1

数理解析研究所講究録
第2073巻 2018年 1-10

ASYMMETRIC TRUNCATED TOEPLITZ OPERATORSAND ITS CHARACTERIZATIONS BY RANK TWO\cdot

OPERATORS

M. CRISTINA CÂMARA 1 : KAMILA KLIŚ‐GARLICKA2:
and MAREK PTAK 2 3

ABSTRACT. When investigating truncated Toeplitz operators, the question of
considering two different model spaces naturally appears. The goal of thispaper is to present asymmetric truncated Toeplitz operators with L^{2} symbols
between two different model spaces given by inner functions such that one
divides the other. Asymmetric truncated Toeplitz operators can be character‐ized in terms of operators of rank at most two. Mainly, the results from [6] arepresented.

1. INTRODUCTION

Toeplitz operators on the Hardy space H^{2} , which are compositions of a multi‐
plication operator and the orthogonal projection from L^{2} onto H^{2} , constitute aclassical topic in operator theory. In the important paper ([2t\}]) Sarason inves‐tigated truncated Toeplitz operators, thus generating huge interest in this classof operators; see, for example [$, t,\mathrm{t}

1

1t ,

12, :{}^{t}\mathrm),{J} \wedge\mathrm{i}\mathrm{f}\cdotInstead of the classicalHardy space H^{2} , they act on a model space K_{ $\theta$}^{2}=H^{2}\ominus $\theta$ H^{2} associated with agiven nonconstant inner function $\theta$ , and a multiplication operator is composedwith the orthogonal projection from H^{2} onto K_{ $\theta$}^{2}.

Asymmetric truncated Toeplitz operators involve the composition of a multi‐

plication operator with two projections from H^{2} onto a model space, associatedwith (possibly different) nonconstant inner functions $\alpha$ and $\theta$ . They are naturalgeneralizations of rectangular Toeplitz matrices, which appear in various con‐texts, such as the study of finite‐time convolution equations, signal processing,

control theory, probability, approximation theory, diffraction problems (see for
instance [2, 3, 4, 17,18, 1, 21]).
Asymmetric truncated Toeplitz operators were introduced (in the context ofthe Hardy space H^{p} on the half‐plane, with 1 <p<\infty ) and studied in the caseof bounded symbols in [\aleph^{-}] . The following review paper presents properties of anasymmetric truncated Toeplitz operator on the unit disc and is mainly based on

the results from

This work was inspired by the work of Sarason ([20]) , where

many interesting properties of truncated Toeplitz operators were given.

Here we consider bounded asymmetric truncated Toeplitz operators with L^{2}
symbols, defined between two model spaces K_{ $\theta$}^{2} and K_{ $\alpha$}^{2} , where $\alpha$ divides $\theta$2010 Mathematics Subject Classification. Primary: 47B35; Secondary: 30H10, 47A15.
Key words and phrases. Model space, truncated Toeplitz operator, kernel functions,conjugation.

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C. CÂMARA, K. KLIS’‐GARLICKA, and M. PTAK

( $\alpha$ \leq $\theta$) . We study various properties of these operators and their relations

with the corresponding symbols, and we present a necessary and sufficient con‐dition for a bounded operator between two model spaces to be an asymmetrictruncated Toeplitz operator in terms of rank two operators, thus generalizing acorresponding result of Sarason for the case where $\alpha$= $\theta$ . In the asymmetric case,however, a more complex connection between the operators and their symbols isrevealed, which is not apparent when the two model spaces involved are the same.
2. MODEL SPACES AND DECOMPOSITIONS

Let L^{2} denote the space L^{2}(\mathbb{T}, m) , where \mathb {T} is the unit circle and m is thenormalized Lebesgue measure on \mathb {T} , and let H^{2} be the Hardy space on the unit
disc \mathbb{D} , identified as usual with a subspace of L^{2} . Similarly, L^{\infty}=L^{\infty}( $\Gamma$, m) and

we denote by H^{\infty} the space of all analytic and bounded functions on \mathbb{D} . Denoting

by H_{0}^{2} the subspace consisting of all functions in H^{2} which vanish at 0 , we haveL^{2}\ominus H^{2}=\overline{H_{\underline{0},^{2}and we denote by P and P^{-} the ortłiogonal projections from L^{2}

onto H^{2} and H_{0}^{2} , respectively.
With any given inner function $\theta$ we associate the so called model space K_{ $\theta$}^{2},
defined by K_{ $\theta$}^{2}=H^{2}\ominus $\theta$ H^{2} . We also have K_{ $\theta$}^{2}=H^{2}\cap $\theta$\overline{H_{0}^{2} , and thus

f\in K_{ $\theta$}^{2} if and only if \overline{ $\theta$}f\in\overline{H_{0}^{2} and f\in H^{2}
In particular, if f \in K_{ $\theta$}^{2} , then $\theta$\overline{f} \in H_{0}^{2} . Let P_{ $\theta$} be the orthogonal projection
P_{ $\theta$}:L^{2}\rightar ow K_{ $\theta$}^{2}.
Model spaces are also equipped with conjugations (antilinear isometric invo‐

lutions), which are important tools in the study of model spaces \mathrm{a}\mathrm{n}\acute{\mathrm{d} truncatedToeplitz operators (see for example [ 14, 15, 19]). For a given inner function $\theta$,
the conjugation C_{ $\theta$} is defined by C_{ $\theta$} : L^{2}\rightarrow L^{2},

