# Characterizations of Matrix valued asymmetric truncated Hankel operators

Rewayat Khan  R. Khan, Abbottabad University of Science and Technology,
Pakistan<EMAIL_ADDRESS>and  Ji Eun Lee  Ji Eun Lee, Department of
Mathematics and Statistics, Sejong University, Seoul, 05006, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.

In this paper we introduce the class of matrix valued asymmetric truncated
Hankel operators. By using characterizations of matrix valued asymmetric
truncated Toeplitz operators, we characterize matrix valued asymmetric
truncated Hankel operators in the case when two involved inner matrices are
$J$-symmetric.

###### Key words and phrases:

model spaces, conjugations, matrix valued asymmetric truncated Toeplitz
operators, matrix valued asymmetric truncated Hankel operators.

###### 2010 Mathematics Subject Classification:

Primary 47B35, Secondary 47B32, 30D20

##  1\. Introduction

As usual, $\mathbb{D}=\\{\lambda\in\mathbb{C}:|\lambda|<1\\}$ and
$\mathbb{T}=\partial\mathbb{D}=\\{{z}\in\mathbb{C}:|{z}|=1\\}$ denote the unit
disc and unit circle, respectively. Let $H^{2}$ denote the classical Hardy
space in the unit disk $\mathbb{D}$. Truncated Hankel operators (THO’s)
(respectively truncated Toeplitz operators (TTO’s)) and asymmetric truncated
Hankel operators (ATHO’s)(respectively asymmetric truncated Toeplitz operators
(ATTO’s)) are compressions of multiplication operator to the backward shift
invariant subspaces of $H^{2}$ (with two possibly different underlying
subspaces called model spaces in the asymmetric case). Each of these subspaces
is of the form $K_{\theta}=(\theta H^{2})^{\perp}=H^{2}\ominus\theta H^{2}$,
where $\theta$ a complex-valued inner function: $\theta\in H^{\infty}$ and
$|\theta({z})|=1$ a.e. on the unit circle $\mathbb{T}$. Since D. Sarasons
paper [19] TTO’s, and later on ATTO’s [5, 4, 2], MTTO’s [15] and MATTO’s [13,
14], have been intensly studied.

The theory of THO’s and ATHO’s in the scalar case can be extended to vector
version, defined on subspaces of a vector valued Hardy space
$H^{2}(\mathcal{H})$ with a finite dimensional complex Hilbert space
$\mathcal{H}$. These subspaces are corresponding to inner functions: An
analytic function $\Theta$ having values in $\mathcal{L}(\mathcal{H})$ (the
algebra of all bounded linear operators on $\mathcal{H}$), bounded and such
that the boundary values $\Theta(z)$ are unitary operators a.e. on
$\mathbb{T}$. We will consider only pure inner functions, that is, $\Theta$
such that $\|\Theta(0)\|_{\mathcal{L}(\mathcal{H})}<1$. The model space

$K_{\Theta}=[\Theta H^{2}(\mathcal{H})]^{\perp}$

corresponding to an inner function $\Theta$ is invariant under the backward
shift $S^{*}$. Moreover, by the vector valued version of Beurling’s invariant
subspace theorem, each closed (nontrivial) $S^{*}$–invariant subspace of
$H^{2}(\mathcal{H})$ is a model space ([1, Chapter 5, Theorem 1.10]). Let
$P_{\Theta}$ be the orthogonal projection from
$L^{2}(\mathcal{L}(\mathcal{H}))$ onto $K_{\Theta}$.

Sections 2 and 3 contain preliminary material on spaces of vector valued
functions (Section 2), model spaces and MATTO’s (Section 3). Section 4
contains some results related to matrix valued asymmetric truncated Hankel
operators. In section 5, we just remind the characterization of matrix valued
asymmetric truncated Toeplitz operators (see [14]). In section 6, connection
between matrix valued asymmetric truncated Toeplitz (MATTO’s) and matrix
valued asymmetric truncated Hankel operators (MATHO’s) is presented. Section 7
and section 8 is devoted to characterizes matrix valued asymmetric truncated
Hankel operators by using compressed shift and modified compressed shift
respectively.

##  2\. Spaces of vector valued functions and their operators

Let $\mathcal{H}$ be a complex separable Hilbert space. In what follows
$\|\cdot\|_{\mathcal{H}}$ and $\langle\cdot,\cdot\rangle_{\mathcal{H}}$ will
denote the norm and the inner product in $\mathcal{H}$, respectively.
Moreover, we will assume that $\dim\mathcal{H}<\infty$. The space
$L^{2}(\mathcal{H})$ can be defined as

$L^{2}(\mathcal{H})=\\{f\colon\mathbb{T}\to\mathcal{H}:f\text{ is measurable
and}\int_{\mathbb{T}}\|f({z})\|_{\mathcal{H}}^{2}\,dm({z})<\infty\\}$

($m$ being the normalized Lebesgue measure on $\mathbb{T}$). As usual, each
$f\in L^{2}(\mathcal{H})$ is interpreted as a class of functions equal to the
representing $f$ a.e. on $\mathbb{T}$ with respect to $m$. The space
$L^{2}(\mathcal{H})$ is a (separable) Hilbert space with the inner product
given by

$\langle f,g\rangle_{L^{2}(\mathcal{H})}=\int_{\mathbb{T}}\langle
f({z}),g({z})\rangle_{\mathcal{H}}\,dm({z}),\quad f,g\in L^{2}(\mathcal{H})$

and the induced norm is given by

$\|f\|_{L^{2}(\mathcal{H})}^{2}=\int_{\mathbb{T}}\|f({z})\|_{\mathcal{H}}^{2}\,dm({z})=\sum\limits_{n=-\infty}^{\infty}\|a_{n}\|_{\mathcal{H}}^{2}.$

The vector valued Hardy space $H^{2}(\mathcal{H})$ is defined as the set of
all the elements of $L^{2}(\mathcal{H})$ whose Fourier coefficients with
negative indices vanish. Each $f\in H^{2}(\mathcal{H})$, $\displaystyle
f({z})=\sum_{n=0}^{\infty}a_{n}{z}^{n}$, can also be identified with a
function

$f(\lambda)=\sum\limits_{n=0}^{\infty}a_{n}\lambda^{n},\quad\lambda\in\mathbb{D},$

analytic in the unit disk $\mathbb{D}$.

We can also consider the spaces of all essentially bounded functions on the
$\mathbb{T}$. This space is denoted by $L^{\infty}(\mathcal{H})$ and

$H^{\infty}(\mathcal{H})=L^{\infty}(\mathcal{H})\cap H^{2}(\mathcal{H}).$

Here we assume that $\dim\mathcal{H}<\infty$ so we can consider
$\mathcal{L}(\mathcal{H})$ as a Hilbert space with the Hilbert–Schmidt norm
and we may also define as above the spaces $L^{2}(\mathcal{L}(\mathcal{H}))$
and $H^{2}(\mathcal{L}(\mathcal{H}))$:

$H^{2}(\mathcal{L}(\mathcal{H}))=\left\\{{\mathbf{F}}\in
L^{2}(\mathcal{L}(\mathcal{H})):{\mathbf{F}}({z})=\sum_{n=0}^{\infty}F_{n}{z}^{n}\right\\}.$

We thus have

$L^{2}(\mathcal{L}(\mathcal{H}))=[zH^{2}(\mathcal{L}(\mathcal{H}))]^{*}\oplus
H^{2}(\mathcal{L}(\mathcal{H})).$

For each $\lambda\in\mathbb{D}$ the function
${\mathbf{k}}_{\lambda}(z)=(1-\bar{\lambda}z)^{-1}I_{\mathcal{H}}$ is bounded
and belongs to $H^{2}(\mathcal{L}(\mathcal{H}))$, and has the following
reproducing property

$\langle f,{\mathbf{k}}_{\lambda}x\rangle_{L^{2}(\mathcal{H})}=\langle
f(\lambda),x\rangle_{\mathcal{H}},\quad f\in H^{2}(\mathcal{H}).$

To each $\Phi\in L^{\infty}(\mathcal{L}(\mathcal{H}))$ there corresponds a
multiplication operator $M_{\Phi}:L^{2}(\mathcal{H})\to L^{2}(\mathcal{H})$:
for $f\in L^{2}(\mathcal{H})$,

$(M_{\Phi}f)({z})=\Phi({z})f({z})\quad\text{a.e. on }\mathbb{T}.$

Let $P_{+}$ denote the orthogonal projection from $L^{2}(\mathcal{H})$ to
$H^{2}(\mathcal{H})$. Then the Toeplitz operator $T_{\Phi}$ is the compression
of $M_{\Phi}$ to the Hardy space: $T_{\Phi}:H^{2}(\mathcal{H})\to
H^{2}(\mathcal{H})$,

$T_{\Phi}f=P_{+}M_{\Phi}f\quad\text{for }f\in H^{2}(\mathcal{H}).$

It is clear that $(M_{\Phi})^{*}=M_{\Phi^{*}}$ and
$(T_{\Phi})^{*}=T_{\Phi^{*}}$, where $\Phi^{*}({z})=\Phi({z})^{*}$ a.e. on
$\mathbb{T}$. The operators $S$ (forward shift) and $S^{*}$ (backward shift)
are examples of Toeplitz operators with symbols $zI_{\mathcal{H}}$ and
$\overline{z}I_{\mathcal{H}}$, respectively, where $I_{\mathcal{H}}$ is the
identity operator on $\mathcal{H}$.

##  3\.  MATTO’s and MATHO’s

For two operator valued inner functions $\Theta_{1},\Theta_{2}\in
H^{\infty}(\mathcal{L}(\mathcal{H}))$ and $\Phi\in
L^{2}(\mathcal{L}(\mathcal{H}))$ let

(3.1) $A_{\Phi}^{\Theta_{1},\Theta_{2}}f=P_{\Theta_{2}}(\Phi f),\quad f\in
K_{\Theta_{1}}\cap H^{\infty}(\mathcal{L}(\mathcal{H})).$

The operator $A_{\Phi}^{\Theta_{1},\Theta_{2}}$ is called a matrix valued
asymmetric truncated Toeplitz operator (MATTO). When
$\Theta_{1}=\Theta_{2}=\Theta$, then
$A_{\Phi}^{\Theta}=A_{\Phi}^{\Theta,\Theta}$ is called a matrix valued
truncated Toeplitz operator (MTTO’s) [15]. Both are densely defined. Let
$\mathcal{MT}(\Theta_{1},\Theta_{2})$ be the set of all MATTO’s of the form
(3.1) which can be extended boundedly to the whole space $K_{\Theta_{1}}$ and
for $\Theta_{1}=\Theta_{2}=\Theta$ let
$\mathcal{MT}(\Theta)=\mathcal{MT}(\Theta,\Theta)$.

Two important examples of operators from $\mathcal{MT}(\Theta)$ are the model
operators

(3.2)
$S_{\Theta}=A_{z}^{\Theta}=A_{zI_{\mathcal{H}}}^{\Theta}\quad\text{and}\quad
S_{\Theta}^{*}=A_{\bar{z}}^{\Theta}=A_{\bar{z}I_{\mathcal{H}}}^{\Theta}.$

These operators are models for a class of Hilbert space contractions. For
example, each $C_{0}$ contraction with finite defect indices is unitarily
equivalent to $S_{\Theta}$ for some operator valued inner function $\Theta$
(see [18, Chapter IV]).

Truncated Hankel operators were first defined and studied by C. Gu in [9].
Their asymmetric version were introduced in [16].

An asymmetric matrix valued truncated Hankel operator
$B_{\Phi}^{\Theta_{1},\Theta_{2}}$ with symbol $\Phi\in
L^{2}(\mathcal{L}(\mathcal{H}))$ is an operator from $K_{\Theta_{1}}$ into
$K_{\Theta_{2}}$ densely defined by

(3.3)
$B_{\Phi}^{\Theta_{1},\Theta_{2}}f=P_{\Theta_{2}}{\mathbf{J}}(I-P_{+})(\Phi
f),\quad f\in K_{\Theta_{1}}^{\infty},$

where ${\mathbf{J}}:L^{2}(E)\to L^{2}(E)$ is the flip operator given by

${\mathbf{J}}f(z)=\overline{z}f(\overline{z}),\quad|z|=1.$

Let $\mathcal{MH}(\Theta_{1},\Theta_{2})$ be the set of all MATHO’s of the
form (3.3) which can be extended boundedly to the whole space $K_{\Theta_{1}}$
and $\mathcal{MH}(\Theta)$ denote the set of all MTHO’s. Some algebraic
properties of THO’s were studied in [9]. Here we continue the investigation
started in [10, 16].

