Characterizations of Asymmetric Truncated Toeplitz and Hankel Operators

Abstract

It was recently proved that in some special cases asymmetric truncated Toeplitz operators can be characterized in terms of compressed shifts and rank-two operators of special form. In this paper we show that such characterizations hold in all cases. We also show a connection between asymmetric truncated Toeplitz operators and asymmetric truncated Hankel operators. We use this connection to generalize results known for truncated Hankel operators to asymmetric truncated Hankel operators.

1 Introduction

As usual, let $H^2$ denote the classical Hardy space. The space $H^2$ can be seen as a space of functions analytic in the unit disk $\mathbb {D}=\{z:|z|<1\}$ or as a closed subspace of $L^2=L^2(\partial \mathbb {D})$. In the first case $H^2$ consists of functions analytic in $\mathbb {D}$ with square summable Maclaurin coefficients and in the second it consists of functions from $L^2$ such that their Fourier coefficients with negative indices vanish.

The unilateral shift S on $H^2$ is the operator of multiplication by the independent variable, that is,

$$\begin{aligned} Sf(z)=z\cdot f(z). \end{aligned}$$

The adjoint $S^*$ of S is called the backward shift. A simple verification shows that

$$\begin{aligned} S^*f(z)=\frac{f(z)-f(0)}{z}. \end{aligned}$$

The famous Beurling theorem provides a characterization of all $S^*$-invariant subspaces of $H^2$. Namely, a closed nontrivial subspace of $H^2$ is $S^*$-invariant if and only if it is of the form

$$\begin{aligned} K_\alpha =H^2\ominus \alpha H^2, \end{aligned}$$

where $\alpha$ is an inner function, i.e., $\alpha$ belongs to the algebra $H^{\infty }$ of bounded analytic functions and $|\alpha |=1$ a.e. on $\partial \mathbb {D}$. The space $K_{\alpha }$ is called the model space associated with $\alpha$.

The operators S and $S^*$ are two examples of classical Toeplitz operators. Let P denote the orthogonal projection from $L^2$ onto $H^2$. A Toeplitz operator $T_{\varphi }$ with symbol $\varphi \in L^\infty$ is defined on $H^2$ by

$$\begin{aligned} T_{\varphi }f=P(\varphi f). \end{aligned}$$

Clearly, if $\varphi \in H^\infty$, then $T_\varphi$ is just the multiplication by $\varphi$. Also, the operator $T_{\varphi }$ is densely defined on bounded functions whenever $\varphi \in L^2$, and extends to a bounded operator on $H^2$ if and only if $\varphi \in L^\infty$. We have $S=T_z$ and $S^{*}=T_{\overline{z}}$.

Truncated Toeplitz operators are compressions of Toeplitz operators to model spaces. More precisely, a truncated Toeplitz operator $A_{\varphi }^{\alpha }$ with symbol $\varphi \in L^2$ is densely defined on the model space $K_{\alpha }$ by

$$\begin{aligned} A_{\varphi }^{\alpha }f=P_{\alpha }(\varphi f),\quad f\in K_{\alpha }^{\infty }=K_{\alpha }\cap H^{\infty }, \end{aligned}$$

where $P_{\alpha }$ is the orthogonal projection from $L^2$ onto $K_{\alpha }$. In particular, $S_\alpha =A^\alpha _z$ is called the compressed shift. Since $K_\alpha$ is $S^*$-invariant, it easily follows that $S^*_\alpha =S^*_{|K_\alpha }$.

The study of the class of truncated Toeplitz operators was started in 2007 with D. Sarason’s paper [12] (see [6]). Recently, the authors in [1] and [3, 4] initiated the study of so-called asymmetric truncated Toeplitz operators (see also [8, 9]).

Let $\alpha$, $\beta$ be two inner functions. An asymmetric truncated Toeplitz operator $A_{\varphi }^{\alpha ,\beta }$ with symbol $\varphi \in L^2$ is an operator from $K_{\alpha }$ into $K_{\beta }$ densely defined by

$$\begin{aligned} A_{\varphi }^{\alpha ,\beta }f=P_{\beta }(\varphi f),\quad f\in K_{\alpha }^{\infty }. \end{aligned}$$

Clearly, $A_{\varphi }^{\alpha }=A_{\varphi }^{\alpha ,\alpha }$. Set

$$\begin{aligned} \mathscr {T}(\alpha ,\beta )=\{A_{\varphi }^{\alpha ,\beta }:\, \varphi \in L^2\ \mathrm {and}\ A_{\varphi }^{\alpha ,\beta }\ \mathrm {is\ bounded}\}. \end{aligned}$$

Closely related to Toeplitz operators are Hankel operators. A Hankel operator $H_\varphi$ with symbol $\varphi \in L^\infty$ may be defined on $H^2$ as

$$\begin{aligned} H_\varphi f=J(I-P)(\varphi f), \end{aligned}$$

where $J:L^2\rightarrow L^2$ is given by

$$\begin{aligned} Jf(z)=\overline{z}f(\overline{z}),\quad |z|=1. \end{aligned}$$

Note here that J is an isometry that maps $H^2$ onto $\overline{zH^2}$ and $\overline{zH^2}$ onto $H^2$. The operator $H_\varphi$ is densely defined for $\varphi \in L^2$ and clearly $H_\varphi =0$ for $\varphi \in H^2$. It is known that a Hankel operator is bounded if and only if it has a symbol from $L^\infty$.