C_{ $\theta$}f(z)= $\theta$\overline{zf(z)}.
It is worth noting that C_{ $\theta$} preserves the space K_{ $\theta$}^{2} and maps $\theta$ H^{2} onto L^{2}\ominus H^{2}.
Recall that for $\lambda$ \in \mathbb{D} the kernel function in H^{2} denoted by k_{ $\lambda$} is given byk_{ $\lambda$}(z)\displaystyle\frac{1}{1-$\lambda$z} . Similarly, for an inner function $\theta$ , in K_{ $\theta$}^{2} the kernel function k_{$\lambda$}^{$\thea$} isgiven by k_{$\lambda$}^{$\thea$} P_{ $\theta$}k_{ $\lambda$} , i.e., k_{$\lambda$}^{$\thea$} k_{ $\lambda$}(1-\overline{ $\theta$( $\lambda$)} $\theta$) . The set \{k_{ $\lambda$}^{ $\theta$} : $\lambda$ \in \mathb {D}\} is linearly
dense in K_{ $\theta$}^{2} . Since k_{$\lambda$}^{$\thea$} \in K_{ $\theta$}^{\infty} , where K_{ $\theta$}^{\infty} denotes the subspace H^{\infty}\cap K_{ $\theta$}^{2} , thespace K_{ $\theta$}^{\infty} is dense in K_{ $\theta$}^{2} (see [2===Defining \tilde{k}_{$\lambda$}^{$\theta$}=C_{$\theta$}k_{$\lambda$}^{$\theta$} , we have in particulark_{0}^{ $\theta$}(z)=1-\overline{ $\theta$(0)} $\theta$(z) , \tilde{k}_{0}^{ $\theta$}(z)=\overline{z}( $\theta$(z)- $\theta$(0) .
It is easy to see that, for all f\in K_{ $\theta$}^{2},\langle f, k_{0}^{ $\theta$}\rangle =f(0) , \langle f, \tilde{k}_{0}^{ $\theta$}\rangle =(C_{ $\theta$}f)(0) .
Now let us consider two nonconstant inner functions $\alpha$ and $\theta$ . If \overline{$\alpha$}$\theta$ is an inner

function, we say that $\alpha$ divides $\theta$ and we write $\alpha$\leq $\theta$.

Proposition 2.1. Let $\alpha$, $\theta$ be nonconstant inner functions such that $\alpha$\leq $\theta$ . The
following holds:

3
ASYMMETRIC TRUNCATED TOEPLITZ OPERATORS(1) K_{ $\theta$}^{2}=K_{ $\alpha$}^{2}\oplus $\alpha$ K_{\underline{ $\theta$} ^{2},(2) P_{$\theta$}=P_{$\alpha$}+$\alpha$P_{\frac{$\theta$}{$\alpha$}\displaystyle\frac{$\alpha$}{$\alpha$},(3) k_{0}^{ $\theta$}=k_{0}^{ $\alpha$}+\overline{ $\alpha$(0)} $\alpha$ k^{\frac{ $\theta$}{0^{ $\alpha$} ,(4) \displayst le\tilde{k}_{0}^{$\theta$}=\frac{$\theta$}{$\alpha$}(0)\tilde{k}_{0}^{$\alpha$}+$\alpha$\tilde{k}^{\frac{$\theta$}{0^{$\alpha$} ,(5) P_{ $\alpha$}k_{0}^{ $\theta$}=k_{0}^{ $\alpha$}, P_{$\alpha$}\displaystyle\tilde{k}_{0}^{$\theta$}=\frac{$\theta$}{$\alpha$}(0)\tilde{k}_{0}^{$\alpha$}.
The following proposition describes some relations between decompositions andconjugations. Note that, if $\alpha$ \leq $\theta$ , any f \in K_{ $\theta$}^{2} can be uniquely decomposed as
f=f_{1}+ $\alpha$ f_{2}for some f_{1} \in K_{ $\alpha$}^{2} and some f_{2} \inK_{\frac{2$\theta$}{$\alpha$} , or as f=f_{2}+\displaystyle \frac{ $\theta$}{ $\alpha$}f_{1} , for somef_{1} \in K_{ $\alpha$}^{2} and some f_{2} \in K_{\underline{ $\theta$} ^{2} . Then the conjugation C_{ $\theta$} can be seen as C_{ $\theta$} : K_{ $\theta$}^{2}=

K_{$\alpha$}^{2}\displayst le\oplus$\alpha$K_{\frac{2$\theta$}{$\alpha$}\rightarowK_{$\theta$}^{2}=K_{\frac{2$\theta$}{$\alpha$}\oplus\frac{$\alpha$_{$\theta$}{$\alpha$}K_{$\alpha$}^{2}, \mathrm{o}\mathrm{r}\mathrm{a}\mathrm{e}C_{$\theta$} : K_{$\theta$}^{2}=K_{\frac{2$\theta$}{$\alpha$}\displaystyle\oplus\frac{$\theta$}{$\alpha$}K_{$\alpha$}^{2}\rightar owK_{$\theta$}^{2}=K_{$\alpha$}^{2}\oplus$\alpha$K_{\frac{2$\theta$}{$\alpha$}.

Now we have:

Proposition 2.2. Let $\alpha$, $\theta$ be nonconstant inner functions such that $\alpha$\leq $\theta$ . Then,f_{1} \in K_{ $\alpha$}^{2}if

and f_{2}\in K_{\frac{2 $\theta$}{ $\alpha$} ,(1) C_{ $\theta$}(f_{1}+ $\alpha$ f_{2})=C_{\frac{ $\theta$}{ $\alpha$} f_{2}+\displaystyle \frac{ $\theta$}{ $\alpha$}C_{ $\alpha$}f_{1},(2) C_{ $\theta$}(f_{2}+\displaystyle \frac{ $\theta$}{ $\alpha$}f_{1})=C_{ $\alpha$}f_{1}+ $\alpha$ C_{\frac{ $\theta$}{ $\alpha$} f_{2}.
Let S be the unilateral shift on the Hardy. space H^{2} and, for a nonconstantinner function $\theta$ , let, S_{ $\theta$}=P_{ $\theta$}S_{|K_{ $\theta$}^{2} be the compression of S to K_{ $\theta$}^{2} . The space K_{ $\theta$}^{2}
is invariant for S^{*} . thus (S_{ $\theta$})^{*} =S_{1K_{ $\theta$}^{2} ^{*} . Note that, for any f\in K_{ $\theta$}^{2},(2.1)

S_{ $\theta$}f=zf-(C_{ $\theta$}f)(0) $\theta$=Sf-\langle f, \tilde{k}_{0}^{ $\theta$}\} $\theta$,(2.2)

S_{ $\theta$}^{*}f=\overline{z}(f-f(0)) .