For each $\lambda\in\mathbb{D}$ we can consider

${\mathbf{k}}_{\lambda}^{\Theta}(z)=\tfrac{1}{1-\bar{\lambda}z}(I_{\mathcal{H}}-\Theta(z)\Theta(\lambda)^{*})\in
H^{\infty}(\mathcal{L}(\mathcal{H})).$

Then, for each $x\in\mathcal{H}$ it has the following reproducing property

$\langle
f,{\mathbf{k}}_{\lambda}^{\Theta}x\rangle_{L^{2}(\mathcal{H})}=\langle
f(\lambda),x\rangle_{\mathcal{H}}\quad\text{for every }f\in K_{\Theta}.$

It follows in particular that $K_{\Theta}^{\infty}=K_{\Theta}\cap
H^{\infty}(\mathcal{H})$ is a dense subset of $K_{\Theta}$.

Let

$\mathcal{D}_{\Theta}=\\{(I_{\mathcal{H}}-\Theta\Theta(0)^{*})x:\
x\in\mathcal{H}\\}=\\{{\mathbf{k}}_{0}^{\Theta}x:\ x\in\mathcal{H}\\}\subset
K_{\Theta}.$

For each $f\in K_{\Theta}$ we have $f\perp\mathcal{D}_{\Theta}$ if and only if
$f(0)=0$. It follows that

$(S_{\Theta}^{*}f)(z)=\left\\{\begin{array}[]{cl}\bar{z}f(z)&\text{for
}f\perp\mathcal{D}_{\Theta},\\\
-\bar{z}\big{(}\Theta(z)-\Theta(0)\big{)}\Theta(0)^{*}x&\text{for
}f={\mathbf{k}}_{0}^{\Theta}x\in\mathcal{D}_{\Theta}.\end{array}\right.$

The defect operator of $S_{\Theta}$ is denoted by

$D_{\Theta}:=I_{K_{\Theta}}-S_{\Theta}S_{\Theta}^{*}.$

Since for each $f\in H^{2}(\mathcal{H})$ we have
$(I_{H^{2}(\mathcal{H})}-SS^{*})f=f(0)$ (a constant function in
$H^{2}(\mathcal{H})$), it follows that for $f\in K_{\Theta}$,

(3.4)
$\begin{split}D_{\Theta}f&=(I_{K_{\Theta}}-S_{\Theta}S_{\Theta}^{*})f=P_{\Theta}(I_{H^{2}(\mathcal{H})}-SS^{*})f\\\
&=(I_{\mathcal{H}}-\Theta\Theta(0)^{*})f(0)={\mathbf{k}}_{0}^{\Theta}f(0)\in\mathcal{D}_{\Theta}.\end{split}$

More precisely,

$D_{\Theta}f=\left\\{\begin{array}[]{cl}0&\text{for
}f\perp\mathcal{D}_{\Theta},\\\
{\mathbf{k}}_{0}^{\Theta}(I_{\mathcal{H}}-\Theta(0)\Theta(0)^{*})x&\text{for
}f={\mathbf{k}}_{0}^{\Theta}x\in\mathcal{D}_{\Theta}.\end{array}\right.$

##  4\. MATHO’s and Model spaces

In [12] the author considers the generalized Crofoot transform. For a pure
inner function $\Theta\in H^{\infty}(\mathcal{L}(\mathcal{H}))$ and
$W\in\mathcal{L}(\mathcal{H})$ such that $\|W\|_{\mathcal{L}(\mathcal{H})}<1$
define the generalized Crofoot transform $J_{W}^{\Theta}:L^{2}(\mathcal{H})\to
L^{2}(\mathcal{H})$ by

(4.1) $J_{W}^{\Theta}f=D_{W^{*}}(I_{L^{2}(\mathcal{H})}-\Theta
W^{*})^{-1}f,\quad f\in L^{2}(\mathcal{H}).$

The adjoint of $J_{W}^{\Theta}$ is given by the formula

(4.2)
$(J_{W}^{\Theta})^{\ast}g=D_{W^{*}}(I_{L^{2}(\mathcal{H})}+\Theta^{W}W^{*})^{-1}g,\quad
g\in L^{2}(\mathcal{H}).$

Then $J_{W}^{\Theta}$ is unitary and maps $K_{\Theta}$ onto $K_{\Theta^{W}}$
where

(4.3)
$\Theta^{W}(z)=-W+D_{W^{*}}(I_{L^{2}(\mathcal{H})}-\Theta(z)W^{*})^{-1}\Theta(z)D_{W}.$

The following theorem describes the action of the Crofoot transform on
$\mathcal{MH}(\Theta_{1},\Theta_{2})$ (see [12] for the case
$\Theta_{1}=\Theta_{2}$ and [11] for the scalar case):

######  Theorem 4.1.

Let $\Theta_{1},\Theta_{2}\in H^{\infty}(\mathcal{L}(\mathcal{H}))$ be two
pure inner functions and let $W_{1},W_{2}\in\mathcal{L}(\mathcal{H})$ be such
that $\|W_{1}\|_{\mathcal{L}(\mathcal{H})}<1$ and
$\|W_{2}\|_{\mathcal{L}(\mathcal{H})}<1$. A bounded linear operator
$B:K_{\Theta_{1}}\to K_{\Theta_{2}}$ belongs to
$\mathcal{MH}(\Theta_{1},\Theta_{2})$ if and only if
$J_{W_{2}}^{\Theta_{2}}B(J_{W_{1}}^{\Theta_{1}})^{*}$ belongs to
$\mathcal{MH}(\Theta_{1}^{W_{1}},\Theta_{2}^{W_{2}})$. More precisely,
$B=B_{\Phi}^{\Theta_{1},\Theta_{2}}\in\mathcal{MH}(\Theta_{1},\Theta_{2})$ if
and only if

$J_{W_{2}}^{\Theta_{2}}B(J_{W_{1}}^{\Theta_{1}})^{*}=B_{\Psi}^{\Theta_{1}^{W_{1}},\Theta_{2}^{W_{2}}}\in\mathcal{MH}(\Theta_{1}^{W_{1}},\Theta_{2}^{W_{2}})$

where $\Psi$ is given by

$\Psi(z)=D_{W_{2}^{*}}(I_{\mathcal{L}(\mathcal{H})}-\Theta_{2}(z)W_{2}^{*})^{-1}\Phi(z)D_{W_{1}^{*}}(I_{\mathcal{L}(\mathcal{H})}+\Theta^{W_{1}}_{1}(z)W_{1}^{*})^{-1}.$

###### Proof.

Let $f\in K_{\Theta_{1}^{W_{1}}}$ and $x\in\mathcal{H}$. Then

$\displaystyle\langle
J_{W_{2}}^{\Theta_{2}}B_{\Phi}^{\Theta_{1},\Theta_{2}}(J_{W_{1}}^{\Theta_{1}})^{\ast}f,k_{\lambda}^{\Theta^{W_{2}}_{2}}x\rangle$
$\displaystyle=$ $\displaystyle\langle
B_{\Phi}^{\Theta_{1},\Theta_{2}}(J_{W_{1}}^{\Theta_{1}})^{\ast}f,(J_{W_{2}}^{\Theta_{2}})^{\ast}k_{\lambda}^{\Theta^{W_{2}}_{2}}x\rangle$
$\displaystyle=$ $\displaystyle\langle
P_{\Theta_{2}}{\mathbf{J}}(I-P_{+})(\Phi(J_{W_{1}}^{\Theta_{1}})^{\ast}f),(J_{W_{2}}^{\Theta_{2}})^{\ast}k_{\lambda}^{\Theta^{W_{2}}_{2}}x\rangle$
$\displaystyle=$
$\displaystyle\langle{\mathbf{J}}(\Phi(J_{W_{1}}^{\Theta_{1}})^{\ast}f),k_{\lambda}^{\Theta_{2}}(I-W_{2}\Theta_{2}^{*})^{-1}D_{W_{2}^{*}}x\rangle$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}}\langle\bar{z}(\Phi(J_{W_{1}}^{\Theta_{1}})^{\ast}f)(\bar{z}),(J_{W_{2}}^{\Theta_{2}})^{\ast}k_{\lambda}^{\Theta^{W_{2}}_{2}}x\rangle$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}}\langle\bar{z}\Phi(\bar{z})((J_{W_{1}}^{\Theta_{1}})^{\ast}f)(\bar{z}),(J_{W_{2}}^{\Theta_{2}})^{\ast}k_{\lambda}^{\Theta^{W_{2}}_{2}}x\rangle$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}}\langle\bar{z}\Phi(\bar{z})D_{W_{1}^{*}}(I+\Theta^{\prime}_{1}(\bar{z})W_{1}^{*})^{-1}f(\bar{z}),(J_{W_{2}}^{\Theta_{2}})^{\ast}k_{\lambda}^{\Theta^{W_{2}}_{2}}x\rangle$
$\displaystyle=$ $\displaystyle\int_{\mathbb{T}}\langle
J_{W_{2}}^{\Theta_{2}}(\bar{z}\Phi(\bar{z})D_{W_{1}^{*}}(I+\Theta^{\prime}_{1}(\bar{z})W_{1}^{*})^{-1}f(\bar{z})),k_{\lambda}^{\Theta^{W_{2}}_{2}}x\rangle$
$\displaystyle=$ $\displaystyle\int_{\mathbb{T}}\langle
D_{W_{2}^{*}}(I-\Theta_{2}(\bar{z})W_{2}^{*})^{-1}\bar{z}\Phi(\bar{z})D_{W_{1}^{*}}(I+\Theta^{\prime}_{1}(\bar{z})W_{1}^{*})^{-1}f(\bar{z}),k_{\lambda}^{\Theta^{W_{2}}_{2}}x\rangle$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}}\langle\bar{z}D_{W_{2}^{*}}(I-\Theta_{2}(\bar{z})W_{2}^{*})^{-1}\Phi(\bar{z})D_{W_{1}^{*}}(I+\Theta^{\prime}_{1}(\bar{z})W_{1}^{*})^{-1}f(\bar{z}),k_{\lambda}^{\Theta^{W_{2}}_{2}}x\rangle$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}}\langle{\mathbf{J}}(D_{W_{2}^{*}}(I-\Theta_{2}(z)W_{2}^{*})^{-1}\Phi(z)D_{W_{1}^{*}}(I+\Theta^{\prime}_{1}(z)W_{1}^{*})^{-1}f(z)),k_{\lambda}^{\Theta^{W_{2}}_{2}}x\rangle$
$\displaystyle=$ $\displaystyle\int_{\mathbb{T}}\langle
P_{\Theta^{W_{2}}_{2}}{\mathbf{J}}(\Psi(z)f(z)),k_{\lambda}^{\Theta^{W_{2}}_{2}}x\rangle$
$\displaystyle=$ $\displaystyle\langle P_{\Theta^{W_{2}}_{2}}{\mathbf{J}}(\Psi
f),k_{\lambda}^{\Theta^{W_{2}}_{2}}x\rangle$ $\displaystyle=$
$\displaystyle\langle P_{\Theta^{W_{2}}_{2}}{\mathbf{J}}(I-P_{+})(\Psi
f),k_{\lambda}^{\Theta^{W_{2}}_{2}}x\rangle$ $\displaystyle=$
$\displaystyle\langle
B^{\Theta^{W_{1}}_{1},\Theta^{W_{2}}_{2}}_{\Psi}f,k_{\lambda}^{\Theta^{W_{2}}_{2}}x\rangle,$

where $\Psi$ is given by

$\Psi(z)=D_{W_{2}^{*}}(I_{\mathcal{L}(\mathcal{H})}-\Theta_{2}(z)W_{2}^{*})^{-1}\Phi(z)D_{W_{1}^{*}}(I_{\mathcal{L}(\mathcal{H})}+\Theta^{W_{1}}_{1}(z)W_{1}^{*})^{-1}.$

Hence we complete the proof. ∎

Recall that if $\Theta\in H^{\infty}(\mathcal{L}(\mathcal{H}))$ is an inner
function, then so is

$\widetilde{\Theta}(z)=\Theta(\bar{z})^{*}.$

Let us consider the map $\tau_{\Theta}:L^{2}(\mathcal{H})\to
L^{2}(\mathcal{H})$ defined for $f\in L^{2}(\mathcal{H})$ by

