$-symmetric and $Q_{A}(C_{u/b_{a}}f)=Q_{A}(f)$ for every $f\in K_{u/b_{a}}$.

Proof.

Suppose first that $a=0$ and thus $b_{a}=\chi$. If $f\in K_{u}$, then $Sf\in K_{u}$ if and only if $f\in K_{u/\chi}$. For such a function $f$ we have $C_{u}C_{u/\chi}f=\chi f=Sf$ by Lemma 2.1. The two symmetry hypotheses in the statement imply that

[TABLE]

It follows then from Lemma 2.3 that $A\in{\mathcal{T}}_{u}$.

For the general case $a\neq 0$ we use Lemma 2.4. The inner function $v=u\circ b_{-a}$ satisfies $v(0)=0$, and the unitary map $\omega_{a}$ defined in (2.2) yields by restriction unitary maps from $K_{v}$ onto $K_{u}$ and from $K_{v/\chi}$ to $K_{u/b_{a}}$ that intertwine the standard conjugations on these spaces. Therefore $\omega_{a}A\omega_{a}^{*}$ is $C_{v}$-symmetric, and its compression to $K_{v/\chi}$ is $C_{v/\chi}$-symmetric. By the first part of the proof, $\omega_{a}A\omega_{a}^{*}\in{\mathcal{T}}_{v}$. It follows from Lemma 2.4 that $A\in{\mathcal{T}}_{u}$. ∎

We have thus proved a stronger version of the conjecture in case $u$ has a zero in $\mathbb{D}$: an operator $A$ on $K_{u}$ is a truncated Toeplitz operator if and only if the complex symmetry condition is satisfied by $A$ as well as by a single one of its compressions to model spaces.

4. Singular inner functions

Given a positive, singular Borel measure $\nu$ on $\mathbb{T}$, we denote by $e_{\nu}$ the corresponding singular inner function, that is,

[TABLE]

Lemma 4.1**.**

Let $\nu$ be a nonzero, positive, singular Borel measure on $\mathbb{T}$. Then there exist $\zeta\in\mathbb{T}$ and a sequence of nonzero, positive Borel measures $\mu_{n}\leq\nu$, $n\in\mathbb{N}$, such that:

• (i)

$\lim_{n\to\infty}e_{\mu_{n}}(z)=1$* for every $z\in\mathbb{D}$, and*

• (ii)

for every $g\in H^{2}$, the functions

[TABLE]

converge weakly in $H^{2}$ to $(\chi+\eta)g$ as $n\to\infty$.

Proof.

Choose $\eta\in\mathbb{T}$ and a sequence $\{I_{n}\}_{n\in\mathbb{N}}$ of arcs in $\mathbb{T}$, symmetric about $\eta$, with length $|I_{n}|=1/n$, such that $\lim_{n\to\infty}(\nu(I_{n})/|I_{n}|)=+\infty$. This is possible since $\nu$ is singular. Define the measures $\mu_{n}$ by

[TABLE]

By the maximum modulus principle, condition (i) only needs to be verified at $z=0$, and this is immediate because $e_{\mu_{n}}(0)=e^{-\mu_{n}(\mathbb{T})}$. In fact, we have

[TABLE]

Lemma 2.2 shows that (ii) is true as well once we verify that

[TABLE]

Observe first that, if $z\in\mathbb{D}$ and $|z-\eta|<10/n=10|I_{n}|$,

[TABLE]

and the last quantity tends to [math] by the choice of $I_{n}$. If $|z-\eta|\geq 10/n$, we use the inequalities

[TABLE]

and

[TABLE]

to deduce that

[TABLE]

For such values of $z$ we see that

[TABLE]

This concludes the proof of the lemma. ∎

We need one more technical result before establishing Conjecture 3.1 for $u=e_{\nu}$. Recall that $K_{u}^{0}$ consists of those vectors $f\in K_{u}$ with the property that $Sf$ also belongs to $K_{u}$. Clearly, $K_{v}^{0}\subset K_{u}^{0}$ if $v$ is an inner divisor of $u$.

Lemma 4.2**.**

Suppose that $u\in H^{\infty}$ is an inner function and $\{u_{n}\}_{n\in\mathbb{N}}$ is a sequence of inner divisors of $u$ such that $u_{n+1}$ divides $u_{n}$, $n\in\mathbb{N}$, and $\lim_{n\to\infty}|u_{n}(0)|=1$. Then $\bigcup_{n\in\mathbb{N}}K_{u/u_{n}}^{0}$ is dense in $K_{u}^{0}$.