In particular,(2.3)
S_{ $\theta$}^{*}k_{0}^{ $\theta$}=-\overline{ $\theta$(0)}\tilde{k}_{0}^{ $\theta$}, S_{ $\theta$}\tilde{k}_{0}^{ $\theta$} - $\theta$(0)k_{0}^{ $\theta$}.
The function k_{0}^{$\theta$} is a cyclic vector for S_{ $\theta$} and \tilde{k}_{0}^{$\thea$} is a cyclic vector for S_{$\theta$}^{*} (see[_{\sim}^{\rangle}13 , Lemma 2.3]). We can define the defect operators I_{K_{ $\theta$}^{2} -S_{ $\theta$}S_{ $\theta$}^{*} =k_{0}^{ $\theta$}\otimes k_{0}^{ $\theta$} and
I_{K_{ $\theta$}^{2} -S_{ $\theta$}^{*}S_{ $\theta$} =\tilde{k}_{0}^{ $\theta$}\otimes\tilde{k}_{0}^{ $\theta$} , using the notation (x\otimes y)z= \{z, y\rangle x for any x, y, z in a
Hilbert space H ([20, Lemma 2.4]).
3. ASYMMETRIC TRUNCATED TOEPLITZ OPERATORSLet $\alpha$, $\theta$ be nonconstant inner functions. For $\varphi$ \in L^{2} we define an operator
A_{ $\varphi$}^{ $\theta,\ \alpha$}:D\subset K_{ $\theta$}^{2} \rightarrow K_{ $\alpha$}^{2} , as A_{ $\varphi$}^{ $\theta,\ \alpha$}f= P_{ $\alpha$}( $\varphi$ f) having domain D=D(A_{ $\varphi$}^{$\theta$_{:} $\alpha$}) \{f \in
K_{ $\theta$}^{2} : $\varphi$ f \in L^{2}\} . The operator A_{$\varphi$}^{$\theta,\ alpha$} is closed and densely defined in K_{ $\theta$}^{2} . Notethat K_{ $\theta$}^{\infty} \subset D(A_{ $\varphi$}^{ $\theta,\ \alpha$}) . The operator A_{$\varphi$}^{$\theta,\ alpha$} will be called an asymmetric truncated=Toeplitz operator. If this operator is bounded, then it admits a unique bounded

extension to K_{ $\theta$}^{2}, A_{\backslash $\rho$}^{ $\theta,\ \alpha$}:K_{ $\theta$}^{2}\rightar ow K_{ $\alpha$}^{2} . By T( $\theta$, $\alpha$) we denote the space of all boundedasymmetric truncated Toeplitz operators. For $\alpha$= $\theta$ we will write A_{$\varphi$}^{$\theta$} instead of
A_{$\varphi$}^{$\theta,\ theta$} (such operators are called truncated Toeplitz operators and were studied bySarason in [20]) and T( $\theta$) instead of T( $\theta$, $\theta$) .

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C. CÂMARA, K. KLIŚ‐GARLICKA, and M. PTAK

It is easy to see that the following holds.
Proposition 3.1. Let $\alpha$, $\theta$ be any inner functions and $\varphi$\in L^{2}

Then\langle A_{ $\varphi$}^{ $\theta,\ alpha$}f, g\rangle=\{f, A_{\frac{ $\alpha$}{ $\varphi$'} ^{ $\theta$}g\} for all f\in D(A_{(P}^{ $\theta,\ \alpha$}) , g\in \mathcal{D}(A_{\overline{ $\varphi$} ^{ $\alpha,\ \theta$}) .
Moreover, D(A_{\frac{ $\alpha$}{ $\varphi$} ^{ $\theta$})=D( A_{ $\varphi$}^{ $\theta,\ \alpha$})^{*}) and (A_{ $\varphi$}^{ $\theta,\ alpha$})^{*}=A_{\frac{ $\alpha$}{ $\varphi$} :^{ $\theta$}
The following shows the first difference between asymmetric truncated andtruncated Toeplitz operators

Proposition 3.2. Let $\alpha$, $\theta$ be nonconstant inner functions such that $\alpha$\leq $\theta$ . Let
A_{$\psi$}^{$\theta,\ alpha$} be an asymmetric truncated Toeplitz operator with $\psi$\in H^{2} . Then

S_{ $\alpha$}A_{ $\psi$}^{ $\theta,\ \alpha$}f=A_{ $\psi$}^{ $\theta,\ \alpha$}S_{ $\theta$}f

for all f\in K_{ $\theta$}^{\infty}

Remark 3.3. Theorem 3.1.16
implies that for nonconstant inner functions $\alpha$, $\theta$
$
\
t
h
e
t
a
$
such that $\alpha$ \leq , if a bounded operator A : K_{ $\theta$}^{2} \rightarrow K_{ $\alpha$}^{2} intertwines S_{ $\alpha$}, S_{ $\theta$} , i.e.,
S_{ $\alpha$}A=AS_{ $\theta$} , then A=A_{ $\psi$}^{ $\theta,\ \alpha$} for some $\psi$\in H^{\infty}.
Example 3.4. One can ask, whether a similar result as in Proposition 3.2 canbe obtained for A_{ $\psi$}^{ $\alpha,\ theta$} with $\alpha$ \leq $\theta$ and $\psi$ \in H^{2} , but the answer is negative. For

example, let $\alpha$=z^{2}, $\theta$=z^{n}, n > 5, $\psi$=z^{3} and f=z . Then S_{ $\theta$}A_{ $\psi$}^{$\alpha$_{:} $\theta$}f=z^{5} but

A_{ $\psi$}^{ $\alpha,\ \theta$}S_{ $\alpha$}f=0.

The next theorem shows a necessary and sufficient condition for a boundedasymmetric truncated Toeplitz operator to be the zero operator in terms of its
symbol.

Theorem 3.5. Let $\alpha$, $\theta$ be nonconstant inner functions such that $\alpha$ \leq $\theta$ . Let
A_{$\varphi$}^{$\theta,\ alpha$} : K_{ $\theta$}^{2}\rightar ow K_{ $\alpha$}^{2} be a bounded asymmetnc truncated Toeplitz operator with $\varphi$\in L^{2}
Then A_{ $\varphi$}^{ $\theta,\ \alpha$}=0 if and only if $\varphi$\in $\alpha$ H^{2}+\overline{ $\theta$ H^{2} .
Corollary 3.6. Let $\alpha$\leq $\theta$ be nonconstant inner functions and let A_{ $\varphi$}^{ $\theta,\ \alpha$}\in T( $\theta$, $\alpha$) .