(4.4)
$(\tau_{\Theta}f)(z)=\bar{z}\Theta(\bar{z})^{*}f(\bar{z})=\bar{z}\widetilde{\Theta}(z)f(\bar{z})\quad\text{a.e.
on }\mathbb{T}.$

The map $\tau_{\Theta}$ is an isometry and its adjoint
$\tau_{\Theta}^{*}=\tau_{\widetilde{\Theta}}$ is also its inverse. Hence
$\tau_{\Theta}$ is unitary. Moreover, it is easy to verify that

$\tau_{\Theta}(\Theta H^{2}(\mathcal{H}))\subset
H^{2}(\mathcal{H})^{\perp}\quad\text{and}\quad\tau_{\Theta}(H^{2}(\mathcal{H})^{\perp})\subset\widetilde{\Theta}H^{2}(\mathcal{H}),$

which implies that

$\tau_{\Theta}(K_{\Theta})=K_{\widetilde{\Theta}}.$

######  Theorem 4.2.

Let $\Theta_{1},\Theta_{2}\in H^{\infty}(\mathcal{L}(\mathcal{H}))$ be two
pure inner functions. A bounded linear operator $B:K_{\Theta_{1}}\to
K_{\Theta_{2}}$ belongs to $\mathcal{MH}(\Theta_{1},\Theta_{2})$ if and only
if $\tau_{\Theta_{2}}B\,\tau_{\Theta_{1}}^{*}$ belongs to
$\mathcal{MH}(\widetilde{\Theta}_{1},\widetilde{\Theta}_{2})$. More precisely,
$B=B_{\Phi}^{\Theta_{1},\Theta_{2}}\in\mathcal{MH}(\Theta_{1},\Theta_{2})$ if
and only if
$\tau_{\Theta_{2}}B\,\tau_{\Theta_{1}}^{*}=B_{\Psi}^{\widetilde{\Theta}_{1},\widetilde{\Theta}_{2}}\in\mathcal{MH}(\widetilde{\Theta}_{1},\widetilde{\Theta}_{2})$
with

(4.5)
$\begin{split}\Psi(z)&=\Theta_{2}(z)^{*}\Phi(\bar{z})\widetilde{\Theta}_{1}(z)^{*}\\\
&=\widetilde{\Theta}_{2}(\bar{z})\Phi(\bar{z})\Theta_{1}(\bar{z})\quad\text{a.e.
on }\mathbb{T}.\end{split}$

###### Proof.

Let $B:K_{\Theta_{1}}\to K_{\Theta_{2}}$ be a bounded linear operator. Assume
that
$B=B_{\Phi}^{\Theta_{1},\Theta_{2}}\in\mathcal{MT}(\Theta_{1},\Theta_{2})$
with some $\Phi\in L^{2}(\mathcal{L}(\mathcal{H}))$, and take $f\in
K_{\widetilde{\Theta}_{1}}^{\infty}$ and $g\in
K_{\widetilde{\Theta}_{2}}^{\infty}$. Note that
$\tau_{\widetilde{\Theta}_{1}}f\in K_{{\Theta}_{1}}^{\infty}$ and
$\tau_{\widetilde{\Theta}_{2}}g\in K_{{\Theta}_{2}}^{\infty}$. Then

$\displaystyle\langle\tau_{\Theta_{2}}B\,\tau_{\Theta_{1}}^{*}f,g\rangle_{L^{2}(\mathcal{H})}$
$\displaystyle=\langle
B_{\Phi}^{\Theta_{1},\Theta_{2}}\tau_{\widetilde{\Theta}_{1}}f,\tau_{{\Theta}_{2}}^{*}g\rangle_{L^{2}(\mathcal{H})}$
$\displaystyle=\langle
P_{\Theta_{2}}{\mathbf{J}}(I-P_{+})({\Phi}\,\tau_{\widetilde{\Theta}_{1}}f),\tau_{{\Theta}_{2}}^{*}g\rangle_{L^{2}(\mathcal{H})}$
$\displaystyle=\langle
P_{\Theta_{2}}{\mathbf{J}}({\Phi}\,\tau_{\widetilde{\Theta}_{1}}f),\tau_{{\Theta}_{2}}^{*}g\rangle_{L^{2}(\mathcal{H})}$
$\displaystyle=\langle\tau_{{\Theta}_{2}}{\mathbf{J}}({\Phi}\,\tau_{\widetilde{\Theta}_{1}}f),g\rangle_{L^{2}(\mathcal{H})}$
$\displaystyle=\int_{\mathbb{T}}\langle\tau_{{\Theta}_{2}}(\bar{z}\Phi(\bar{z})(\tau_{\widetilde{\Theta}_{1}}f)(\bar{z})),g(z)\rangle_{\mathcal{H}}\,dm(z)$
$\displaystyle=\int_{\mathbb{T}}\langle\bar{z}\Theta_{2}(\bar{z})^{*}(z\Phi(z)(\tau_{\widetilde{\Theta}_{1}}f)(z)),g(z)\rangle_{\mathcal{H}}\,dm(z)$
$\displaystyle=\int_{\mathbb{T}}\langle\Theta_{2}(\bar{z})^{*}\Phi(z)\widetilde{\Theta}_{1}(\bar{z})^{*}\bar{z}f(\bar{z}),g(z)\rangle_{\mathcal{H}}\,dm(z)$
$\displaystyle=\int_{\mathbb{T}}\langle\bar{z}\Theta_{2}(\bar{z})^{*}\Phi(z)\widetilde{\Theta}_{1}(\bar{z})^{*}f(\bar{z}),g(z)\rangle_{\mathcal{H}}\,dm(z)$
$\displaystyle=\int_{\mathbb{T}}\langle{\mathbf{J}}(\Theta_{2}(z)^{*}\Phi(\bar{z})\widetilde{\Theta}_{1}(z)^{*}f(z)),g(z)\rangle_{\mathcal{H}}\,dm(z)$
$\displaystyle=\int_{\mathbb{T}}\langle
P_{\Theta_{2}}{\mathbf{J}}(I-P_{+})(\Theta_{2}(z)^{*}\Phi(\bar{z})\widetilde{\Theta}_{1}(z)^{*}f(z)),g(z)\rangle_{\mathcal{H}}\,dm(z)$
$\displaystyle=\int_{\mathbb{T}}\langle
P_{\widetilde{\Theta}_{2}}{\mathbf{J}}(I-P_{+})(\Psi(z)f(z)),g(z)\rangle_{\mathcal{H}}\,dm(z)$
$\displaystyle=\langle
B_{\Psi}^{\widetilde{\Theta}_{1},\widetilde{\Theta}_{2}}f,g\rangle,$

where $\Psi\in L^{2}(\mathcal{L}(\mathcal{H}))$ is given by (4.5).

Now, if
$\tau_{\Theta_{2}}B\,\tau_{\Theta_{1}}^{*}=B_{\Psi}^{\widetilde{\Theta}_{1},\widetilde{\Theta}_{2}}\in\mathcal{MH}(\widetilde{\Theta}_{1},\widetilde{\Theta}_{2})$
for some $\Psi\in L^{2}(\mathcal{L}(\mathcal{H}))$, then
$B=\tau_{\widetilde{\Theta}_{2}}B_{\Psi}^{\widetilde{\Theta}_{1},\widetilde{\Theta}_{2}}\tau_{\widetilde{\Theta}_{1}}^{*}$
by the first part of the proof of
$B=B_{\Phi}^{\Theta_{1},\Theta_{2}}\in\mathcal{MH}(\Theta_{1},\Theta_{2})$
with

(4.6)
$\begin{split}\Phi(z)=\widetilde{\Theta}_{2}(z)^{*}\Psi(\bar{z})\Theta_{1}(z)^{*}\quad\text{a.e.
on }\mathbb{T}.\end{split}$

∎

Denote

$\widetilde{D}_{\Theta}=I-S_{\Theta}^{*}S_{\Theta}.$

Applying Theorem 4.2 to the model operator $S_{\Theta}$
($\Theta_{1}=\Theta_{2}=\Theta$) we obtain that

(4.7)
$\tau_{\Theta}S_{\Theta}\tau_{\Theta}^{*}=\tau_{\Theta}S_{\Theta}\tau_{\widetilde{\Theta}}=S_{\widetilde{\Theta}}^{*}$

(see [15, p. 1001]). It follows that

(4.8)
$\widetilde{D}_{\Theta}=\tau_{\widetilde{\Theta}}{D}_{\widetilde{\Theta}}\tau_{\Theta}=\tau_{\widetilde{\Theta}}{D}_{\widetilde{\Theta}}\tau_{\widetilde{\Theta}}^{*}$

and by (3.4),

$\widetilde{D}_{\Theta}f=\tau_{\widetilde{\Theta}}\big{(}{\mathbf{k}}_{0}^{\widetilde{\Theta}}(\tau_{\Theta}f)(0)\big{)}\quad\text{for
all }f\in K_{\Theta}.$

For $\lambda\in\mathbb{D}$, let
$\widetilde{{\mathbf{k}}}_{\lambda}^{\Theta}(z)x=\tau_{\widetilde{\Theta}}({\mathbf{k}}_{\bar{\lambda}}^{\widetilde{\Theta}}(z)x)$
for $x\in\mathcal{H}$. Then (a.e. on $\mathbb{T}$)

$\displaystyle\widetilde{{\mathbf{k}}}_{\lambda}^{\Theta}(z)x$
$\displaystyle=\tau_{\widetilde{\Theta}}({\mathbf{k}}_{\bar{\lambda}}^{\widetilde{\Theta}}(z)x)=\bar{z}\Theta(z){\mathbf{k}}_{\bar{\lambda}}^{\widetilde{\Theta}}(\bar{z})x$
$\displaystyle=\tfrac{\bar{z}}{1-\lambda\bar{z}}\Theta(z)(I_{\mathcal{H}}-\widetilde{\Theta}(\bar{z})\widetilde{\Theta}(\bar{\lambda})^{*})x$
$\displaystyle=\tfrac{1}{z-\lambda}\Theta(z)(I_{\mathcal{H}}-{\Theta}(z)^{*}{\Theta}(\lambda))x$
$\displaystyle=\tfrac{1}{z-\lambda}(\Theta(z)-{\Theta}(\lambda))x\in
K_{\Theta}\quad\text{for each }x\in\mathcal{H}.$

In particular,

$\widetilde{{\mathbf{k}}}_{0}^{\Theta}(z)x=\bar{z}(\Theta(z)-{\Theta}(0))x~{}\quad\text{for
each }x\in\mathcal{H}.$

and

$\widetilde{D}_{\Theta}f=\widetilde{{\mathbf{k}}}_{0}^{\Theta}(\tau_{\Theta}f)(0)\in\widetilde{\mathcal{D}}_{\Theta},$

where

$\widetilde{\mathcal{D}}_{\Theta}=\tau_{\widetilde{\Theta}}\mathcal{D}_{\widetilde{\Theta}}=\\{\widetilde{{\mathbf{k}}}_{0}^{{\Theta}}x:x\in\mathcal{H}\\}=\\{\bar{z}(\Theta(z)-\Theta(0))x:x\in\mathcal{H}\\}.$

Observe that for $f\in K_{\Theta}$, $x\in\mathcal{H}$,

$\displaystyle\langle
f,\widetilde{{\mathbf{k}}}_{\lambda}^{\Theta}x\rangle_{L^{2}(\mathcal{H})}$
$\displaystyle=\langle
f,\tau_{\widetilde{\Theta}}({\mathbf{k}}_{\bar{\lambda}}^{\widetilde{\Theta}}x)\rangle_{L^{2}(\mathcal{H})}$
$\displaystyle=\langle\tau_{\Theta}f,{\mathbf{k}}_{\bar{\lambda}}^{\widetilde{\Theta}}x\rangle_{L^{2}(\mathcal{H})}=\langle(\tau_{\Theta}f)(\bar{\lambda}),x\rangle_{\mathcal{H}}.$

Then it follows that for $f\in K_{\Theta}$ we have $M_{z}f\in K_{\Theta}$ if
and only if $f\perp\widetilde{\mathcal{D}}_{\Theta}$. Indeed, $M_{z}f\in
K_{\Theta}$ if and only if $\Theta P_{+}(\Theta^{*}M_{z}f)=0$. Since

$(\Theta^{*}M_{z}f)(z)=\Theta(z)^{*}zf(z)=(\tau_{\Theta}f)(\overline{z}),$

we have $P_{+}(\Theta^{*}M_{z}f)=(\tau_{\Theta}f)(0)$ and so $M_{z}f\in
K_{\Theta}$ if and only if