Proof.

We can, and do, assume without loss of generality that $u_{n}(0)\geq 0$, $n\in\mathbb{N}$. The least common inner multiple of the functions $\{u/u_{n}\}_{n\in\mathbb{N}}$ is equal to $u$, and thus $\bigcap_{n\in\mathbb{N}}(u/u_{n})H^{2}=uH^{2}$. It follows that $\bigcup_{n\in\mathbb{N}}K_{u/u_{n}}$ is dense in $K_{u}$ and therefore the sequence $\{P_{u/u_{n}}\}_{n\in\mathbb{N}}$ converges to $P_{u}$ in the strong operator topology.

It was noted earlier that the space $K_{u/u_{n}}\ominus K_{u/u_{n}}^{0}$ is generated by the vector $S^{*}(u/u_{n})$. It follows from Lemma 2.2 that the sequence $\{u/u_{n}\}_{n\in\mathbb{N}}$ converges in the $H^{2}$ norm to $u$, and thus $\lim_{n\to\infty}S^{*}(u/u_{n})=S^{*}u$. If we denote by $P_{n}$ and $P$ the orthogonal projections onto the spaces $K_{u/u_{n}}\ominus K_{u/u_{n}}^{0}$ and $K_{u}\ominus K_{u}^{0}$, respectively, it follows that the sequence $\{P_{n}\}_{n\in\mathbb{N}}$ converges to $P$ in the strong operator topology. Given an arbitrary vector $f\in K_{u}^{0}$, we have $f_{n}=P_{u/u_{n}}f-P_{n}P_{u/u_{n}}f\in K_{u/u_{n}}^{0}$ and $\lim_{n\to\infty}f_{n}=f-Pf=f$. The lemma follows. ∎

We may now give the solution of the conjecture for singular inner functions.

Theorem 4.3**.**

Suppose that $\nu$ is a positive, singular Borel measure on $\mathbb{T}$. Let $A$ be an operator on $K_{e_{\nu}}$ that is $C_{e_{\nu}}$-symmetric, and such that for every positive Borel measure $\mu\leq\nu$, the compression of $A$ to $K_{e_{\mu}}$, is $C_{e_{\mu}}$-symmetric. Then $A\in{\mathcal{T}}_{e_{\nu}}$.

Proof.

Let $\eta\in\mathbb{T}$ and $\{\mu_{n}\}_{n\in\mathbb{N}}$ be as in Lemma 4.1. By Lemmas 2.1 and 4.2, it suffices to show that $Q_{A}(Sf)=Q_{A}(f)$ for every $f\in\bigcup_{n\in\mathbb{N}}K_{e_{\mu}/e_{\mu_{n}}}$. Fix $n\in\mathbb{N}$, $f\in K^{0}_{e_{\nu}/e_{\mu_{n}}}$, and observe that then $(\chi-\eta)f\in K_{e_{\nu}/e_{\mu_{n}}}$. If $m\geq n$, we also have $e_{\mu_{m}}|e_{\mu_{n}}$ and $(\chi-\eta)f\in K_{e_{\nu}/e_{\mu_{m}}}$. Lemma 2.1 yields

[TABLE]

The complex symmetry of $A$ and of its compression to $K_{e_{\nu}/e_{\mu_{m}}}$ shows that

[TABLE]

and therefore

[TABLE]

By Lemma 4.1(ii) $(1/\mu_{m}(\mathbb{T}))({e_{\mu_{m}}-1})(\chi-\eta)f$ tends weakly in $H^{2}$ to $(\chi+\eta)f$. On the other hand, $e_{\mu_{m}}(\chi-\eta)f$ tends pointwise on $\mathbb{D}$ to $(\chi-\eta)f$ and $\|e_{\mu_{m}}(\chi-\eta)f\|=\|(\chi-\eta)f\|$. By Lemma 2.2 $e_{\mu_{m}}(\chi-\eta)f$ tends to $(\chi-\eta)f$ in norm. We obtain then, by letting $m\to\infty$ the last equality,

[TABLE]

A simple calculation yields then $Q_{A}(Sf)=Q_{A}(f),$ thereby concluding the proof. ∎

We observe that the argument above only requires that $A$ and its compressions to $K_{e_{\nu}/e_{\mu_{n}}}$, $n\in\mathbb{N}$, be complex symmetric.