For $\varphi$ \in L^{2} there are functions $\psi$ \in K_{ $\alpha$}^{2} and $\chi$ \in K_{ $\theta$}^{2} such that A_{ $\varphi$}^{ $\theta,\ alpha$}=

A_{$\psi$+\overline{$\chi$}^{$\theta,\ alpha$}.

Moreover, A_{ $\psi$+\overline{x} ^{ $\theta,\ alpha$}=A_{$\psi$_{1}+\overline{x}_{1} ^{ $\theta,\ alpha$} iff $\psi$_{1}= $\psi$+ck_{0}^{ $\alpha$}, $\chi$_{1}= $\chi$-\overline{c}k_{0}^{ $\theta$} for some constant G.

The following properties can be immediately obtained from the previous resultsby taking adjoint.
Corollary 3.7. Let A_{ $\varphi$}^{ $\alpha,\ theta$} : K_{ $\alpha$}^{2} \rightarrow K_{ $\theta$}^{2}, A_{ $\varphi$}^{ $\alpha,\ theta$} \in T( $\alpha$,\cdot $\theta$) , $\alpha$ \leq $\theta$, $\varphi$ \in L^{2} .
A_{ $\varphi$}^{ $\alpha,\ \theta$}=0 iff $\varphi$\in $\theta$ H^{2}+\overline{ $\alpha$ H^{2} .
Corollary 3.8. Let A_{ $\varphi$}^{ $\alpha,\ theta$} : K_{ $\alpha$}^{2}\rightar ow K_{ $\theta$}^{2}, A_{\ell}^{ $\alpha,\ \theta$}\in T( $\alpha$, $\theta$) , $\alpha$\leq $\theta$, $\varphi$\in L^{2}

Then

Then there

are functions $\psi$\in K_{ $\alpha$}^{2}, $\chi$\in K_{ $\theta$}^{2} such that A_{ $\varphi$}^{ $\alpha,\ theta$}=A_{+x}^{ $\theta$}\displaystyle \frac{ $\alpha$}{ $\psi$},.

4. CHARACTERIZATIONS IN TERMS OF RANK‐TWO OPERATORSIn [2 [\backslh\prime , Theorem 4.1] a characterization of truncated Toeplitz operators in T( $\theta$)

was presented using rank two operators defined in terms of the kernel function k_{0}^{ $\theta$}.

Following [6], an analogous result for asymmetric truncated Toeplitz operators
T( $\theta$, $\alpha$) using the kernel functions k_{0}^{ $\alpha$} and k_{0}^{$\theta$} can be presented.

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ASYMMETRIC TRUNCATED TOEPLITZ OPERATORS

Theorem 4.1 (Theorem 5.1 [6]). Let $\alpha$, $\theta$ be nonconstant inner functions suchthat $\alpha$\leq $\theta$ and let A:K_{ $\theta$}^{2}\rightar ow K_{ $\alpha$}^{2} be a bounded operator. Then A\in T( $\theta$, $\alpha$) if andonly if there are $\psi$\in K_{ $\alpha$}^{2}, $\chi$\in K_{ $\theta$}^{2} such that

(4.1)

A-S_{ $\alpha$}AS_{ $\theta$}^{*}= $\psi$\otimes k_{0}^{ $\theta$}+k_{0}^{ $\alpha$}\otimes $\chi$.

It can be obtained a similar characterization for operators from T( $\alpha$, $\theta$) by
taking adjoints in (4.1).

Corollary 4.2 (Corollary 5.2 [i,] ). Let $\alpha$, $\theta$ be nonconstant inner functions suchthat $\alpha$\leq $\theta$ and let A : K_{ $\alpha$}^{2}\rightar ow K_{ $\theta$}^{2} be a bounded operator. Then A\in T( $\alpha$, $\theta$) if andonly if there are $\psi$\in K_{ $\alpha$}^{2}, $\chi$\in K_{ $\theta$}^{2} such that

A-S_{ $\theta$}AS_{ $\alpha$}^{*}=k_{0}^{ $\theta$}\otimes $\psi$+ $\chi$\otimes k_{0}^{ $\alpha$}.
Sarason obtained also a characterization for truncated Toeplitz operators be‐longing to T( $\theta$) using the function \tilde{k}_{0}^{ $\theta$}=C_{ $\theta$}k_{0}^{ $\theta$} instead of k_{0}^{$\theta$} , by a simple applicationof the conjugation C_{ $\theta$} to the result of Theorem 4, 1 in the case $\alpha$= $\theta$ . Here, follow‐

ing [6] we will present that an analogous result holds for operators belonging toT( $\theta$, $\alpha$) , $\alpha$\leq $\theta$ . However, in the case of asymmetric truncated Toeplitz operators
the situation is more complex. The relation between a symbol of an asymmetrictruncated Toeplitz operator and a rank two operator appearing in (4.2) is more
involved.

Theorem 4.3 (Theorem 6.1
Let $\alpha$, $\theta$ be nonconstant inner functions suchA$\alpha$ \leq $\theta$_{f}\rightarrow
K_{ $\alpha$}^{2} be a bounded operator. Then A \in T( $\theta$, $\alpha$) if
that
and let : K_{ $\theta$}^{2}
and only if there are $\mu$\in K_{ $\alpha$}^{2} and $\iota$/\in K_{ $\theta$}^{2} such that

(4.2)

A-S_{ $\alpha$}^{*}AS_{ $\theta$}= $\mu$\otimes\tilde{k}_{0}^{ $\theta$}+\tilde{k}_{0}^{ $\alpha$}\otimes \mathrm{v}.

Moreover, if A=A_{ $\psi$+\overline{x} ^{ $\theta,\ \alpha$} with $\psi$\in K_{ $\alpha$}^{2} and $\chi$\in K_{ $\theta$}^{2} , then A satisfies (4.2) with

(4.3)$\mu$=C_{ $\alpha$}P_{ $\alpha$}(\displaystyle \frac{\overline{ $\theta$} {\overline{ $\alpha$} $\chi$) , v=C_{ $\alpha$} $\psi$+S^{*}( $\alpha$ P_{\frac{ $\theta$}{ $\alpha$} $\chi$) .