$0=\langle(\tau_{\Theta}f)(0),x\rangle=\langle
f,\widetilde{{\mathbf{k}}}_{0}^{\Theta}x\rangle_{L^{2}(\mathcal{H})}\quad\text{for
every}\quad x\in\mathcal{H},$

i.e, $f\perp\widetilde{\mathcal{D}}_{\Theta}$. Therefore we get

$(S_{\Theta}f)(z)=\left\\{\begin{array}[]{cl}zf(z)&\text{for
}f\perp\widetilde{\mathcal{D}}_{\Theta},\\\
-\big{(}I_{\mathcal{H}}-\Theta(z)\Theta(0)^{*}\big{)}\Theta(0)x&\text{for
}f=\widetilde{{\mathbf{k}}}_{0}^{\Theta}x\in\widetilde{\mathcal{D}}_{\Theta}.\end{array}\right.$

Hence

$\widetilde{{D}}_{\Theta}f=\left\\{\begin{array}[]{cl}0&\text{for
}f\perp\widetilde{\mathcal{D}}_{\Theta},\\\
\widetilde{{\mathbf{k}}}_{0}^{\Theta}(I_{\mathcal{H}}-\Theta(0)\Theta(0)^{*})x&\text{for
}f=\widetilde{{\mathbf{k}}}_{0}^{\Theta}x\in\widetilde{\mathcal{D}}_{\Theta}.\end{array}\right.$

Let us now consider conjugations. A conjugation $J$ on a Hilbert space
$\mathcal{H}$ is an antilinear map $J:\mathcal{H}\longrightarrow\mathcal{H}$
such that $J^{2}=I_{\mathcal{H}}$ and

$\langle Jf,Jg\rangle=\langle g,f\rangle\quad\text{for all}\quad
f,g\in\mathcal{H}$

Recall that a bounded linear operator
$T:\mathcal{H}\longrightarrow\mathcal{H}$ is said to be $J$-symmetric ($J$
being a conjugation on $\mathcal{H}$) if $JTJ=T^{*}$. We say that $T$ is
complex symmetric if it is $J$-symmetric with respect to some conjugation $J$
(see [8]).

In [3] the authors consider certain classes of conjugations in
$L^{2}(\mathcal{H})$. One such conjugation is
${\mathbf{J}}^{*}:L^{2}(\mathcal{H})\to L^{2}(\mathcal{H})$ defined for a
fixed conjugation $J$ on $\mathcal{H}$ by

(4.9) $({\mathbf{J}}^{*}f)(z)=J(f(\overline{z}))\quad\text{a.e.
on}~{}~{}\mathbb{T}.$

It is not difficult to verify that for
$f(z)=\sum_{n=-\infty}^{\infty}a_{n}z^{n}\in L^{2}(\mathcal{L}(\mathcal{H}))$,
we have

$({\mathbf{J}}^{*}f)(z)=\sum_{n=-\infty}^{\infty}J(a_{n})z^{n}.$

Hence, ${\mathbf{J}}^{*}$ is an $M_{z}$-commuting conjugation, i.e,
${\mathbf{J}}^{*}M_{z}=M_{z}{\mathbf{J}}^{*}$,
${\mathbf{J}}^{*}(H^{2}(\mathcal{H}))=H^{2}(\mathcal{H})$, and
${\mathbf{J}}^{*}P_{+}=P_{+}{\mathbf{J}}^{*}$ (see [3, Section 4]).

For $\Phi\in L^{\infty}(\mathcal{L}(\mathcal{H}))$ and an arbitrary
conjugation $J$ on $\mathcal{H}$ let

(4.10) $\Phi_{J}(z)=J\Phi(z)J\quad\text{a.e on}~{}~{}~{}\mathbb{T}.$

Then $\Phi_{J}\in L^{\infty}(\mathcal{L}(\mathcal{H}))$. As observed in [3],
$\Phi_{J}\in H^{\infty}(\mathcal{L}(\mathcal{H}))$ if and only if $\Phi\in
H^{\infty}(\mathcal{L}(\mathcal{H}))$, and $\Phi_{J}$ is an inner function if
and only if $\Phi$ is. Clearly, $(\Phi_{J})_{J}=\Phi.$ Let us also observe
that if $\Phi$ is $J$-symmetric, that is, $J\Phi(z)J=\Phi(z)^{*}$ a.e on
$\mathbb{T}$ (or equivalently $\Phi(\lambda)$ is $J$-symmetric for $\lambda$
in $\mathbb{D}$, see [3]), then $\Phi_{J}=\widetilde{\Phi}$.

For two conjugations $J_{1}$ and $J_{2}$ on $\mathcal{H}$, let
${\mathbf{J}}_{1}^{*}$ and ${\mathbf{J}}_{2}^{*}$ denote the corresponding
conjugations on $L^{2}(\mathcal{H})$ given by $\eqref{J}$. For each $f\in
L^{2}(\mathcal{H})$ we have

(4.11)
$({\mathbf{J}}_{2}^{*}M_{\Phi}f)(z)=J_{2}(\Phi(\overline{z})f(\overline{z}))=J_{2}\Phi(\overline{z})J_{1}J_{1}f(\overline{z})=(M_{\mathbf{G}}{\mathbf{J}}^{*}_{1}f)(z)$

where ${\mathbf{G}}(z)=J_{2}\Phi(\overline{z})J_{1}$ a.e. on $\mathbb{T}$. In
particular

(4.12) ${\mathbf{J}}^{*}M_{\Phi}=M_{\Phi}{\mathbf{J}}^{*}$

(see [3, Lemma 8.3]).

Note that $\Phi_{J}$ is also defined for $\Phi\in
L^{2}(\mathcal{L}(\mathcal{H}))$. In that case (4.11) and (4.12) hold for
$L^{\infty}(\mathcal{H})$.

For $f\in L^{2}(\mathcal{H})$ we have the following equality:

(4.13)
$({\mathbf{J}}^{*}{\mathbf{J}}f)(z)={\mathbf{J}}^{*}(\bar{z}f(\bar{z}))=J(zf(z))=\bar{z}J(f(z))$

and

(4.14)
$({\mathbf{J}}{\mathbf{J}}^{*}f)(z)={\mathbf{J}}(Jf(\bar{z}))=\bar{z}J(f(z))=({\mathbf{J}}^{*}{\mathbf{J}}f)(z).$

######  Proposition 4.3.

[3] Let $\Theta\in H^{\infty}(\mathcal{L}(\mathcal{H}))$ be a pure inner
function and let $J$ be a conjugation on $\mathcal{H}$. Then the following
identities hold.

  1. (a)

${\mathbf{J}}^{*}(\Theta H(\mathcal{H}))=\Theta_{J}H^{2}(\mathcal{H})$;

  2. (b)

${\mathbf{J}}^{*}P_{\Theta}=P_{\Theta_{J}}{\mathbf{J}}^{*}$;

  3. (c)

${\mathbf{J}}^{*}(K_{\Theta})=K_{\Theta_{J}}$;

  4. (d)

${\mathbf{J}}^{*}({\mathbf{k}}_{\lambda}^{\Theta}x)={\mathbf{k}}_{\overline{\lambda}}^{\Theta_{{\mathbf{J}}}}Jx$.

If $\Theta$ is $J$-symmetric, we obtain [3, Proposition 7.7]:

######  Proposition 4.4.

[3] Let $J$ be a conjugation on $\mathcal{H}$ and let $\Theta$ be a
$J$-symmetric inner function. Then the following identities hold.

  1. (a)

${\mathbf{J}}^{*}M_{\Theta}=M_{\widetilde{\Theta}}{\mathbf{J}}^{*}$;

  2. (b)

${\mathbf{J}}^{*}(\Theta
H^{2}(\mathcal{H}))=\widetilde{\Theta}H^{2}(\mathcal{H})$;

  3. (c)

${\mathbf{J}}^{*}(K_{\Theta})=K_{\widetilde{\Theta}}$;

  4. (d)

${\mathbf{J}}^{*}({\mathbf{k}}_{0}^{\Theta}x)={\mathbf{k}}_{0}^{\widetilde{\Theta}}Jx$.

######  Theorem 4.5.

Let $\Theta_{1},\Theta_{2}\in H^{\infty}(\mathcal{L}(\mathcal{H}))$ be two
pure inner functions and let $J_{1},J_{2}$ be two conjugations on
$\mathcal{H}$. A bounded linear operator $B:K_{\Theta_{1}}\to K_{\Theta_{2}}$
belongs to $\mathcal{MH}(\Theta_{1},\Theta_{2})$ if and only if
${\mathbf{J}}_{2}^{*}B{\mathbf{J}}_{1}^{*}$ belongs to
$\mathcal{MH}((\Theta_{1})_{J_{1}},(\Theta_{2})_{J_{2}})$. More precisely,
$B=B_{\Phi}^{\Theta_{1},\Theta_{2}}\in\mathcal{MH}(\Theta_{1},\Theta_{2})$ if
and only if
${\mathbf{J}}_{2}^{*}B_{\Phi}{\mathbf{J}}_{1}^{*}=B_{\Psi}^{(\Theta_{1})_{J_{1}},(\Theta_{2})_{J_{2}}}\in\mathcal{MH}((\Theta_{1})_{J_{1}},(\Theta_{2})_{J_{2}})$
with

(4.15) $\Psi(z)=J_{2}\Phi(\overline{z})J_{1}\quad\text{a.e. on
}~{}~{}\mathbb{T}.$

###### Proof.

Assume that
$B=B_{\Phi}^{\Theta_{1},\Theta_{2}}\in\mathcal{MH}(\Theta_{1},\Theta_{2})$
with $\Phi\in L^{2}(\mathcal{L}(\mathcal{H}))$. Let $f\in
K_{(\Theta_{1})_{J_{1}}}^{\infty}$. Note that ${\mathbf{J}}_{1}^{*}f\in
K_{\Theta_{1}}^{\infty}$. Therefore, by Proposition 4.3(b), (4.12) and (4.14),

$\displaystyle{\mathbf{J}}_{2}^{*}B{\mathbf{J}}_{1}^{*}f$
$\displaystyle={\mathbf{J}}_{2}^{*}P_{\Theta_{2}}{\mathbf{J}}(I-P)(\Phi{\mathbf{J}}_{1}^{*}f)$
$\displaystyle=P_{(\Theta_{2})_{J_{2}}}{\mathbf{J}}_{2}^{*}{\mathbf{J}}(\Phi{\mathbf{J}}_{1}^{*}f)$
$\displaystyle=P_{(\Theta_{2})_{J_{2}}}{\mathbf{J}}{\mathbf{J}}_{2}^{*}(\Phi{\mathbf{J}}_{1}^{*}f)$
$\displaystyle=P_{(\Theta_{2})_{J_{2}}}{\mathbf{J}}{\mathbf{J}}_{2}^{*}(\Phi
J_{1}f(\bar{z}))$
$\displaystyle=P_{(\Theta_{2})_{J_{2}}}{\mathbf{J}}(J_{2}\Phi(\bar{z})J_{1}f(z))$
$\displaystyle=P_{(\Theta_{2})_{J_{2}}}{\mathbf{J}}(I-P_{+})(J_{2}\Phi(\bar{z})J_{1}f(z))$
$\displaystyle=B_{\Psi}^{(\Theta_{1})_{J_{1}},(\Theta_{2})_{J_{2}}}f$

where $\Psi$ is given by $\eqref{equi}$. Thus
${\mathbf{J}}_{2}^{*}B{\mathbf{J}}_{1}^{*}\in\mathcal{MH}((\Theta_{1})_{J_{1}},(\Theta_{2})_{J_{2}})$.

On the other hand, if
$B={\mathbf{J}}_{2}^{*}B_{\Psi}^{(\Theta_{1})_{J_{1}},(\Theta_{2})_{J_{2}}}{\mathbf{J}}_{1}^{*}\in\mathcal{MH}((\Theta_{1})_{J_{1}},(\Theta_{2})_{J_{2}})$
with some $\Psi\in L^{2}(\mathcal{L}(\mathcal{H}))$, then
$B={\mathbf{J}}_{2}^{*}B_{\Psi}^{(\Theta_{1})_{J_{1}},(\Theta_{2})_{J_{2}}}{\mathbf{J}}_{1}^{*}$
and as above, $B=B_{\Phi}^{\Theta_{1},\Theta_{2}}$ with

$\Phi(z)=J_{2}\Psi(\bar{z})J_{1}\quad\text{a.e. on}~{}~{}\mathbb{T}.$

∎

For the scalar case, Theorem 4.5 can be found in [10] (see also [7] for the
symmetric case).