Truncated Hankel operators were first defined and studied by Gu in his manuscript [7]. Their asymmetric versions were introduced in [11].

An asymmetric truncated Hankel operator $B_\varphi ^{\alpha ,\beta }$ with symbol $\varphi \in L^2$ is an operator from $K_\alpha$ into $K_\beta$ densely defined by

$$\begin{aligned} B_\varphi ^{\alpha ,\beta } f=P_\beta J(I-P)(\varphi f),\quad f\in K_{\alpha }^{\infty }. \end{aligned}$$

A truncated Hankel operator is the operator $B_\varphi ^{\alpha }= B_\varphi ^{\alpha ,\alpha }$. Denote

$$\begin{aligned} \mathscr {H}(\alpha ,\beta )=\{B_{\varphi }^{\alpha ,\beta }:\, \varphi \in L^2\ \mathrm {and}\ B_{\varphi }^{\alpha ,\beta }\ \mathrm {is\ bounded}\}. \end{aligned}$$

A bounded linear operator T on $H^2$ is a Toeplitz operator if and only if $T-S^* TS=0$ and is a Hankel operator if and only if $TS-S^* T=0$. Analogues of these characterizations are known for truncated Toeplitz and Hankel operators. For example, Sarason [12] proved that a bounded linear operator A on $K_\alpha$ is a truncated Toeplitz operator if and only if $A-S_{\alpha }^* AS_\alpha$ is an operator of a special kind and rank at most two. Gu showed in [7] that a bounded linear operator B on $K_\alpha$ is a truncated Hankel operator if and only if $BS_\alpha -S_{\alpha }^* B$ is a special kind of operator of rank at most two. Similar characterizations for the operators from $\mathscr {T}(\alpha ,\beta )$ were proved for the case when $\beta$ divides $\alpha$ (that is, $\alpha /\beta$ is an inner function) [1], and for the case when $\alpha$ and $\beta$ are finite Blaschke products (in other words, when $K_\alpha$ and $K_\beta$ are finitely dimensional) [10].

In this paper we provide such characterizations of the operators from $\mathscr {T}(\alpha ,\beta )$ for all $\alpha , \beta$. This is done in Sect. 2. In Sect. 3 we point out a connection between operators from $\mathscr {T}(\alpha ,\beta )$ and from $\mathscr {H}(\alpha ,\beta )$. We then use this connection in Sect. 4 to translate to $\mathscr {H}(\alpha ,\beta )$ some known results on $\mathscr {T}(\alpha ,\beta )$.

2 Characterizations of Asymmetric Truncated Toeplitz Operators

Recall that a model space $K_{\alpha }$ is a reproducing kernel Hilbert space. That is to say that for every f in the model space $K_{\alpha }$ and each $w\in \mathbb {D}$,

$$\begin{aligned} f(w)=\langle f, k_{w}^{\alpha }\rangle , \end{aligned}$$

where the reproducing kernel function $k_{w}^{\alpha }$ is of the form

$$\begin{aligned} k_{w}^{\alpha }(z)=\frac{1-\overline{\alpha (w)}\alpha (z)}{1-\overline{w}z}. \end{aligned}$$

Observe that since $k_{w}^{\alpha }\in H^{\infty }$, the set $K_{\alpha }^{\infty }=K_{\alpha }\cap H^{\infty }$ is dense in $K_{\alpha }$.

A conjugate kernel is the function $\widetilde{k}_{w}^{\alpha }=C_{\alpha }{k}_{w}^{\alpha }$, where $C_{\alpha }:L^2\rightarrow L^2$ is given by

$$\begin{aligned} C_{\alpha }f(z)=\widetilde{f}(z)=\alpha (z)\overline{z}\overline{f(z)},\quad |z|=1. \end{aligned}$$

It can be seen that $C_\alpha$, as defined on $L^2$, is an antilinear isometric involution (a map with these properties is called a conjugation). It can also be verified that the conjugation $C_\alpha$ preserves $K_\alpha$. Therefore $\widetilde{k}_{w}^{\alpha }\in K_\alpha$ for all $w\in \mathbb {D}$ and a simple computation gives

$$\begin{aligned} \widetilde{k}_{w}^{\alpha }(z)=\frac{\alpha (z)-\alpha (w)}{z-w}. \end{aligned}$$

Another conjugation that will be important in what follows is the operator $J^{\#}:L^2\rightarrow L^2$ defined by

$$\begin{aligned} J^{\#}f(z)=f^{\#}(z)=\overline{f(\overline{z})}. \end{aligned}$$

It is known that $J^{\#}$ transforms $K_{\alpha }$ onto $K_{\alpha ^{\#}}$ (see [5, Lemma 4.4]).

Recall that the rank-one operator $f\otimes g$ is defined by $f\otimes g(h)=\langle h,g\rangle f$.

Theorem 2.1

Let A be a bounded linear operator from $K_\alpha$ into $K_\beta$. Then $A\in \mathscr {T(\alpha ,\beta )}$ if and only if there exist $\psi \in K_\beta$ and $\chi \in K_\alpha$ such that

$$\begin{aligned} A-S_\beta A S^*_\alpha =\psi \otimes k^\alpha _0 +k^\beta _0\otimes \chi . \end{aligned}$$

(2.1)

The proof of Theorem 2.1 follows the proof given by D. Sarason for truncated Toeplitz operators [12, Theorem 4.1] and requires some auxiliary lemmas.