By taking adjoints in (1.2) we obtain a similar characterization for operatorsfrom T( $\alpha$, $\theta$) :Corollary 4.4 (Corollary 6.2 [6]). Let $\alpha$, $\theta$ be nonconstant inner functions suchthat $\alpha$ \leq $\theta$ , and let A : K_{ $\alpha$}^{2} \rightarrow K_{ $\theta$}^{2} be a bounded operator. Then A \in T( $\alpha$, $\theta$) if

and only if there are $\mu$\in K_{ $\alpha$}^{2}, u\in K_{ $\theta$}^{2} such that

A-S_{ $\theta$}^{*}AS_{ $\alpha$}=\tilde{k}_{0}^{ $\theta$}\otimes $\mu$+v\otimes\tilde{k}_{0}^{ $\alpha$}.
It is clear that if an asymmetric truncated Toeplitz operator A satisfies equation(4.2) with some $\mu$, v , then that equation is also satisfied if $\mu$, $\iota$/ are replaced by(4.4)$\mu$'= $\mu$+\overline{b}\tilde{k}_{0}^{ $\alpha$}, $\iota$/'= $\nu$-b\tilde{k}_{0}^{ $\theta$},respectively, for any b\in \mathbb{C} . On the other hand, it is also true that the symbol of

A=A_{ $\psi$+\overline{x} ^{ $\theta,\ \alpha$}\in T( $\theta$, $\alpha$) is not unique, and by Corollary 3.\acute{6}\rangle we can replace $\psi$\in K_{ $\alpha$}^{2}
and $\chi$\in K_{ $\theta$}^{2} by(4.5)$\psi$'= $\psi$+ck_{0}^{ $\alpha$}\in K_{ $\alpha$}^{2}, $\chi$'= $\chi$-\overline{c}k_{0}^{ $\theta$}\in K_{ $\theta$}^{2},

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C. CÂMARA, K. KLIŚ‐GARLICKA, and M. PTAK

respectively, for any c\in \mathbb{C} . Using (\cdot\prime/1.3) , it is easy to see that the following relation
between the freedom of choice of $\mu$, $\nu$ on the one hand, and $\psi$, $\chi$ on the other,holds.

Corollary 4.5 (Corollary 6.3 [:}]). Let $\mu$\in K_{ $\alpha$}^{2} and $\nu$\in K_{ $\theta$}^{2} be defined by (4.3) for
given $\psi$\in K_{ $\alpha$}^{2} and $\chi$\in K_{ $\theta$}^{2} , and let $\mu$' \in K_{ $\alpha$}^{2} and $\iota$/'\in K_{ $\theta$}^{2} be defined analogously
for $\psi$'\in K_{ $\alpha$}^{2} and $\chi$'\in K_{ $\theta$}^{2} . If (4.5) holds, then$\mu$'= $\mu$-c\displaystyle \frac{ $\theta$}{ $\alpha$}(0)\tilde{k}_{0}^{ $\alpha$} , $\nu$'= $\nu$+\overline{c}\overline{\frac{ $\theta$}{ $\alpha$}(0)}\tilde{k}_{0}^{ $\theta$}.
The examples below illustrate the result of Theorem 4.3 in the case of Toeplitz
matrices.

Example 4.6 (Example 6.4 [6]). Let us consider $\alpha$=z^{2}, $\theta$=z^{5} and a Toeplitz
operator A A_{ $\psi$+\overline{x} ^{z^{5} . Assume that $\psi$ a_{0}+a_{1}z and $\chi$ \overline{b}_{0}+\overline{b}_{-1}z+\overline{b}_{-2}z^{2}+\overline{b}_{-3}z^{3}+\overline{b}_{-4}z^{4}(\overline{b}_{0}+\overline{b}_{-1}z+\overline{b}_{-2}z^{2})+z^{3}(\overline{b}_{-3}+\overline{b}_{-4}z) . Then C_{z^{2} $\psi$ \overline{a}_{1} +\overline{a}_{0}z,C_{z^{2}}P_{z^{2}}\overline{z}^{3} $\chi$=b_{-4}+b_{-3}z and S^{*}(z^{2}(\overline{b}_{0}+\overline{b}_{-1}z+\overline{b}_{-2}z^{2}))=\overline{b}_{0}z+\overline{b}_{-1}z^{2}+\overline{b}_{-2}z^{3} . Notethat A-S_{z^{2}}^{*}AS_{z^{5}} has a matrix representation
=

=

=

=

=

\begin{ar ay}{l l} 0 & 0 & 0 & 0 & b_{-4}\ a_{1} & a_{0}+b_{0} & b_{-1} & b_{-2} & b_{-3} \end{ar ay},

which can be expressed as(b_{-4}+b_{-3}z)\otimes z^{4}+z\otimes(\overline{a}_{1}+(\overline{a}_{0}+\overline{b}_{0})z+\overline{b}_{-1}z^{2}+\overline{b}_{-2}z^{3}) .
On the other hand, let A-S_{z^{2}}^{*}AS_{z^{5}} have a matrix representation\begin{ar ay}{l l} 0 & 0 & 0 & 0 & b_{0}\ a_{0} & a_{1} & a_{2} & a_{3} & a_{4}+b_{1} \end{ar ay},

which can be expressed as$\mu$\otimes z^{4}+z\otimes $\nu$=(b_{0}+b_{1}z)\otimes z^{4}+z\otimes(\overline{a}_{0}+\overline{a}_{1}z+\overline{a}_{2}z^{2}+\overline{a}_{3}z^{3}+\overline{a}_{4}z^{4}) .
Note that $\nu$=$\nu$_{z^{2} +z^{2}$\nu$_{z^{3} =(\overline{a}_{0}+\overline{a}_{1}z)+z^{2}(\overline{a}_{2}+\overline{a}_{3}z+\overline{a}_{4}z^{2}) . Then $\psi$=C_{z^{2}}P_{z^{2}} $\nu$=a_{1}+a_{0}z and $\chi$=\overline{a}_{2}z+\overline{a}_{3}z^{2}+(\overline{b}_{1}+\overline{a}_{4})z^{3}+\overline{b}_{0}z^{4} . Hence by Theorem ’1.1 we have