In the scalar case each model space $K_{\theta}$ is equipped with a natural
conjugation $C_{\theta}$ defined in terms of boundary functions by
$(C_{\theta}f)(z)=\theta(z)\overline{z}\overline{f(z)}$. If $\Theta\in
H^{\infty}(\mathcal{L}(\mathcal{H}))$ is an inner function and $J$ is a
conjugation on $\mathcal{H}$ we can similarly define
${\mathbf{C}}_{\Theta}:L^{2}(\mathcal{H})\to L^{2}(\mathcal{H})$ by

$({\mathbf{C}}_{\Theta}f)(z)=\Theta(z)\overline{z}J(f(z))\quad\text{a.e. on
}~{}~{}\mathbb{T}.$

Although ${\mathbf{C}}_{\Theta}$ is obviously an antilinear isometry, it is
not in general an involution. A simple computation shows that
${\mathbf{C}}_{\Theta}$ is an involution (and so a conjugation) if and only if
$\Theta(z)J\Theta(z)J=I_{\mathcal{H}}$ a.e. on $\mathbb{T}$, i.e., if and only
if $\Theta$ is $J$-symmetric.

If $\Theta$ is $J$-symmetric, then ${\mathbf{C}}_{\Theta}(\Theta
H^{2}(\mathcal{H}))=H^{2}(\mathcal{H})^{\perp}$ and so

${\mathbf{C}}_{\Theta}(K_{\Theta})=K_{\Theta}.$

Note that in that case

${\mathbf{C}}_{\Theta}={\mathbf{J}}^{*}\tau_{\Theta}.$

By Theorem 4.2 and Theorem 4.5 we get the following.

######  Theorem 4.6.

Let $\Theta_{1},\Theta_{2}\in H^{\infty}(\mathcal{L}(\mathcal{H}))$ be two
pure inner functions and let $J_{1},J_{2}$ be two conjugations on
$\mathcal{H}$ such that $\Theta_{1}$ is $J_{1}$-symmetric and $\Theta_{2}$ is
$J_{2}$-symmetric. A bounded linear operator $B:K_{\Theta_{1}}\to
K_{\Theta_{2}}$ belongs to $\mathcal{MH}(\Theta_{1},\Theta_{2})$ if and only
if $C_{\Theta_{2}}BC_{\Theta_{1}}$ belongs to
$\mathcal{MH}(\Theta_{1},\Theta_{2})$. More precisely,
$B=B_{\Phi}^{\Theta_{1},\Theta_{2}}\in\mathcal{MH}(\Theta_{1},\Theta_{2})$ if
and only if

$C_{\Theta_{2}}B_{\Phi}^{\Theta_{1},\Theta_{2}}C_{\Theta_{1}}=B_{\Psi}^{\Theta_{1},\Theta_{2}}\in\mathcal{MH}(\Theta_{1},\Theta_{2})$

with

(4.16)
$\Psi(z)=J_{2}\Theta_{2}(z)\Phi(z)\Theta_{1}(\bar{z})^{*}J_{1}\quad\text{a.e.
on }\mathbb{T}$

###### Proof.

Let $\Phi\in L^{2}({\mathcal{L}(\mathcal{H})})$ and
$B_{\Phi}^{\Theta_{1},\Theta_{2}}\in\mathcal{MH}(\Theta_{1},\Theta_{2})$.
Consider

$\displaystyle C_{\Theta_{2}}B_{\Phi}^{\Theta_{1},\Theta_{2}}C_{\Theta_{1}}$
$\displaystyle=C_{\Theta_{2}}B_{\Phi}^{\Theta_{1},\Theta_{2}}C_{\Theta_{1}}^{*}={\mathbf{J}}_{2}^{*}\tau_{\Theta_{2}}B_{\Phi}^{\Theta_{1},\Theta_{2}}\tau^{*}_{\Theta_{1}}{\mathbf{J}}_{1}^{*}={\mathbf{J}}_{2}^{*}B_{\chi}^{\widetilde{\Theta}_{1},\widetilde{\Theta}_{2}}{\mathbf{J}}^{*}_{1}$

where the symbol $\chi(z)$ is given by (as Theorem 4.2)

$\chi(z)=\widetilde{\Theta}_{2}({z})^{*}\Phi(\bar{z})\widetilde{\widetilde{\Theta}}_{1}(z)^{*}=\Theta_{2}(\bar{z})\Phi(\bar{z})\Theta_{1}(z)^{*}$

and

$\displaystyle
C_{\Theta_{2}}B_{\Phi}^{\Theta_{1},\Theta_{2}}C_{\Theta_{1}}={\mathbf{J}}_{2}^{*}B_{\chi}^{\widetilde{\Theta}_{1},\widetilde{\Theta}_{2}}{\mathbf{J}}^{*}_{1}=B_{\Psi}^{(\widetilde{\Theta}_{1})_{J_{1}},(\widetilde{\Theta}_{2})_{J_{2}}}=B_{\Psi}^{\Theta_{1},\Theta_{2}}$

with the symbol $\Psi$ is given by

$\Psi(z)=J_{2}\chi(\bar{z})J_{1}=J_{2}\Theta_{2}(z)\Phi(z)\Theta_{1}(\bar{z})^{*}J_{1}$

from Theorem 4.5. ∎

For the scalar version of Theorem 4.6, please see [10].

######  Remark 4.7.

Recall that in the scalar case $\mathcal{H}=\mathbb{C}$ every TTO on the model
space $K_{\theta}$ is $C_{\theta}$-symmetric, i.e.,

$C_{\theta}A_{\varphi}^{\theta}C_{\theta}=(A_{\varphi}^{\theta})^{*}=A_{\overline{\varphi}}^{\theta}$

(see, e.g., [19]). In that case however the only conjugation in $\mathcal{H}$
we need to consider is $J(z)=\overline{z}$ (and each $\varphi\in L^{2}$ is
$J$-symmetric). In the vector valued case, the equality

(4.17) $C_{\Theta}A_{\Phi}^{\theta}C_{\Theta}=A_{\Phi^{*}}^{\Theta}.$

is not necessarily true for an arbitrary $\Phi\in
L^{2}(\mathcal{L}(\mathcal{H}))$ (even though we assume here that $\Theta$ is
$J$-symmetric). It is however satisfied if also $\Phi$ is $J$-symmetric and
commute with $\Theta$ (see [15]).

##  5\. Characterizations of Matrix valued asymmetric truncated Toeplitz
operators

In [15, Theorem 5.2 and Remark 5.4] characterizations of matrix valued
truncated Toeplitz operators in $\mathcal{MT}(\Theta)$ were given by using the
model operators $S_{\Theta}$, $S_{\Theta}^{*}$ and the defect operators
$D_{\Theta}$, $\widetilde{D}_{\Theta}$. These characterizations generalized D.
Sarason’s results for the scalar case (see [19]). We obtain analogous results
for matrix valued asymmetric truncated Toeplitz operators in [14].

###  5.1. Characterizations by Compressed Shift

######  Lemma 5.1.

[14] If $\Phi\in H^{2}(\mathcal{L}(\mathcal{H}))$, then

$A_{\Phi}^{{\Theta_{1}},{\Theta_{2}}}-S_{\Theta_{2}}A_{\Phi}^{{\Theta_{1}},{\Theta_{2}}}S_{\Theta_{1}}^{*}=P_{\Theta_{2}}M_{\Phi}(I_{H^{2}(\mathcal{H})}-SS^{*})\quad\text{on
}K_{\Theta_{1}}^{\infty}.$

Recall that

$D_{\Theta}=I_{K_{\Theta}}-S_{\Theta}S_{\Theta}^{*},\quad\widetilde{D}_{\Theta}=I_{K_{\Theta}}-S_{\Theta}^{*}S_{\Theta}$

and

$\mathcal{D}_{\Theta}=\\{(I_{\mathcal{H}}-\Theta(z)\Theta(0)^{*})x:x\in\mathcal{H}\\},\quad\widetilde{\mathcal{D}}_{\Theta}=\\{\bar{z}(\Theta(z)-\Theta(0))x:x\in\mathcal{H}\\},$

while the operator
$\Omega_{\Theta}:\mathcal{D}_{\Theta}\rightarrow\mathcal{H}\subset
H^{2}(\mathcal{H})$ is defined by

$\Omega_{\Theta}({\mathbf{k}}_{0}^{\Theta}x)=x.$

######  Theorem 5.2.

[14] Let ${\Theta_{1}},{\Theta_{2}}\in H^{\infty}(\mathcal{L}(\mathcal{H}))$
be two pure inner functions and let $A:K_{\Theta_{1}}\rightarrow
K_{\Theta_{2}}$ be a bounded linear operator. Then $A$ belongs to
$\mathcal{MT}(\Theta_{1},\Theta_{2})$ if and only if there exist bounded
linear operators $B_{1}:\mathcal{D}_{\Theta_{1}}\to K_{\Theta_{2}}$ and
$B_{2}:\mathcal{D}_{\Theta_{2}}\to K_{\Theta_{1}}$, such that

(5.1)
$A-S_{\Theta_{2}}AS_{\Theta_{1}}^{*}=B_{1}D_{\Theta_{1}}+D_{\Theta_{2}}B_{2}^{*}.$

If a bounded operator $A:K_{\Theta_{1}}\to K_{\Theta_{2}}$ satisfies (5.1),
then
$A=A_{\Psi+\Xi^{*}}^{\Theta_{1},\Theta_{2}}\in\mathcal{MT}(\Theta_{1},\Theta_{2})$
with $\Psi,\Xi\in H^{2}(\mathcal{L}(\mathcal{H}))$ are given below.

######  Corollary 5.3.

[14] Let ${\Theta_{1}},{\Theta_{2}}\in H^{\infty}(\mathcal{L}(\mathcal{H}))$
be two pure inner functions and let $A:K_{\Theta_{1}}\rightarrow
K_{\Theta_{2}}$ be a bounded linear operator.

  1. (a)

If $A$ satisfies (5.1), then
$A=A_{\Psi+\Xi^{*}}^{{\Theta_{1}},{\Theta_{2}}}\in\mathcal{MT}(\Theta_{1},\Theta_{2})$
with

(5.2)
$\Psi(z)x=\big{(}B_{1}{\mathbf{k}}_{0}^{\Theta_{1}}x\big{)}(z)\quad\text{and}\quad\Xi(z)x=\big{(}B_{2}{\mathbf{k}}_{0}^{\Theta_{2}}x\big{)}(z)~{}\mbox{for}~{}x\in\mathcal{H}.$

  2. (b)

If
$A=A_{\Psi+\Xi^{*}}^{{\Theta_{1}},{\Theta_{2}}}\in\mathcal{MT}(\Theta_{1},\Theta_{2})$,
then $A$ satisfies (5.1) with

(5.3) $B_{1}=P_{\Theta_{2}}M_{\Psi}\Omega_{\Theta_{1}}\quad\text{and}\quad
B_{2}=P_{\Theta_{1}}M_{\Xi}\Omega_{\Theta_{2}}.$

######  Remark 5.4.

  1. (a)

For an inner function $\Theta\in H^{\infty}(\mathcal{L}(\mathcal{H}))$ denote

$\mathcal{M}_{\Theta}=H^{2}(\mathcal{L}(\mathcal{H}))\ominus\Theta
H^{2}(\mathcal{L}(\mathcal{H})).$

Therefore, if a bounded linear operator $A:K_{\Theta_{1}}\to K_{\Theta_{2}}$
satisfies (5.1), then
$A=A_{\Psi+\Xi^{*}}^{\Theta_{1},\Theta_{2}}\in\mathcal{MT}(\Theta_{1},\Theta_{2})$
with $\Psi\in\mathcal{M}_{\Theta_{2}}$ and $\Xi\in\mathcal{M}_{\Theta_{1}}$
given by (5.2).

  2. (b)

Recall that $A_{\Phi}^{\Theta_{1},\Theta_{2}}=0$ if and only if

$\Phi\in\Theta_{2}H^{2}(\mathcal{L}(\mathcal{H}))+(\Theta_{1}H^{2}(\mathcal{L}(\mathcal{H})))^{*}~{}(\mbox{see
\cite[cite]{[\@@bibref{Number}{RK2}{}{}]}}).$

Clearly, if $A=A_{\Psi+\Xi^{*}}^{\Theta_{1},\Theta_{2}}$ with $\Psi,\Xi\in
H^{2}(\mathcal{L}(\mathcal{H}))$, then the operators $B_{1}$ and $B_{2}$ given
by (5.3) do not depend on the parts of $\Psi$ and $\Xi$ that belong to
$\Theta_{2}H^{2}(\mathcal{L}(\mathcal{H}))$ and
$\Theta_{1}H^{2}(\mathcal{L}(\mathcal{H}))$, respectively.