Lemma 2.2

For every $\varphi \in L^2$ the equality

$$\begin{aligned} A^{\alpha ,\beta }_\varphi -S_\beta A^{\alpha ,\beta }_\varphi S^*_\alpha =\psi \otimes k^\alpha _0 +k^\beta _0\otimes \chi \end{aligned}$$

holds on $K_{\alpha }^{\infty }$, where

$$\begin{aligned}&\psi =S_\beta P_\beta \left( \overline{z}\varphi \right) \in K_\beta ,\\&\chi =P_\alpha \left( \overline{\varphi }\right) \in K_\alpha . \end{aligned}$$

Proof

Let $f\in K_\alpha ^\infty$ and $g\in K_\beta ^\infty$. The functions

$$\begin{aligned} S_\alpha ^* f(z)=S^* f(z) =\frac{f(z)-f(0)}{z} \end{aligned}$$

and

$$\begin{aligned} S_\beta ^* g(z)=S^* g(z)=\frac{g(z)-g(0)}{z} \end{aligned}$$

clearly belong to $H^2\cap L^\infty =H^\infty$. Therefore,

$$\begin{aligned} \left\langle S_\beta A_\varphi ^{\alpha ,\beta }S_\alpha ^* f,g\right\rangle&=\left\langle P_\beta (\varphi S_\alpha ^*f),S_\beta ^* g\right\rangle =\left\langle \varphi \cdot \frac{f-f(0)}{z},S_\beta ^* g\right\rangle \\&=\left\langle \overline{z}\varphi f,S_\beta ^* g\right\rangle -f(0)\left\langle \overline{z}\varphi ,S_\beta ^* g\right\rangle \\&=\left\langle \overline{z}\varphi f,\frac{g-g(0)}{z}\right\rangle -\left\langle f,k^\alpha _0 \right\rangle \left\langle S_\beta P_\beta (\overline{z}\varphi ), g\right\rangle \\&=\left\langle \overline{z}\varphi f,\overline{z}g\right\rangle -\overline{g(0)}\left\langle \overline{z}\varphi f,\overline{z}\right\rangle -\left\langle (S_\beta P_\beta (\overline{z}\varphi )\otimes k^\alpha _0)f, g\right\rangle \\&=\left\langle A^{\alpha ,\beta }_\varphi f, g\right\rangle -\langle k^\beta _0,g \rangle \left\langle f,P_\alpha (\overline{\varphi })\right\rangle -\left\langle (S_\beta P_\beta (\overline{z}\varphi )\otimes k^\alpha _0)f, g\right\rangle \\&=\left\langle A^{\alpha ,\beta }_\varphi f, g\right\rangle -\langle (k^\beta _0\otimes P_\alpha (\overline{\varphi }))f,g\rangle -\left\langle (S_\beta P_\beta (\overline{z}\varphi )\otimes k^\alpha _0)f, g\right\rangle . \end{aligned}$$

From this and the density of $K_\beta ^\infty$ in $K_\beta$,

$$\begin{aligned} A^{\alpha ,\beta }_\varphi f -S_\beta A^{\alpha ,\beta }_\varphi S^*_\alpha f =(S_\beta P_\beta (\overline{z}\varphi )\otimes k^\alpha _0)f+(k^\beta _0\otimes P_\alpha (\overline{\varphi }))f \end{aligned}$$

for all $f\in K_\alpha ^\infty$. $\square$

Lemma 2.3

If $\varphi =\overline{\chi }+\psi$, where $\chi \in K_\alpha$ and $\psi \in K_\beta$, $\psi (0)=0$, then the equality

$$\begin{aligned} \langle A_\varphi ^{\alpha ,\beta }f,g\rangle =\sum _{n=0}^\infty \langle (S^n_\beta \psi \otimes S^n_\alpha k^\alpha _0 +S^n_\beta k^\beta _0\otimes S^n_\alpha \chi )f,g\rangle \end{aligned}$$

holds for all $f\in K_\alpha ^\infty$ and $g\in K_\beta ^\infty$.

Proof

Note that if $\varphi =\overline{\chi }+\psi$, where $\chi \in K_\alpha$ and $\psi \in K_\beta$, $\psi (0)=0$, then

$$\begin{aligned} P_\alpha (\overline{\varphi })=\chi \qquad \text {and }\qquad S_\beta P_\beta (\overline{z}\varphi ) =S_\beta P_\beta (\overline{z}\psi )=S_\beta S^*_\beta \psi =\psi . \end{aligned}$$

By Lemma 2.2, on $K_{\alpha }^{\infty }$,

$$\begin{aligned} A^{\alpha ,\beta }_\varphi -S_\beta A^{\alpha ,\beta }_\varphi S^*_\alpha =\psi \otimes k^\alpha _0+k^\beta _0\otimes \chi , \end{aligned}$$

and for every $n\ge 0$,

$$\begin{aligned}&S^n_\beta A_\varphi ^{\alpha ,\beta } (S^*_\alpha )^n -S^{n+1}_\beta A_\varphi ^{\alpha ,\beta } (S^*_\alpha )^{n+1} =S^n_\beta \psi \otimes S^n_\alpha k^\alpha _0+S^n_\beta k^\beta _0\otimes S^n_\alpha \chi . \end{aligned}$$

From this

$$\begin{aligned} A_\varphi ^{\alpha ,\beta } =\sum _{n=0}^N (S^n_\beta \psi \otimes S^n_\alpha k^\alpha _0 +S^n_\beta k^\beta _0\otimes S^n_\alpha \chi ) +S^{N+1}_\beta A_\varphi ^{\alpha ,\beta } (S^*_\alpha )^{N+1} \end{aligned}$$

on $K_{\alpha }^{\infty }$ for every $N\ge 0$.