(4.6)

A-S_{z^{2}}\cdot AS_{z^{5}}^{*}=(a_{1}+a_{0}z)\otimes 1+1\otimes(\overline{a}_{2}z+\overline{a}_{3}z^{2}+(\overline{b}_{1}+\overline{a}_{4})z^{3}+\overline{b}_{0}z^{4})

Requiring that \mathrm{v}_{z^{3} is orthogonal to z^{2} (see Theorem 6.3) determines that a_{4}=0.
On the other hand, we have some freedom in defining $\psi$ and $\chi$ ; namely $\psi$_{1}=s+a_{0}z and $\chi$_{1}=\overline{t}+\overline{a}_{2}z+\overline{a}_{3}z^{2}+(\overline{b}_{1}+\overline{a}_{4})z^{3}+\overline{b}_{0}z^{4} also satisfy (\angle 1.6) if we assume

that t+s=a_{1}.

Let us now \mathrm{t}_{\partial}\mathrm{J}<\mathrm{e} $\alpha$\cdot=z^{3}, $\theta$=z^{3}(( $\lambda$-z)/(1and consider the operator A=A_{ $\psi$+\overline{x} ^{ $\theta,\ \alpha$} , where $\psi$=a_{0}+a_{1}z+a_{2}z^{2}\in K_{ $\alpha$}^{2}

Example 4.7 (Example 6.5\overline{ $\lambda$}z) ^{2}, $\lambda$\in \mathbb{D}and $\chi$=(\overline{b}_{0}+\overline{b}_{1}z+\overline{b}_{2}z^{2}+\overline{b}_{3}z^{3}+\overline{b}_{4}z^{4})(1-\overline{ $\lambda$}z)^{-2}\in K_{ $\theta$}^{2} (see [11‘$, Corollary 5.7.3]).

Then by Theorem 4.3

A-S_{ $\alpha$}^{*}AS_{ $\theta$}= $\mu$\otimes($\lambda$^{2}z^{2}-2 $\lambda$ z^{3}+z^{4})(1-\overline{ $\lambda$}z)^{-2}+z^{2}\otimes $\nu$,where $\mu$=b_{4}+(b_{3}+2\overline{ $\lambda$}b_{4})z+(b_{2}+3\overline{ $\lambda$}^{2}b_{4}+\overline{ $\lambda$}b_{3})z^{2} and(\overline{a}_{2}+(\overline{a}_{1}-2\overline{ $\lambda$}\overline{a}_{2})z+(\overline{b}_{0}+\overline{a}_{0}-2\overline{ $\lambda$}\overline{a}_{1}+\overline{ $\lambda$}\overline{a}_{2})z^{2}+(\overline{b}_{1}+\overline{ $\lambda$}^{2}\overline{a}_{1}-2\overline{ $\lambda$}\overline{a}_{0})z^{3}+\overline{ $\lambda$}^{2}\overline{a}_{0}z^{4})(1-\overline{ $\lambda$}z)^{-2}.
\mathrm{v}=

7
ASYMMETRIC TRUNCATED TOEPLITZ OPERATORS

5. CHARACTERIZATIONS IN TERMS OF RANK—ONE OPERATORS

Our aim now is to describe the classes of symbols of an operator A \in T( $\theta$, $\alpha$)
for which the right hand side of (4.2) is a rank one operator. The corresponding
question regarding the equation (4.1) is trivial by Corollary 3.6, since the rightside of (/ł.1) is a rank one operator if and only if $\psi$ c\cdot k_{0}^{ $\alpha$} or X c\cdot k_{0}^{ $\theta$} with$\theta$ the question regarding the equality (\cdot\prime 1.2) also has anc\in \mathbb{C} . In the case $\alpha$easy answer, since the relation between the symbols in (4.1) and (/1.2) is $\psi$=C_{ $\theta$} $\nu$===

ànd $\chi$=C_{ $\theta$} $\mu$.

Theorem 5.1 (Theorem 7.2 [6]). Let $\alpha$\leq $\theta$ be nonconstant inner functions andlet A_{ $\psi$+\overline{ $\chi$} ^{ $\theta,\ \alpha$}\in T( $\theta$, $\alpha$) , where $\psi$\in K_{ $\alpha$}^{2} and $\chi$\in K_{ $\theta$}^{2} . Then
(1) A_{ $\psi$+\overline{x} ^{ $\theta,\ alpha$}-S_{ $\alpha$}^{*}A_{ $\psi$+\overline{x} ^{ $\theta,\ alpha$}S_{ $\theta$}= $\mu$\otimes\tilde{k}_{0}^{ $\theta$} for $\mu$\in K_{ $\alpha$}^{2} if and only if there is s\in \mathbb{C} suchthat $\psi$=sk_{0}^{ $\alpha$}, P_{\frac{$\theta$}{$\alpha$} $\chi$=-\overline{s}k^{\frac{$\theta$}{0^{$\alpha$} )(2) A_{ $\psi$+\overline{x} ^{ $\theta,\ alpha$}-S_{ $\alpha$}^{*}A_{ $\psi$+\overline{x} ^{ $\theta,\ alpha$}S_{ $\theta$}=\tilde{k}_{0}^{ $\alpha$}\mathrm{X}u for $\nu$\in K_{ $\theta$}^{2} if and only if P_{$\alpha$}($\chi$\displayst le\frac{\overline{$\theta$}{\overline{$\alpha$})= const. k_{0}^{$\alpha$}.
Remark 5.2 (Remark 7.3 [6]). When the right hand side of the characterization(4.1) reduces to a rank one operator const. k_{0}^{ $\alpha$}\otimes k_{0}^{ $\theta$} it is immediate that thisoperator can be expressed in terms of the symbol $\psi$+\overline{ $\chi$} asconst. k_{0}^{ $\alpha$}\otimes k_{0}^{ $\theta$}=P_{\mathb {C}k_{0}^{ $\alpha$} $\psi$\otimes k_{0}^{ $\theta$}+k_{0}^{ $\alpha$}\otimes P_{\mathb {C}k_{0}^{ $\theta$} $\chi$= ( $\psi$(0)\Vert\tilde{k}_{0}^{ $\alpha$}\Vert^{-2}+\overline{ $\chi$(0)}\Vert\tilde{k}_{0}^{ $\theta$}\Vert^{-2})k_{0}^{ $\alpha$}\otimes k_{0}^{ $\theta$}.
It might be of independent interest to consider the case when the right hand sidein the equation (4.2) reduces to a rank one operator const. \tilde{k}_{0}^{$\alpha$}\otimes\tilde{k}_{0}^{$\theta$} . In fact thisoperator can be expressed in terms of the symbol $\psi$+\overline{ $\chi$} as