As in [15] we can use the unitary operator $\tau_{\Theta}$ defined by (4.4)
and obtain the following theorem.

######  Theorem 5.5.

[14] Let ${\Theta_{1}},{\Theta_{2}}\in H^{\infty}(\mathcal{L}(\mathcal{H}))$
be two pure inner functions and let $A:K_{\Theta_{1}}\rightarrow
K_{\Theta_{2}}$ be a bounded linear operator. Then $A$ belongs to
$\mathcal{MT}(\Theta_{1},\Theta_{2})$ if and only if there exist bounded
linear operators
$\widetilde{B}_{1}:\widetilde{\mathcal{D}}_{\Theta_{1}}\rightarrow
K_{\Theta_{2}}$ and
$\widetilde{B}_{2}:\widetilde{\mathcal{D}}_{\Theta_{2}}\rightarrow
K_{\Theta_{1}}$, such that

(5.4)
$A-S_{\Theta_{2}}^{*}AS_{\Theta_{1}}=\widetilde{B}_{1}\widetilde{D}_{\Theta_{1}}+\widetilde{D}_{\Theta_{2}}\widetilde{B}_{2}^{*}.$

Note from the proof of Theorem 5.5 that if $A:K_{\Theta_{1}}\to
K_{\Theta_{2}}$ satisfies (5.4) with some
$\widetilde{B}_{1}:\widetilde{\mathcal{D}}_{\Theta_{1}}\to K_{\Theta_{2}}$ and
$\widetilde{B}_{2}:\widetilde{\mathcal{D}}_{\Theta_{2}}\to K_{\Theta_{1}}$
where $B_{1}$ and $B_{2}$ are given by (5.3), then

$\widetilde{A}=\tau_{\Theta_{2}}A\tau_{\Theta_{1}}^{*}.$

By Corollary 5.3,
$\widetilde{A}=A_{\Psi+\Xi^{*}}^{\widetilde{\Theta}_{1},\widetilde{\Theta}_{2}}$
with

$\Psi(z)x=(B_{1}{\mathbf{k}}_{0}^{\widetilde{\Theta}_{1}}x)(z)=(\tau_{\Theta_{2}}\widetilde{B}_{1}\tau^{*}_{\Theta_{1}}{\mathbf{k}}_{0}^{\widetilde{\Theta}_{1}}x)(z)=(\tau_{\Theta_{2}}\widetilde{B}_{1}\widetilde{{\mathbf{k}}}_{0}^{\Theta_{1}}x)(z)$

and

$\Xi(z)x=(B_{2}{\mathbf{k}}_{0}^{\widetilde{\Theta}_{1}}x)(z)=(\tau_{\Theta_{1}}\widetilde{B}_{2}\tau^{*}_{\Theta_{2}}{\mathbf{k}}_{0}^{\widetilde{\Theta}_{2}}x)(z)=(\tau_{\Theta_{1}}\widetilde{B}_{2}\widetilde{{\mathbf{k}}}_{0}^{\Theta_{2}}x)(z).$

Moreover (see Remark 5.4), $\Psi\in\mathcal{M}_{\widetilde{\Theta}_{2}}$ and
$\Xi\in\mathcal{M}_{\widetilde{\Theta}_{1}}$.

As in the scalar case, we can use Theorem 5.2 and Theorem 5.5 to get the
following.

######  Corollary 5.6.

[14] Let $\Theta_{1},\Theta_{2}\in H^{\infty}(\mathcal{L}(\mathcal{H}))$ be
two pure inner functions and let $A:K_{\Theta_{1}}\to K_{\Theta_{2}}$ be a
bounded linear operator. Then $A$ belongs to
$\mathcal{MT}(\Theta_{1},\Theta_{2})$ if and only if the following hold:

  1. (a)

there exist bounded linear operators
$\widehat{B}_{1}:\mathcal{D}_{\Theta_{1}}\to K_{\Theta_{2}}$ and
$\widehat{B}_{2}:\widetilde{{\mathcal{D}}}_{\Theta_{2}}\to K_{\Theta_{1}}$
such that

$S_{\Theta_{2}}^{*}A-AS_{\Theta_{1}}^{*}=\widehat{B}_{1}D_{\Theta_{1}}+\widetilde{D}_{\Theta_{2}}\widehat{B}_{2}^{*}.$

  2. (b)

there exist bounded linear operators
$\widehat{B}_{1}:\widetilde{\mathcal{D}}_{\Theta_{1}}\to K_{\Theta_{2}}$ and
$\widehat{B}_{2}:\mathcal{D}_{\Theta_{2}}\to K_{\Theta_{1}}$ such that

$S_{\Theta_{2}}A-AS_{\Theta_{1}}=\widehat{B}_{1}\widetilde{D}_{\Theta_{1}}+D_{\Theta_{2}}\widehat{B}_{2}^{*}.$

###  5.2. By Shift invariance

In the scalar case the notion of shift invariance for TTO’s was introduced in
[19]. D. Sarason proved that a bounded linear operators $A:K_{\theta}\to
K_{\theta}$ is a TTO if and only if it is shift invariant, i.e.,

$\langle ASf,Sf\rangle_{L^{2}}=\langle Af,f\rangle_{L^{2}}$

for each $f\in K_{\theta}$ such that $Sf\in K_{\theta}.$ In [15] we prove that
the same is true for MATTO’s.

For ATTO’s the notion of shift invariance was introduced in [6] ( see also
[17]). Shift invariance of matrix valued truncated Toeplitz operators is
presented in [15]. Here we consider shift invariance of MATTO’s. As in the
scalar case (see [10]), we characterize MATTO’s in term of four (equivalent)
types of shift invariance.

Recall that for an operator valued inner function $\Theta\in
H^{\infty}(\mathcal{L}(\mathcal{H}))$ and for $f\in K_{\Theta}$ we have

$Sf=M_{z}f\in K_{\Theta}~{}~{}\text{if and only
if}~{}~{}f\perp\widetilde{\mathcal{D}}_{\Theta}~{}~{}((\tau_{\Theta}f(0))=0)$

and

$S^{*}f=M_{\overline{z}}f\in K_{\Theta}~{}~{}\text{if and only
if}~{}~{}f\perp\mathcal{D}_{\Theta}~{}~{}(f(0)=0).$

######  Theorem 5.7.

[14] Let $\Theta_{1},\Theta_{2}\in H^{\infty}(\mathcal{L}(\mathcal{H}))$ be
two pure inner functions and let $A:K_{\Theta_{1}}\to K_{\Theta_{2}}$ be a
bounded linear operator. Then $A$ belongs to
$\mathcal{MT}(\Theta_{1},\Theta_{2})$ if and only if it has one (and all) of
the following properties:

  1. (a)

$\langle AS^{*}f,S^{*}g\rangle_{L^{2}(\mathcal{H})}=\langle
Af,g\rangle_{L^{2}(\mathcal{H})}$ for all $f\in K_{\Theta_{1}}$, $g\in
K_{\Theta_{2}}$ such that $f\perp\mathcal{D}_{\Theta_{1}}$,
$g\perp\mathcal{D}_{\Theta_{2}}$;

  2. (b)

$\langle AS^{*}f,g\rangle_{L^{2}(\mathcal{H})}=\langle
Af,Sg\rangle_{L^{2}(\mathcal{H})}$ for all $f\in K_{\Theta_{1}}$, $g\in
K_{\Theta_{2}}$ such that $f\perp\mathcal{D}_{\Theta_{1}}$,
$g\perp\widetilde{\mathcal{D}}_{\Theta_{2}}$;

  3. (c)

$\langle ASf,Sg\rangle_{L^{2}(\mathcal{H})}=\langle
Af,g\rangle_{L^{2}(\mathcal{H})}$ for all $f\in K_{\Theta_{1}}$, $g\in
K_{\Theta_{2}}$ such that $f\perp\widetilde{\mathcal{D}}_{\Theta_{1}}$,
$g\perp\widetilde{\mathcal{D}}_{\Theta_{2}}$;

  4. (d)

$\langle ASf,g\rangle_{L^{2}(\mathcal{H})}=\langle
Af,S^{*}g\rangle_{L^{2}(\mathcal{H})}$ for all $f\in K_{\Theta_{1}}$, $g\in
K_{\Theta_{2}}$ such that $f\perp\widetilde{\mathcal{D}}_{\Theta_{1}}$,
$g\perp\mathcal{D}_{\Theta_{2}}$.

###  5.3. Characterization by modified compressed shift

The notion of modified compressed shift was introduced in [15]. For any
nonconstant inner function $\Theta$, suppose that
$X_{\Theta}:\widetilde{\mathcal{D}}_{\Theta}\rightarrow\mathcal{D}_{\Theta}$,
and consider $\widehat{X}_{\Theta}\in\mathcal{L}(K_{\Theta})$ defined by
$\widehat{X}_{\Theta}f=X_{\Theta}P_{\widetilde{\mathcal{D}}_{\Theta}}f$. The
operator modified shift is defined by

$S_{\Theta,X_{\Theta}}=S_{\Theta}+(\widehat{X}_{\Theta}-S_{\Theta})P_{\widetilde{\mathcal{D}}_{\Theta}},$

or

$S_{\Theta,X_{\Theta}}=S_{\Theta}+P_{\mathcal{D}_{\Theta}}Y_{\Theta}P_{\widetilde{\mathcal{D}}_{\Theta}},$

implies that

$S_{\Theta}=S_{\Theta,X_{\Theta}}-P_{\mathcal{D}_{\Theta}}Y_{\Theta}P_{\widetilde{\mathcal{D}}_{\Theta}}$

where $Y_{\Theta}=\widehat{X}_{\Theta}-S_{\Theta}$.

From equation (3.8) of [15], there is an operator
$J_{\Theta_{1}}\in\mathcal{L}(K_{\Theta_{1}})$ such that

(5.5)
$P_{\mathcal{D}_{\Theta_{1}}}=(I-S_{\Theta_{1}}S_{\Theta_{1}}^{*})J_{\Theta_{1}}=D_{\Theta_{1}}J_{\Theta_{1}}=J_{\Theta_{1}}^{*}D_{\Theta_{1}},$

and similarly there is $J_{\Theta_{2}}\in\mathcal{L}(K_{\Theta_{2}})$ such
that

(5.6)
$P_{\mathcal{D}_{\Theta_{2}}}=(I-S_{\Theta_{2}}S_{\Theta_{2}}^{*})J_{\Theta_{2}}=D_{\Theta_{2}}J_{\Theta_{2}}=J_{\Theta_{2}}^{*}D_{\Theta_{2}}.$

######  Theorem 5.8.

[14] Let ${\Theta_{1}},{\Theta_{2}}\in H^{\infty}(\mathcal{L}(\mathcal{H}))$
be two pure inner functions and let $A:K_{\Theta_{1}}\rightarrow
K_{\Theta_{2}}$ be a bounded operator. Then
$A\in\mathcal{MT}(\Theta_{1},\Theta_{2})$ if and only if

$\displaystyle
A-S_{\Theta_{2},X_{\Theta_{2}}}AS_{\Theta_{1},X_{\Theta_{1}}}^{*}=BD_{\Theta_{1}}+D_{\Theta_{2}}B^{{}^{\prime}*},$

with $B_{1}:\mathcal{D}_{\Theta_{1}}\to K_{\Theta_{2}}$ and
$B_{2}^{\prime}:\mathcal{D}_{\Theta_{2}}\to K_{\Theta_{1}}$.

##  6\. Connection between MATTO’s and MATHO’s

Let $J_{1},J_{2}$ be conjugations on $\mathcal{H}$ such that $\Theta_{1}$ is
$J_{1}$-symmetric and $\Theta_{2}$ is $J_{2}$-symmetric. Then we have the
following relations between MATTO’s and MATHO’s.

######  Proposition 6.1.

For every $\Phi\in L^{2}(\mathcal{L}(\mathcal{H}))$, we have

$C_{\Theta_{2}}A_{\Phi}^{\Theta_{1},\Theta_{2}}C_{\Theta_{1}}=A^{\Theta_{1},\Theta_{2}}_{J_{2}\Theta_{2}^{*}\Phi\Theta_{1}J_{1}}$.