If $f\in K_\alpha ^\infty$ and $g\in K_\beta ^\infty$, then

$$\begin{aligned} \begin{aligned}&\left\langle S^{N+1}_\beta \!\!\!\right. \left. A_\varphi ^{\alpha ,\beta }(S^*_\alpha )^{N+1} f,g\right\rangle \\&\quad =\left\langle A_{\overline{\chi }}^{\alpha ,\beta }(S^*_\alpha )^{N+1} f, (S^*_\beta )^{N+1} g\right\rangle +\left\langle A_\psi ^{\alpha ,\beta }(S^*_\alpha )^{N+1} f,(S^*_\beta )^{N+1} g\right\rangle \\&\quad =\left\langle T_{\overline{\chi }}(S^*)^{N+1} f, (S^*)^{N+1} g\right\rangle +\left\langle (S^*)^{N+1} f,T_{\overline{\psi }}(S^*)^{N+1} g\right\rangle \\&\quad =\left\langle P(\overline{z}^{N+1}\overline{\chi }f) , (S^*)^{N+1}g\right\rangle +\left\langle (S^*)^{N+1}f, P(\overline{z}^{N+1}\overline{\psi }g)\right\rangle \rightarrow 0\ \text {as } N\rightarrow \infty \end{aligned} \end{aligned}$$

since

$$\begin{aligned} \Vert P(\overline{z}^{N+1}\overline{\chi }f)\Vert _2\le \Vert f\Vert _{\infty }\cdot \Vert \chi \Vert _2,\qquad \Vert P(\overline{z}^{N+1}\overline{\psi }g)\Vert _2\le \Vert g\Vert _{\infty }\cdot \Vert \psi \Vert _2 \end{aligned}$$

and $(S^*)^N\rightarrow 0$ in the strong operator topology. Therefore

$$\begin{aligned} \langle A_\varphi ^{\alpha ,\beta }f,g\rangle =\sum _{n=0}^\infty \langle (S^n_\beta \psi \otimes S^n_\alpha k^\alpha _0 +S^n_\beta k^\beta _0\otimes S^n_\alpha \chi )f,g\rangle \end{aligned}$$

for all $f\in K_{\alpha }^{\infty }$ and $g\in K_{\beta }^{\infty }$. $\square$

Proof of Theorem 2.1

If $A\in \mathscr {T}(\alpha ,\beta )$, then $A=A_\varphi ^{\alpha ,\beta }$ for some $\varphi \in L^2$ and it satisfies (2.1) by Lemma 2.2.

Assume now that A is a bounded linear operator from $K_\alpha$ into $K_\beta$ such that (2.1) holds for $\psi \in K_\beta$ and $\chi \in K_\alpha$. Without any loss of generality we can assume that $\psi (0)=0$. Indeed, if this was not the case we would replace $\psi$ and $\chi$ with $\psi -c k^\beta _0$ and $\chi +\overline{c} k^\alpha _0$, respectively, for $c=\psi (0)/(1-|\beta (0)|^2)$.

Define $\varphi =\overline{\chi }+\psi$. By Lemma 2.3, for every $f\in K_{\alpha }^{\infty }$ and $g\in K_{\beta }^{\infty }$,

$$\begin{aligned} \langle A_\varphi ^{\alpha ,\beta }f,g\rangle =\sum _{n=0}^\infty \langle (S^n_\beta \psi \otimes S^n_\alpha k^\alpha _0 +S^n_\beta k^\beta _0\otimes S^n_\alpha \chi )f,g\rangle . \end{aligned}$$

But an argument similar to the one given in the proof of Lemma 2.3 shows that

$$\begin{aligned} \langle Af,g\rangle =\sum _{n=0}^\infty \langle (S^n_\beta \psi \otimes S^n_\alpha k^\alpha _0 +S^n_\beta k^\beta _0\otimes S^n_\alpha \chi )f,g\rangle , \end{aligned}$$

and so $A=A_\varphi ^{\alpha ,\beta }\in \mathscr {T}(\alpha ,\beta )$. $\square$

Corollary 2.4

Let A be a bounded linear operator from $K_\alpha$ into $K_\beta$. Then $A\in \mathscr {T}(\alpha ,\beta )$ if and only if one (and all) of the following conditions holds

• (a) $A-S_\beta A S^*_\alpha =\psi \otimes k^\alpha _0 +k^\beta _0\otimes \chi$;

• (b) $S^*_\beta A-A S^*_\alpha =\psi \otimes k^\alpha _0 +\widetilde{k}^\beta _0\otimes \chi$;

• (c) $A-S^*_\beta A S_\alpha =\psi \otimes \widetilde{k}^\alpha _0 +\widetilde{k}^\beta _0\otimes \chi$;

• (d) $S_\beta A-A S_\alpha =\psi \otimes \widetilde{k}^\alpha _0 +k^\beta _0\otimes \chi$;

for some $\psi \in K_\beta$ and $\chi \in K_\alpha$ (possibly different for different conditions).