CO7 $\iota$ st\cdot\tilde{k}_{0}^{ $\alpha$}\otimes\tilde{k}_{0}^{ $\theta$}=P_{\mathb {C}\tilde{k}_{0}^{ $\alpha$} $\mu$\otimes\tilde{k}_{0}^{ $\theta$}+\tilde{k}_{0}^{ $\alpha$}\otimes P_{\mathrm{C}\overline{k}_{0}^{ $\theta$} $\nu$=(\displaystyle\overline{$\chi$_{$\alpha$}(0)}\Vert\ ilde{k}_{0}^{$\alpha$}\Vert^{-2}+\frac{$\theta$}{$\alpha$}(0)($\psi$(0)-\overline{$\chi$_{\frac{$\theta$}{\mathrm{Q} (0)}\Vert\ ilde{k}_{0}^{$\theta$}\Vert^{-2}) \tilde{k}_{0}^{$\alpha$}\otimes\tilde{k}_{0}^{$\theta$}.
A similar question can be asked regarding the case when the right hand side ofthe equation (4.2) reduces to a rank one operator const. \tilde{k}_{0}^{$\alpha$}\otimes\tilde{k}_{0}^{$\alpha$} . We haveconst. \tilde{k}_{0}^{$\alpha$}\otimes\tilde{k}_{0}^{$\alpha$}=P_{\mathb {C}\overline{k}_{0}^{$\alpha$} $\mu$\otimes\tilde{k}_{0}^{$\alpha$}+\tilde{k}_{0}^{$\alpha$}\otimesP_{\mathb {C}\overline{k}_{0}^{\mathrm{Q} $\nu$=(\overline{ $\chi$(0)}-\overline{$\chi$_{\frac{ $\theta$}{ $\alpha$} (0)}| $\alpha$(0)|^{2}+ $\psi$(0) \Vert k_{0}^{ $\alpha$}\Vert^{-2}\tilde{k}_{0}^{ $\alpha$}\otimes\tilde{k}_{0}^{ $\alpha$}.
6. FROM THE OPERATOR TO THE SYMBOLIn the case of a classical Toeplitz operator T_{ $\varphi$} on H^{2} , the (unique) symbol$\varphi$can be obtained from the operator by the formula \displaystyle \lim_{n\rightar ow\infty}\overline{z}^{n}T_{ $\varphi$}z^{n} . In the$\theta$case of a truncated Toeplitz operator, i.e., of the form A_{$\varphi$}^{$\alpha,\ theta$} with, oneH^{2}+\overline{H^{2}}
k
_
{
0
}
^
{
$
\
t
h
e
t
a
$
can obtain a symbol belonging tofrom the action of A_{$\varphi$}^{$\theta$} on } and\tilde{k}_{0}^{$\thea$} ([*f]) . A similar result can be obtain\ve \mathrm{e}\mathrm{d} for an asymmetric truncated Toeplitz$\alpha$=operator A\in T( $\theta$, $\alpha$) by considering the action of the operator A and its adjoint
on reproducing kernel functions of, the same kind, see [6]. Here we concentrateon the question wether the characterizations of asymmetric truncated Toeplitz
operators in terms of operators of rank two at most, presented in the previous
sections, allow us also to obtain a symbol for the operator.

8
C. CÀMARA, K. KLIS’‐GARLICKA, and M. PTAK

Regarding the first characterization, it follows from Theorem 4.1 that, if A is abounded operator and satisfies the equality (4.1), then A=A_{ $\psi$+\overline{x} ^{ $\theta,\ \alpha$} . Remark thatby Corollary 3.6 we know that $\psi$ and $\chi$ are not unique and we can adjust thevalue of either $\psi$ or $\chi$ at the origin.

For $\alpha$= $\theta$ the characterization (4.2) of truncated Toeplitz operators in Theorem
4.3 reduces to Sarason’s ([20, Remark, p. 501 In that case the relation between$\psi$, $\chi$ in the symbol of A_{ $\psi$+\overline{ $\chi$} ^{ $\theta$} and $\mu$, $\nu$ is given by the conjugation C_{ $\theta$} , namely
C_{ $\theta$} $\psi$ . Thus one can also immediately associate a symbol of
C_{ $\theta$} $\chi$ and v$\mu$
the form $\psi$+\overline{ $\chi$} to a truncated Toeplitz operator satisfying that equality. In
the asymmetric case, however, Theorem 4.3 unveils a more complex connection==

between the rank‐two operator on the right‐hand side of (4^{\underline{\prime $\gam a$} ) and the symbolsof A_{$\psi$+\overline{x}^{$\theta,\ alpha$} , and finding a symbol in terms of $\mu$ and for an operator A satisfying$\nu$equality (4.2) is more difficult.

To solve that problem in the case of asymmetric truncated Toeplitz operators
we start with two auxiliary results.