$C_{\Theta_{2}}B_{\Phi}^{\Theta_{1},\Theta_{2}}C_{\Theta_{1}}=B^{\Theta_{1},\Theta_{2}}_{J_{2}\widetilde{\Theta}_{2}\Phi\Theta_{1}J_{1}}$.

${\mathbf{J}}_{2}^{*}A_{\Phi}^{\Theta_{1},\Theta_{2}}{\mathbf{J}}_{1}^{*}=A_{J_{2}\Phi(\bar{z})J_{1}}^{\widetilde{\Theta}_{1},\widetilde{\Theta}_{2}}$.

${\mathbf{J}}_{2}^{*}B_{\Phi}^{\Theta_{1},\Theta_{2}}{\mathbf{J}}_{1}^{*}=B_{J_{2}\Phi(\bar{z})J_{1}}^{\widetilde{\Theta}_{1},\widetilde{\Theta}_{2}}$.

$C_{\Theta_{2}}A_{\Phi}^{\Theta_{1},\Theta_{2}}{\mathbf{J}}_{1}^{*}=B_{J_{2}\Theta_{2}(\bar{z})^{*}\Phi(\bar{z})J_{1}}^{\widetilde{\Theta}_{1},\Theta_{2}}$.

${\mathbf{J}}_{2}^{*}B_{\Phi}^{\Theta_{1},\Theta_{2}}C_{\Theta_{1}}=A_{J_{2}\Phi\Theta_{1}J_{1}}^{\Theta_{1},\widetilde{\Theta}_{2}}$.

###### Proof.

(a) Let $f\in K_{\Theta_{1}}$ and $g\in K_{\Theta_{2}}$ and let
$J_{1}\Theta_{1}J_{1}=\Theta_{1}^{*}$ and
$J_{2}\Theta_{2}J_{2}=\Theta_{2}^{*}$. Then

$\displaystyle\langle
C_{\Theta_{2}}A_{\Phi}^{\Theta_{1},\Theta_{2}}C_{\Theta_{1}}f,g\rangle$
$\displaystyle=\langle
C_{\Theta_{2}}g,A_{\Phi}^{\Theta_{1},\Theta_{2}}C_{\Theta_{1}}f\rangle=\langle\Theta_{2}\bar{z}J_{2}g,\Phi\Theta_{1}\bar{z}J_{1}f\rangle$
$\displaystyle=\langle\Theta_{2}J_{2}g,\Phi\Theta_{1}J_{1}f\rangle=\langle
J_{2}g,\Theta_{2}^{*}\Phi\Theta_{1}J_{1}f\rangle$ $\displaystyle=\langle
J_{2}\Theta_{2}^{*}\Phi\Theta_{1}J_{1}f,g\rangle=\langle
A_{J_{2}\Theta_{2}^{*}\Phi\Theta_{1}J_{1}}f,g\rangle.$

(b) Let $f\in K_{\Theta_{1}}$ and $g\in K_{\Theta_{2}}$. Then

$\displaystyle\langle
C_{\Theta_{2}}B_{\Phi}^{\Theta_{1},\Theta_{2}}C_{\Theta_{1}}f,g\rangle$
$\displaystyle=\langle
C_{\Theta_{2}}g,B_{\Phi}^{\Theta_{1},\Theta_{2}}C_{\Theta_{1}}f\rangle=\langle
C_{\Theta_{2}}g,{\mathbf{J}}(I-P)(\Phi C_{\Theta_{1}}f)\rangle$
$\displaystyle=\langle{\mathbf{J}}C_{\Theta_{2}}g,\Phi
C_{\Theta_{1}}f\rangle=\langle{\mathbf{J}}(\Theta_{2}\bar{z}J_{2}g),\Phi\Theta_{1}\bar{z}J_{1}f\rangle$
$\displaystyle=\langle\bar{z}\Theta_{2}(\bar{z})zJ_{2}(g(\bar{z})),\bar{z}\Phi\Theta_{1}J_{1}f\rangle$
$\displaystyle=\langle
J_{2}{\mathbf{J}}g,\Theta_{2}(\bar{z})^{*}\Phi\Theta_{1}J_{1}f\rangle=\langle
J_{2}\widetilde{\Theta}_{2}\Phi\Theta_{1}J_{1}f,{\mathbf{J}}g\rangle$
$\displaystyle=\langle
P_{\Theta_{2}}{\mathbf{J}}(I-P)(J_{2}\widetilde{\Theta}_{2}\Phi\Theta_{1}J_{1}f),g\rangle$
$\displaystyle=\langle
B_{J_{2}\widetilde{\Theta}_{2}\Phi\Theta_{1}J_{1}}^{\Theta_{1},\Theta_{2}}f,g\rangle.$

(c) For $f\in K_{\widetilde{\Theta}_{1}}$ and $g\in
K_{\widetilde{\Theta}_{2}}$, it holds that

$\displaystyle\langle{\mathbf{J}}_{2}^{*}A_{\Phi}^{\Theta_{1},\Theta_{2}}{\mathbf{J}}_{1}^{*}f,g\rangle$
$\displaystyle=\langle{\mathbf{J}}^{*}_{2}g,A_{\Phi}^{\Theta_{1},\Theta_{2}}{\mathbf{J}}_{1}^{*}f\rangle=\langle{\mathbf{J}}^{*}_{2}g,\Phi{\mathbf{J}}_{1}^{*}f\rangle$
$\displaystyle=\langle J_{2}g(\bar{z}),\Phi J_{1}f(\bar{z})\rangle=\langle
J_{2}\Phi J_{1}f(\bar{z}),g(\bar{z})\rangle$ $\displaystyle=\langle
J_{2}\Phi(\bar{z})J_{1}f,g\rangle=\langle
A_{J_{2}\Phi(\bar{z})J_{1}}^{\widetilde{\Theta}_{1},\widetilde{\Theta}_{2}}f,g\rangle.$

(d) Let $f\in K_{\widetilde{\Theta}_{1}}$ and $g\in
K_{\widetilde{\Theta}_{2}}$. Then

$\displaystyle\langle{\mathbf{J}}_{2}^{*}B_{\Phi}^{\Theta_{1},\Theta_{2}}{\mathbf{J}}_{1}^{*}f,g\rangle$
$\displaystyle=\langle{\mathbf{J}}_{2}^{*}g,B_{\Phi}^{\Theta_{1},\Theta_{2}}{\mathbf{J}}_{1}^{*}f\rangle=\langle{\mathbf{J}}_{2}^{*}g,{\mathbf{J}}\Phi{\mathbf{J}}_{1}^{*}f\rangle$
$\displaystyle=\langle{\mathbf{J}}{\mathbf{J}}_{2}^{*}g,\Phi{\mathbf{J}}_{1}^{*}f\rangle=\langle{\mathbf{J}}_{2}^{*}{\mathbf{J}}g,\Phi{\mathbf{J}}_{1}^{*}f\rangle$
$\displaystyle=\langle{\mathbf{J}}_{2}^{*}\Phi{\mathbf{J}}_{1}^{*}f,{\mathbf{J}}g\rangle=\langle
J_{2}\Phi(\bar{z})J_{1}f(z),{\mathbf{J}}g\rangle$
$\displaystyle=\langle{\mathbf{J}}(I-P_{+})(J_{2}\Phi(\bar{z})J_{1}f(z)),g\rangle$
$\displaystyle=\langle
B_{J_{2}\Phi(\bar{z})J_{1}}^{\widetilde{\Theta}_{1},\widetilde{\Theta}_{2}}f,g\rangle.$

(e) For $f\in K_{\widetilde{\Theta}_{1}}$ and $g\in K_{\Theta_{2}}$, it holds
that

$\displaystyle\langle
C_{\Theta_{2}}A_{\Phi}^{\Theta_{1},\Theta_{2}}{\mathbf{J}}_{1}^{*}f,g\rangle$
$\displaystyle=\langle
C_{\Theta_{2}}g,A_{\Phi}^{\Theta_{1},\Theta_{2}}{\mathbf{J}}_{1}^{*}f\rangle=\langle\Theta_{2}\bar{z}J_{2}g,\Phi{\mathbf{J}}_{1}^{*}f\rangle$
$\displaystyle=\langle\Theta_{2}{\mathbf{J}}^{*}Jg,\Phi{\mathbf{J}}^{*}f\rangle=\langle
f,{\mathbf{J}}_{1}^{*}\Phi^{*}\Theta_{2}\bar{z}J_{2}g(z)\rangle$
$\displaystyle=\langle
f,J_{1}\Phi(\bar{z})^{*}\Theta_{2}(\bar{z}){z}J_{2}g(\bar{z})\rangle$
$\displaystyle=\langle
f,J_{1}\Phi(\bar{z})^{*}\Theta_{2}(\bar{z})J_{2}(\bar{z}g(\bar{z}))\rangle$
$\displaystyle=\langle
f,J_{1}\Phi(\bar{z})^{*}\Theta_{2}(\bar{z})J_{2}{\mathbf{J}}g\rangle$
$\displaystyle=\langle
J_{2}\Theta_{2}(\bar{z})^{*}\Phi(\bar{z})J_{1}f,\bar{z}g(\bar{z})\rangle$
$\displaystyle=\langle
zJ_{2}\Theta_{2}(\bar{z})^{*}\Phi(\bar{z})J_{1}f,g(\bar{z})\rangle$
$\displaystyle=\langle\bar{z}J_{2}\Theta_{2}(z)^{*}\Phi(z)J_{1}f(\bar{z}),g(z)\rangle$
$\displaystyle=\langle{\mathbf{J}}J_{2}\Theta_{2}(\bar{z})^{*}\Phi(\bar{z})J_{1}f(z),g(z)\rangle$
$\displaystyle=\langle{\mathbf{J}}(I-P_{+})(J_{2}\Theta_{2}(\bar{z})^{*}\Phi(\bar{z})J_{1}f),g\rangle$
$\displaystyle=\langle
B_{J_{2}\Theta_{2}(\bar{z})^{*}\Phi(\bar{z})J_{1}}^{\widetilde{\Theta}_{1},\Theta_{2}}f,g\rangle.$

(f) Let $f\in K_{\Theta_{1}}$ and $g\in K_{\widetilde{\Theta}_{2}}$. Then

$\displaystyle\langle{\mathbf{J}}_{2}^{*}B_{\Phi}^{\Theta_{1},\Theta_{2}}C_{\Theta_{1}}f,g\rangle$
$\displaystyle=\langle{\mathbf{J}}_{2}^{*}g,B_{\Phi}^{\Theta_{1},\Theta_{2}}C_{\Theta_{1}}f\rangle=\langle
J_{2}g(\bar{z}),{\mathbf{J}}\Phi\Theta_{1}\bar{z}J_{1}f\rangle$
$\displaystyle=\langle
J_{2}g(\bar{z}),\bar{z}\Phi(\bar{z})\Theta_{1}(\bar{z})zJ_{1}f(\bar{z})\rangle$
$\displaystyle=\langle
J_{2}\Phi(\bar{z})\Theta_{1}(\bar{z})J_{1}f(\bar{z}),g(\bar{z})\rangle$
$\displaystyle=\langle J_{2}\Phi(z)\Theta_{1}(z)J_{1}f,g\rangle=\langle
P_{\widetilde{\Theta}_{2}}(J_{2}\Phi\Theta_{1}J_{1}f),g\rangle$
$\displaystyle=\langle
A_{J_{2}\Phi\Theta_{1}J_{1}}^{\Theta_{1},\widetilde{\Theta}_{2}}f,g\rangle.$

∎

######  Corollary 6.2.

Let $A$ and $B$ be bounded linear operators from $K_{\Theta_{1}}$ into
$K_{\Theta_{2}}$ for some inner functions $\Theta_{1}$ and $\Theta_{2}$. Then

$A\in\mathcal{MT}(\Theta_{1},\Theta_{2})$ if and only if
$C_{\Theta_{2}}AC_{\Theta_{1}}\in\mathcal{MT}(\Theta_{1},\Theta_{2})$.

$B\in\mathcal{MH}(\Theta_{1},\Theta_{2})$ if and only if
$C_{\Theta_{2}}BC_{\Theta_{1}}\in\mathcal{MH}(\Theta_{1},\Theta_{2})$.

$A\in\mathcal{MT}(\Theta_{1},\Theta_{2})$ if and only if
${\mathbf{J}}_{2}^{*}A{\mathbf{J}}_{1}^{*}\in\mathcal{MT}(\widetilde{\Theta}_{1},\widetilde{\Theta}_{2})$.