Proof

By Theorem 2.1, $A\in \mathscr {T}(\alpha ,\beta )$ if and only if (a) holds for some $\psi \in K_\beta$ and $\chi \in K_\alpha$. We only need to prove that each of the remaining conditions is equivalent to (a).

(a)$\Rightarrow$(b). Assume that

$$\begin{aligned} A-S_\beta A S^*_\alpha =\psi \otimes k^\alpha _0 +k^\beta _0\otimes \chi . \end{aligned}$$

Then

$$\begin{aligned} S^*_\beta A- S^*_\beta S_\beta A S^*_\alpha =(S^*_\beta \psi )\otimes k^\alpha _0 +(S^*_\beta k^\beta _0)\otimes \chi . \end{aligned}$$

Using the following equalities (see [12, Lemma 2.2 and Lemma 2.4])

$$\begin{aligned} S^*_\beta S_\beta =I_{K_\beta }-\widetilde{k}^\beta _0\otimes \widetilde{k}^\beta _0 \quad \text {and} \quad S^*_\beta k^\beta _0=-\overline{\beta (0)}\widetilde{k}^\beta _0, \end{aligned}$$

(by $I_{K_\beta }$ we denote the identity map on $K_{\beta }$) we get

$$\begin{aligned} S^*_\beta A- A S^*_\alpha&=(S^*_\beta \psi )\otimes k^\alpha _0 -\widetilde{k}^\beta _0\otimes (\beta (0)\chi ) -\widetilde{k}^\beta _0\otimes (S_\alpha A^* \widetilde{k}^\beta _0)\\&=(S^*_\beta \psi )\otimes k^\alpha _0 +\widetilde{k}^\beta _0\otimes (-\beta (0)\chi -S_\alpha A^* \widetilde{k}^\beta _0). \end{aligned}$$

That is, (b) holds.

Implications (b)$\Rightarrow$(c), (c)$\Rightarrow$(d) and (d)$\Rightarrow$(a) can be proved similarly using the equalities

$$\begin{aligned} S_\beta S^*_\beta =I_{K_\beta }-{k}^\beta _0\otimes {k}^\beta _0 \quad \text {and} \quad S_\beta \widetilde{k}^\beta _0=-{\beta (0)}{k}^\beta _0 \end{aligned}$$

and their analogues for $K_{\alpha }$ (again, see [12, Lemma 2.2 and Lemma 2.4]). The details are left to the reader. $\square$

Conditions from Corollary 2.4 can also be formulated in terms of the so-called modified compressed shifts.

Corollary 2.5

Let A be a bounded linear operator from $K_\alpha$ into $K_\beta$. Then $A\in \mathscr {T}(\alpha ,\beta )$ if and only if one (and all) of the following conditions holds

• (a) $A-S_{\beta ,b} A S^*_{\alpha ,a}=\psi \otimes k^\alpha _0 +k^\beta _0\otimes \chi$;

• (b) $S^*_{\beta ,b} A-A S^*_{\alpha ,a}=\psi \otimes k^\alpha _0 +\widetilde{k}^\beta _0\otimes \chi$;

• (c) $A-S^*_{\beta ,b} A S_{\alpha ,a}=\psi \otimes \widetilde{k}^\alpha _0 +\widetilde{k}^\beta _0\otimes \chi$;

• (d) $S_{\beta ,b} A-A S_{\alpha ,a}=\psi \otimes \widetilde{k}^\alpha _0 +k^\beta _0\otimes \chi$;

for some $a,b\in \mathbb {C}$, $\psi \in K_\beta$ and $\chi \in K_\alpha$ (possibly different for different conditions), where

$$\begin{aligned} S_{\alpha ,a}=S_\alpha +a (k^\alpha _0\otimes \widetilde{k}^\alpha _0),\qquad S_{\beta ,b}=S_\beta + b(k^\beta _0\otimes \widetilde{k}^\beta _0) \end{aligned}$$

are the modified compressed shifts.

Proof

The proof uses Corollary 2.4 and is analogous to the proof given in [12, Theorem 10.1]. The details are therefore left to the reader. $\square$

Corollary 2.6

$\mathscr {T}(\alpha ,\beta )$ is closed in the weak operator topology.

Proof

See the proof from [12, Theorem 4.2] (see also [10]). $\square$

Using Corollary 2.4 we can now characterize the operators from $\mathscr {T}(\alpha ,\beta )$ in terms of four types of shift invariance. Part (c) of the following corollary was first noted in [2] (for the case when $\beta$ divides $\alpha$).

Corollary 2.7

Let A be a bounded linear operator from $K_\alpha$ into $K_\beta$. Then $A\in \mathscr {T}(\alpha ,\beta )$ if and only if it has one (and all) of the following properties

• (a) $\left\langle AS^* f, S^*g\right\rangle =\left\langle A f,g \right\rangle$ for all $f\in K_\alpha$, $g\in K_\beta$ such that $f(0)=g(0)=0$;

• (b) $\left\langle AS^* f, g\right\rangle =\left\langle A f,S g \right\rangle$ for all $f\in K_\alpha$, $g\in K_\beta$ such that $f(0)=0$, $Sg\in K_\beta$;

• (c) $\left\langle AS f, S g\right\rangle =\left\langle A f,g \right\rangle$ for all $f\in K_\alpha$, $g\in K_\beta$ such that $Sf\in K_\alpha$, $Sg\in K_\beta$;

• (d) $\left\langle AS f, g\right\rangle =\left\langle A f,S^*g \right\rangle$ for all $f\in K_\alpha$, $g\in K_\beta$ such that $Sf\in K_\alpha$, $g(0)=0$.