Lemma 6.1 (Lemma 8.3 [6]). Let $\psi$\in K_{ $\alpha$}^{2}, $\chi$\in K_{ $\theta$}^{2} . Assume that $\chi$=$\chi$_{\frac{$\theta$}{$\alpha$}+\displaystyle\frac{$\theta$}{$\alpha$} \chi$_{$\alpha$}

according to the decomposition K_{$\theta$}^{2}=K_{\frac{2$\theta$}{$\alpha$}\displaystyle\oplus\frac{$\theta$}{$\alpha$}K_{$\alpha$}^{2} . If$\mu$=C_{ $\alpha$}P_{ $\alpha$}(\displaystyle \frac{\overline{ $\theta$} {\overline{ $\alpha$} $\chi$)+\overline{b}\tilde{k}_{0}^{ $\alpha$}, \mathrm{v}=C_{ $\alpha$} $\psi$+S^{*}( $\alpha$ P_{\frac{ $\theta$}{ $\alpha$} $\chi$)-b\tilde{k}_{0}^{ $\theta$}
for fixed b\in \mathbb{C} , then$\psi$=C_{$\alpha$^{l/} $\alpha$}-(\overline{$\chi$_{\frac{ $\theta$}{ $\alpha$} (0)}-\overline{b}\overline{\frac{ $\theta$}{ $\alpha$}(0)} k_{0:}^{ $\alpha$}$\chi$_{$\alpha$}=C_{$\alpha$}$\mu$-bk_{0}^{$\alpha$},$\chi$_{\frac{$\theta$}{$\alpha$} =S_{\frac{$\theta$}{a} $\nu$e$\alpha$+($\chi$_{\frac{$\theta$}{$\alpha$} (0)-b\displaystyle\frac{$\theta$}{$\alpha$}(0) k^{\frac{$\theta$}{0^{$\alpha$} ,where $\nu$=\displaystyle\mathrm{v}_{$\alpha$}+$\alpha$l/\frac{$\theta$}{$\alpha$} according to the decomposition K_{ $\theta$}^{2}=K_{ $\alpha$}^{2}\oplus $\alpha$ K_{\frac{2 $\theta$}{ $\alpha$} .
Lemma 6.2 (Lemma 8.4 [6]) . Let A\in T( $\theta$, $\alpha$) satisfy the equation

A-S_{ $\alpha$}^{*}AS_{ $\theta$}= $\mu$\otimes\tilde{k}_{0}^{ $\theta$}+\tilde{k}_{0}^{ $\alpha$}\otimes \mathrm{v}

for $\mu$\in K_{ $\alpha$}^{2}, $\nu$\in K_{ $\theta$}^{2} . Then $\mu$ and can be chosen such that P_{\frac{$\theta$}{$\alpha$}(\overline{$\alpha$}$\nu$) is orthogonal\mathrm{v}

to \tilde{k}^\frac{$\thea$}{0^ \alph$} . In this case, $\mu$ and are uniquely determined.
\mathrm{v}

When investigating symbols of the asymmetric truncated Toeplitz operator, it
is worth to have in mind Corollary 3.6 saying that it is enough to find one ofthem. Following [6] we have

Theorem 6.3 (Theorem 8.5 [6]). Let $\alpha$, $\theta$ be nonconstant inner functions suchthat $\alpha$\leq $\theta$ and let A be a bounded operator satisfying(6.1)

A-S_{ $\alpha$}^{*}AS_{ $\theta$}= $\mu$\otimes\tilde{k}_{0}^{ $\theta$}+\tilde{k}_{0}^{ $\alpha$}\otimes $\nu$

9
ASYNIMETRIC TRUNCATED TOEPLITZ OPERATORS

for $\mu$\in K_{ $\alpha$}^{2}, \mathrm{v}\in K_{ $\theta$}^{2} . Then A=A_{ $\psi$+\overline{x} ^{ $\theta,\ \alpha$} , where$\psi$=C_{ $\alpha$}P_{ $\alpha$}(\mathrm{v}-c\tilde{k}_{0}^{ $\theta$})=C_{ $\alpha$}P_{ $\alpha$} $\nu$-\overline{c}\overline{\frac{ $\theta$}{ $\alpha$}(0)}k_{0}^{ $\alpha$}\in K_{ $\alpha$}^{2} and$\chi$=S_{\frac{$\theta$}{$\alpha$} P_{\frac{$\theta$}{$\alpha$} \displaystyle\overline{$\alpha$}($\nu$-c\tilde{k}_{0}^{$\theta$})+\frac{$\theta$}{$\alpha$}C_{$\alpha$}($\mu$+\overline{c}\tilde{k}_{0}^{$\alpha$})=(S_{\frac{$\theta$}{\circ}P_{\frac{$\theta$}{$\alpha$}\displaystyle\overline{$\alpha$}$\nu$+c\frac{$\theta$}{$\alpha$}(0)k^{\frac{$\theta$}{0^{$\alpha$} )+\frac{$\theta$}{$\alpha$}(C_{$\alpha$}$\mu$+ck_{0}^{$\alpha$})\inK_{$\theta$}^{2}=K_{\frac{2$\theta$}{$\alpha$}\oplus\frac{$\theta$}{$\alpha$}K_{$\alpha$}^{2} with

c=\langleP_{\frac{$\thea$}{ \alpha$}\overline{$\alpha$} \iota$/,\tilde{k}^{\frac{$\thea$}{0$\alpha$}\rangle\Vert\ilde{k}^{\frac{$\thea$}{0^ $\alpha$} \Vert^{-2}.

Acknowledgments.Tłie work of the first author was partially supported by

Fundaqão para a Ciência \mathrm{e} a Tecnologia (FCT/Portugal), through Project UID/MAT/04459/2013.

The research of the third and the fourth authors was financed by the Ministry of
Science and Higher Education of the Republic of Poland.
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1 CENTER FOR MATHEMATICAL ANALYSIS, GEOMETRY AND DYNAMICAL SYSTEMS, MATH‐EMATICS DEPARTMENT, INSTITUTO SUPERIOR TECNICO, UNIVERSIDADE DE LISBOA, Av.
Rovlsco PAis, 1049‐ 001 LISBOA, PORTUGAL.
E‐mail address: ccamaraQmath. ist. utl. pt

2 DEPARTMENT OF APPLIED MATHEMATICS, UNIVEHSITY OF AGRICULTURE, UL. BALICKA
253c,, 30‐198 KRAKóW, POLAND.
E‐mail address: rmklisQcyfronet. pl

3INSTITUTE OF MATHEMATICS, PEDAGOGICAL UNivERsiTY, UL. PODCHORAZYCH 2,, 30‐

084 KRAKóW, POLAND.
E‐mail address: rmptak@cyf‐kr.edu.pl