Proof

See [10, Theorem 3.4] for a short proof of (c). The rest of the conditions can be proved analogously. $\square$

3 A Connection with Asymmetric Truncated Hankel Operators

The following technical lemma is easy to prove.

Lemma 3.1

For every $f,g\in L^2$ we have

• (a) $J J^{\#}f=J^{\#} J f=\overline{z f}$ ($|z|=1$);

• (b) $J(f\cdot g)=\overline{J^{\#} f}\cdot J g=J f\cdot \overline{J^{\#} g}$;

• (c) $J^{\#}(f\cdot g)=J^{\#} f\cdot J^{\#} g$.

It was noted in [10] that $A\in \mathscr {T}(\alpha ,\beta )$ if and only if $C_\beta A C_\alpha \in \mathscr {T}(\alpha ,\beta )$. In particular, if $A=A_\varphi ^{\alpha ,\beta }$, then $C_\beta AC_\alpha =A_{\overline{\alpha }\beta \overline{\varphi }}^{\alpha ,\beta }$. Some generalizations of this relation are shown in the following proposition.

Proposition 3.2

For every $\varphi \in L^2$ we have

• (a) $C_\beta A_\varphi ^{\alpha ,\beta } C_\alpha =A_{\overline{\alpha }\beta \overline{\varphi }}^{\alpha ,\beta }$;

• (b) $C_\beta B_\varphi ^{\alpha ,\beta } C_\alpha =B_{\overline{\alpha \beta ^{\#}\varphi }}^{\alpha ,\beta }$;

• (c) $J^{\#} A_\varphi ^{\alpha ,\beta } J^{\#} =A_{\varphi ^{\#}}^{\alpha ^{\#},\beta ^{\#}}$;

• (d) $J^{\#} B_\varphi ^{\alpha ,\beta } J^{\#} =B_{\varphi ^{\#}}^{\alpha ^{\#},\beta ^{\#}}$;

• (e) $C_\beta A_\varphi ^{\alpha ,\beta } J^{\#} =B_{\overline{\beta ^{\#}}\varphi ^{\#}}^{\alpha ^{\#},\beta }$;

• (f) $J^{\#} B_\varphi ^{\alpha ,\beta } C_\alpha =A_{\overline{\alpha \varphi }}^{\alpha ,\beta ^{\#}}$.

Proof

(a) For completness we include the proof from [10]. Let $f\in K_\alpha ^\infty$, $g\in K_\beta ^\infty$. Then

$$\begin{aligned} \left\langle C_\beta A_\varphi ^{\alpha ,\beta } C_\alpha f,g\right\rangle&=\left\langle C_\beta g, \varphi \cdot C_\alpha f\right\rangle =\left\langle \beta \overline{zg}, \varphi \cdot \alpha \overline{z f}\right\rangle =\left\langle \beta \overline{\alpha }\overline{\varphi }f,g\right\rangle \\&=\left\langle A_{\overline{\alpha }\beta \overline{\varphi }}^{\alpha ,\beta } f,g \right\rangle . \end{aligned}$$

(b) Let $f\in K_\alpha ^\infty$, $g\in K_\beta ^\infty$. Then $C_{\alpha }f\in K_{\alpha }^{\infty }$, $C_{\beta }g\in K_{\beta }^{\infty }$ and

$$\begin{aligned} \left\langle C_\beta B_\varphi ^{\alpha ,\beta } C_\alpha f, g \right\rangle&=\left\langle JC_\beta g,\varphi \cdot C_\alpha f \right\rangle =\left\langle \overline{z}\overline{\beta ^{\#}}z g^{\#}, \varphi \alpha \overline{z}\overline{f}\right\rangle =\left\langle \overline{\alpha \beta ^{\#}\varphi } f, Jg\right\rangle \\&=\left\langle B_{\overline{\alpha \beta ^{\#}\varphi }}^{\alpha ,\beta } f, g\right\rangle . \end{aligned}$$

(c) Let $f\in K_{\alpha ^{\#}}^\infty$, $g\in K_{\beta ^{\#}}^\infty$. Then $J^{\#} f\in K_\alpha ^\infty$, $J^{\#} g\in K_\beta ^\infty$ and using Lemma 3.1(c) we get

$$\begin{aligned} \left\langle J^{\#} A_\varphi ^{\alpha ,\beta } J^{\#} f,g\right\rangle =\left\langle J^{\#} g, \varphi J^{\#} f\right\rangle =\left\langle J^{\#} g, J^{\#}(\varphi ^{\#} f)\right\rangle =\left\langle A_{\varphi ^{\#}}^{\alpha ^{\#},\beta ^{\#}} f, g\right\rangle . \end{aligned}$$

(d) Let $f\in K_{\alpha ^{\#}}^\infty$, $g\in K_{\beta ^{\#}}^\infty$. Then $J^{\#} f\in K_\alpha ^\infty$, $J^{\#} g\in K_\beta ^\infty$ and using Lemma 3.1(a)(c) we have

$$\begin{aligned} \left\langle J^{\#} B_\varphi ^{\alpha ,\beta } J^{\#} f,g \right\rangle&=\left\langle J^{\#} g,B_\varphi ^{\alpha ,\beta } J^{\#} f\right\rangle =\left\langle JJ^{\#} g, \varphi \cdot J^{\#} f\right\rangle \\&=\left\langle J^{\#}(\varphi \cdot J^{\#} f), J g\right\rangle =\left\langle \varphi ^{\#}\cdot f, J g\right\rangle =\left\langle B_{\varphi ^{\#}}^{\alpha ^{\#},\beta ^{\#}} f,g\right\rangle . \end{aligned}$$

(e) Let $f\in K_{\alpha ^{\#}}^\infty$, $g\in K_{\beta }^\infty$. Then $J^{\#} f\in K_\alpha ^\infty$, $C_\beta g\in K_\beta ^\infty$ and by Lemma 3.1(a)(c),

$$\begin{aligned} \left\langle C_\beta A_\varphi ^{\alpha ,\beta } J^{\#} f,g \right\rangle&=\left\langle C_\beta g, A_\varphi ^{\alpha ,\beta } J^{\#} f\right\rangle =\left\langle \beta \overline{z g}, \varphi \cdot J^{\#} f\right\rangle =\left\langle f,J^{\#}(\beta \overline{\varphi }J^{\#}J g)\right\rangle \\&=\left\langle f,\beta ^{\#}\overline{\varphi ^{\#} }J g\right\rangle =\left\langle \overline{\beta ^{\#}}\varphi ^{\#} f,J g\right\rangle =\left\langle B_{\overline{\beta ^{\#}}\varphi ^{\#}}^{\alpha ^{\#},\beta } f, g\right\rangle . \end{aligned}$$

(f) Let $f\in K_{\alpha }^\infty$, $g\in K_{\beta ^{\#}}^\infty$. Then $C_\alpha f\in K_\alpha ^\infty$, $J^{\#} g\in K_\beta ^\infty$ and using Lemma 3.1(a),

$$\begin{aligned} \left\langle J^{\#} B_\varphi ^{\alpha ,\beta } C_\alpha f,g\right\rangle&=\left\langle J^{\#} g, B_\varphi ^{\alpha ,\beta } C_\alpha f\right\rangle =\left\langle JJ^{\#} g,\varphi \cdot C_\alpha f\right\rangle =\left\langle \overline{z g}, \varphi \alpha \overline{z f}\right\rangle \\&=\left\langle \overline{\alpha \varphi }f,g\right\rangle =\left\langle A_{\overline{\alpha \varphi }}^{\alpha ,\beta ^{\#}} f,g\right\rangle . \end{aligned}$$

$\square$

It should be mentioned that part (b) of Proposition 3.2 is a generalization of a result obtained for truncated Hankel operators in [7] while part (c) generalizes a result obtained for truncated Toeplitz operators in [5].

Corollary 3.3

Let A and B be bounded linear operators from $K_\alpha$ into $K_\beta$ for some inner functions $\alpha$ and $\beta$. Then

• (a) $A\in \mathscr {T}(\alpha ,\beta )$ if and only if $C_\beta A C_\alpha \in \mathscr {T}(\alpha ,\beta )$;

• (b) $B\in \mathscr {H}(\alpha ,\beta )$ if and only if $C_\beta B C_\alpha \in \mathscr {H}(\alpha ,\beta )$;

• (c) $A\in \mathscr {T}(\alpha ,\beta )$ if and only if $J^{\#} AJ^{\#} \in \mathscr {T}(\alpha ^{\#},\beta ^{\#})$;

• (d) $B\in \mathscr {H}(\alpha ,\beta )$ if and only if $J^{\#} BJ^{\#} \in \mathscr {H}(\alpha ^{\#},\beta ^{\#})$;

• (e) $A\in \mathscr {T}(\alpha ,\beta )$ if and only if $C_{\beta } AJ^{\#} \in \mathscr {H}(\alpha ^{\#},\beta )$;

• (f) $B\in \mathscr {H}(\alpha ,\beta )$ if and only if $J^{\#} BC_{\alpha } \in \mathscr {T}(\alpha ,\beta ^{\#})$.

4 Consequences for Asymmetric Truncated Hankel Operators

The connection between asymmetric truncated Hankel and Toeplitz operators enables us to transfer the results known for asymmetric truncated Toeplitz operators to their Hankel analogues. The first result is the description of symbols of asymmetric truncated Hankel operators equal to the zero operator. This was also proved in [11] by a somewhat more direct method.

Proposition 4.1

Let $\alpha ,\beta$ be two nonconstant inner functions and let $\varphi \in L^2$. Then $B_{\varphi }^{\alpha ,\beta }=0$ if and only if $\varphi \in H^2+\overline{\alpha \beta ^{\#}H^2}$.

Proof

Proposition 3.2(f) implies that $B_\varphi ^{\alpha ,\beta }=0$ if and only if $A_{\overline{\alpha \varphi }}^{\alpha ,\beta ^{\#}}=0$. By [8, Theorem 2.1] the latter holds if and only if $\overline{\alpha \varphi }\in \overline{\alpha H^2}+\beta ^{\#}H^2$ or equivalently $\varphi \in H^2+\overline{\alpha \beta ^{\#} H^2}$. $\square$

The following proposition generalizes results obtained for truncated Hankel operators in [7].

Proposition 4.2

Let B be a bounded linear operator from $K_\alpha$ into $K_\beta$. Then $B\in \mathscr {H}(\alpha ,\beta )$ if and only if one (and all) of the following conditions holds

• (a) $B- S_\beta B S_\alpha =\psi \otimes \widetilde{k}^\alpha _0 + k^\beta _0\otimes \chi$;

• (b) $S^*_\beta B- B S_\alpha =\psi \otimes \widetilde{k}^\alpha _0 + \widetilde{k}^\beta _0\otimes \chi$;

• (c) $B- S_\beta ^* B S^*_\alpha =\psi \otimes {k}^\alpha _0 + \widetilde{k}^\beta _0\otimes \chi$;

• (d) $S_\beta B- B S^*_\alpha =\psi \otimes {k}^\alpha _0 + k^\beta _0\otimes \chi$;

for some $\psi \in K_\beta$ and $\chi \in K_\alpha$ (possibly different for different conditions).

Proof

We only prove that $B\in \mathscr {H}(\alpha ,\beta )$ if and only if (a) holds. Conditions (b)–(d) can be proved in a similar way. The equivalence of conditions (a)–(d) can also be shown as in the proof of Corollary 2.4.

To prove that $B\in \mathscr {H}(\alpha ,\beta )$ if and only if (a) holds note that by Corollary 3.3(f), $B\in \mathscr {H}(\alpha ,\beta )$ if and only if $J^{\#} BC_\alpha$ belongs to $\mathscr {T}(\alpha ,\beta ^{\#})$. By Theorem 2.1 the latter happens if and only if there are functions $\psi _0\in K_\beta ^{\#}$, $\chi _0\in K_\alpha$ such that

$$\begin{aligned} J^{\#}BC_\alpha - S_{\beta ^{\#}} J^{\#}B C_\alpha S_\alpha ^*=\psi _0\otimes {k}^\alpha _0 + k^{\beta ^{\#}}_0\otimes \chi _0. \end{aligned}$$

(4.1)

Recall that $S_\alpha$ is $C_\alpha$-symmetric, that is, $S_\alpha =C_\alpha S_\alpha ^* C_\alpha$. Moreover, by Proposition 3.2(c), $J^{\#}S_{\beta ^{\#}}J^{\#}=S_\beta$. Applying $J^{\#}$ (from the left) and $C_\alpha$ (from the right) to both sides of (4.1) we thus get

$$\begin{aligned} B- S_{\beta } B S_\alpha =J^{\#}\psi _0\otimes C_\alpha {k}^\alpha _0 + J^{\#}k^{\beta ^{\#}}_0\otimes C_\alpha \chi _0. \end{aligned}$$

It is easy to verify that $J^{\#}k^{\beta ^{\#}}_0=k^\beta _0$ and so (4.1) is just (a) with $\psi =J^{\#}\psi _0$ and $\chi =C_\alpha \chi _0$. $\square$

Corollary 4.3

Let B be a bounded linear operator from $K_\alpha$ into $K_\beta$. Then $B\in \mathscr {H}(\alpha ,\beta )$ if and only if one (and all) of the following conditions holds

• (a) $B-S_{\beta ,b} B S_{\alpha ,a}=\psi \otimes \widetilde{k}^\alpha _0 +k^\beta _0\otimes \chi$;

• (b) $S^*_{\beta ,b} B-B S_{\alpha ,a}=\psi \otimes \widetilde{k}^\alpha _0 +\widetilde{k}^\beta _0\otimes \chi$;

• (c) $B-S^*_{\beta ,b} B S^*_{\alpha ,a}=\psi \otimes k^\alpha _0 +\widetilde{k}^\beta _0\otimes \chi$;

• (d) $S_{\beta ,b} B-B S^*_{\alpha ,a}=\psi \otimes {k}^\alpha _0 +k^\beta _0\otimes \chi$;

for some $a,b\in \mathbb {C}$, $\psi \in K_\beta$ and $\chi \in K_\alpha$ (possibly different for different conditions), where

$$\begin{aligned} S_{\alpha ,a}=S_\alpha +a (k^\alpha _0\otimes \widetilde{k}^\alpha _0),\qquad S_{\beta ,b}=S_\beta + b(k^\beta _0\otimes \widetilde{k}^\beta _0) \end{aligned}$$

are the modified compressed shifts.

Proof

The proof is analogous to the proof of Corollary 2.5 and is therefore left to the reader. $\square$

Corollary 4.4

$\mathscr {H}(\alpha ,\beta )$ is closed in the weak operator topology.

Corollary 4.5

Let B be a bounded linear operator from $K_\alpha$ into $K_\beta$. Then $B\in \mathscr {H}(\alpha ,\beta )$ if and only if it has one (and all) of the following properties

• (a) $\left\langle BS f, S^*g\right\rangle =\left\langle B f,g \right\rangle$ for all $f\in K_\alpha$, $g\in K_\beta$ such that $Sf\in K_\alpha$, $g(0)=0$;

• (b) $\left\langle B f, S g\right\rangle =\left\langle BS f, g \right\rangle$ for all $f\in K_\alpha$, $g\in K_\beta$ such that $Sf\in K_\alpha$, $Sg\in K_\beta$;

• (c) $\left\langle BS^* f, S g\right\rangle =\left\langle B f,g \right\rangle$ for all $f\in K_\alpha$, $g\in K_\beta$ such that $f(0)=0$, $Sg\in K_\beta$;

• (d) $\left\langle B f, S^* g\right\rangle =\left\langle BS^* f,g \right\rangle$ for all $f\in K_\alpha$, $g\in K_\beta$ such that $f(0)=g(0)=0$.