$B\in\mathcal{MH}(\Theta_{1},\Theta_{2})$ if and only if
${\mathbf{J}}_{2}^{*}B{\mathbf{J}}_{1}^{*}\in\mathcal{MH}(\widetilde{\Theta}_{1},\widetilde{\Theta}_{2})$.

$A\in\mathcal{MT}(\Theta_{1},\Theta_{2})$ if and only if
$C_{\Theta_{2}}A{\mathbf{J}}_{1}^{*}\in\mathcal{MH}(\widetilde{\Theta}_{1},\Theta_{2})$.

$B\in\mathcal{MH}(\Theta_{1},\Theta_{2})$ if and only if
${\mathbf{J}}_{2}^{*}BC_{\Theta_{1}}\in\mathcal{MT}(\Theta_{1},\widetilde{\Theta}_{2})$.

###### Proof.

The proof follows from Proposition 6.1. ∎

The connection between MATTO’s and MATHO’s makes it easy to to transform the
results from MATTO’s to MATHO’s. The following proposition is a consequence of
this connection.

######  Proposition 6.3.

Let $\Theta_{1}$ and $\Theta_{2}$ be two nonconstant inner functions and let
$\Phi\in L^{2}(\mathcal{L}(\mathcal{H}))$. Then
$B_{\Phi}^{\Theta_{1},\Theta_{2}}=0$ if and only if $\Phi\in
J_{2}[H^{2}(\mathcal{L}(\mathcal{H}))]^{*}J_{1}+J_{2}\widetilde{\Theta}_{2}H^{2}(\mathcal{L}(\mathcal{H}))\Theta_{1}J_{1}$.

###### Proof.

Proposition 6.1$(f)$ implies that $B_{\Phi}^{\Theta_{1},\Theta_{2}}=0$ if and
only if $A_{J_{2}\Phi\Theta_{1}J_{1}}^{\Theta_{1},\widetilde{\Theta}_{2}}=0$.
By [13, Theorem 4.2],
$A_{J_{2}\Phi\Theta_{1}J_{1}}^{\Theta_{1},\widetilde{\Theta}_{2}}=0$ if and
only if

$\displaystyle
J_{2}\Phi\Theta_{1}J_{1}\in[\Theta_{1}H^{2}(\mathcal{L}(\mathcal{H}))]^{*}+\widetilde{\Theta}_{2}H^{2}(\mathcal{L}(\mathcal{H}))$
$\displaystyle\Leftrightarrow$ $\displaystyle\Phi\Theta_{1}\in
J_{2}[\Theta_{1}H^{2}(\mathcal{L}(\mathcal{H}))]^{*}J_{1}+J_{2}\widetilde{\Theta}_{2}H^{2}(\mathcal{L}(\mathcal{H}))J_{1}$
$\displaystyle\Leftrightarrow$ $\displaystyle\Phi\in
J_{2}[\Theta_{1}H^{2}(\mathcal{L}(\mathcal{H}))]^{*}J_{1}\Theta_{1}^{*}+J_{2}\widetilde{\Theta}_{2}H^{2}(\mathcal{L}(\mathcal{H}))J_{1}\Theta_{1}^{*}$
$\displaystyle\Leftrightarrow$ $\displaystyle\Phi\in
J_{2}[H^{2}(\mathcal{L}(\mathcal{H}))]^{*}\Theta_{1}^{*}\Theta_{1}J_{1}+J_{2}\widetilde{\Theta}_{2}H^{2}(\mathcal{L}(\mathcal{H}))\Theta_{1}J_{1}$
$\displaystyle\Leftrightarrow$ $\displaystyle\Phi\in
J_{2}[H^{2}(\mathcal{L}(\mathcal{H}))]^{*}J_{1}+J_{2}\widetilde{\Theta}_{2}H^{2}(\mathcal{L}(\mathcal{H}))\Theta_{1}J_{1}.$

Hence we complete the proof. ∎

##  7\. Characterization of MATHO’s

We will use the results of matrix matrix valued asymmetric truncated Toeplitz
operators to characterize matrix valued asymmetric Hankel operators. Recall
that for the inner function $\Theta$ we have
$S_{\Theta}=C_{\Theta}S_{\Theta}^{*}C_{\Theta}$ and
${\mathbf{J}}^{*}S_{\widetilde{\Theta}}{\mathbf{J}}^{*}=S_{\Theta}$.
Furthermore, the defect operators are given by
$D_{\Theta}=I-S_{\Theta}S_{\Theta}^{*}$ and
$\widetilde{D}_{\Theta}=I-S_{\Theta}^{*}S_{\Theta}$.

A simple computation shows that the following relation holds

(7.1)
$D_{\Theta}C_{\Theta}=C_{\Theta}\widetilde{D}_{\Theta}\quad\text{and}\quad{\mathbf{J}}^{*}D_{\widetilde{\Theta}}=D_{\Theta}{\mathbf{J}}^{*}.$

The following theorems are generalization of [9, Proposition 4.2].

######  Theorem 7.1.

Let $\Theta_{1},\Theta_{2}$ be two pure inner functions and let
$B:K_{\Theta_{1}}\to K_{\Theta_{2}}$ be a bounded linear operator. Then
$B\in\mathcal{MH}(\Theta_{1},\Theta_{2})$ if and only if there exist
$B_{1}^{{}^{\prime}}:\widetilde{\mathcal{D}}_{\Theta_{1}}\to K_{\Theta_{2}}$
and $B_{2}^{{}^{\prime}}:\mathcal{D}_{\Theta_{2}}\to K_{\Theta_{1}}$ such that

(7.2)
$B-S_{\Theta_{2}}BS_{\Theta_{1}}=B_{1}^{{}^{\prime}}\widetilde{D}_{\Theta_{1}}+D_{\Theta_{2}}B^{{}^{\prime}*}_{2}.$

###### Proof.

Let
$B=B_{\Phi}^{\Theta_{1},\Theta_{2}}\in\mathcal{MH}(\Theta_{1},\Theta_{2})$. By
Proposition 6.1(f) we have

${\mathbf{J}}_{2}^{*}BC_{\Theta_{1}}=A_{J_{2}\Phi\Theta
J_{1}}^{\Theta_{1},\widetilde{\Theta}_{2}}\in\mathcal{MT}(\Theta_{1},\widetilde{\Theta}_{2}).$

Therefore by Theorem 5.2 we have

$\displaystyle{\mathbf{J}}_{2}^{*}BC_{\Theta_{1}}-S_{\Theta_{2}}{\mathbf{J}}_{2}^{*}BC_{\Theta_{1}}S_{\Theta_{1}}^{*}=B_{1}D_{\Theta_{1}}+{D}_{\widetilde{\Theta}_{2}}B_{2}^{*}$
$\displaystyle\Rightarrow$
$\displaystyle~{}B-{\mathbf{J}}_{2}^{*}S_{\Theta_{2}}{\mathbf{J}}_{2}^{*}BC_{\Theta_{1}}S_{\Theta_{1}}^{*}C_{\Theta_{1}}={\mathbf{J}}_{2}^{*}B_{1}D_{\Theta_{1}}C_{\Theta_{1}}+{\mathbf{J}}_{2}^{*}{D}_{\widetilde{\Theta}_{2}}B_{2}^{*}C_{\Theta_{1}}$
$\displaystyle\Rightarrow$
$\displaystyle~{}B-S_{\Theta_{2}}BS_{\Theta_{1}}={\mathbf{J}}_{2}^{*}B_{1}C_{\Theta_{1}}\widetilde{D}_{\Theta_{1}}+D_{\Theta_{2}}{\mathbf{J}}_{2}^{*}B_{2}^{*}C_{\Theta_{1}}$
$\displaystyle\Rightarrow$
$\displaystyle~{}B-S_{\Theta_{2}}BS_{\Theta_{1}}=B_{1}^{{}^{\prime}}\widetilde{D}_{\Theta_{1}}+D_{\Theta_{2}}B_{2}^{{}^{\prime}*}$

where $B_{1}^{{}^{\prime}}={\mathbf{J}}_{2}^{*}B_{1}C_{\Theta_{1}}$ and
$B_{2}^{{}^{\prime}}=(C_{\Theta_{1}}B_{2}{\mathbf{J}}_{2}^{*})^{*}$. ∎

######  Corollary 7.2.

Let $\Theta_{1},\Theta_{2}$ be two pure inner functions and let
$B:K_{\Theta_{1}}\to K_{\Theta_{2}}$ be a bounded linear operator. Then
$B\in\mathcal{MH}(\Theta_{1},\Theta_{2})$ if and only if there exist
$\widetilde{B}_{1}:\widetilde{\mathcal{D}}_{\Theta_{1}}\to K_{\Theta_{2}}$ and
$\widetilde{B}_{2}:\widetilde{\mathcal{D}}_{\Theta_{2}}\to K_{\Theta_{1}}$
such that

(7.3)
$S_{\Theta_{2}}^{*}B-BS_{\Theta_{1}}=\widetilde{B}_{1}\widetilde{D}_{\Theta_{1}}+\widetilde{D}_{\Theta_{2}}\widetilde{B}^{*}_{2}.$

###### Proof.

Let
$B=B_{\Phi}^{\Theta_{1},\Theta_{2}}\in\mathcal{MH}(\Theta_{1},\Theta_{2})$. By
Theorem 7.1 and using the defect operators
$I-\widetilde{D}_{\Theta_{2}}=S_{\Theta_{2}}^{*}S_{\Theta_{2}}$ we have

$\displaystyle
B-S_{\Theta_{2}}BS_{\Theta_{1}}=B_{1}^{{}^{\prime}}\widetilde{D}_{\Theta_{1}}+D_{\Theta_{2}}B_{2}^{{}^{\prime}*}$
$\displaystyle\Rightarrow$
$\displaystyle~{}S_{\Theta_{2}}^{*}B-S_{\Theta_{2}}^{*}S_{\Theta_{2}}BS_{\Theta_{1}}^{*}=S_{\Theta_{2}}^{*}B_{1}^{{}^{\prime}}\widetilde{D}_{\Theta_{1}}+S_{\Theta_{2}}^{*}D_{\Theta_{2}}B_{2}^{{}^{\prime}*}$
$\displaystyle\Rightarrow$
$\displaystyle~{}S_{\Theta_{2}}^{*}B-(I-\widetilde{D}_{\Theta_{2}})BS_{\Theta_{1}}=S_{\Theta_{2}}^{*}B_{1}^{{}^{\prime}}\widetilde{D}_{\Theta_{1}}+\widetilde{D}_{\Theta_{2}}S_{\Theta_{2}}^{*}B_{2}^{{}^{\prime}*}$
$\displaystyle\Rightarrow$
$\displaystyle~{}S_{\Theta_{2}}^{*}B-BS_{\Theta_{1}}=S_{\Theta_{2}}^{*}B_{1}^{{}^{\prime}}\widetilde{D}_{\Theta_{1}}+\widetilde{D}_{\Theta_{2}}S_{\Theta_{2}}^{*}B_{2}^{{}^{\prime}*}-\widetilde{D}_{\Theta_{2}}BS_{\Theta_{1}}$
$\displaystyle\Rightarrow$
$\displaystyle~{}S_{\Theta_{2}}^{*}B-BS_{\Theta_{1}}=S_{\Theta_{2}}^{*}B_{1}^{{}^{\prime}}\widetilde{D}_{\Theta_{1}}+\widetilde{D}_{\Theta_{2}}(S_{\Theta_{2}}^{*}B_{2}^{{}^{\prime}*}-BS_{\Theta_{1}})$
$\displaystyle\Rightarrow$
$\displaystyle~{}S_{\Theta_{2}}^{*}B-BS_{\Theta_{1}}=\widetilde{B}_{1}\widetilde{D}_{\Theta_{1}}+\widetilde{D}_{\Theta_{2}}\widetilde{B}^{*}_{2}$

where
$\widetilde{B}_{1}=S_{\Theta_{2}}^{*}B_{1}^{{}^{\prime}}:\widetilde{\mathcal{D}}_{\Theta_{1}}\to
K_{\Theta_{2}}$ and
$\widetilde{B}^{*}_{2}=(P_{\widetilde{\mathcal{D}}_{\Theta_{2}}}(S_{\Theta_{2}}^{*}B_{2}^{{}^{\prime}*}-BS_{\Theta_{1}}))^{*}:\widetilde{\mathcal{D}}_{\Theta_{2}}\to
K_{\Theta_{1}}$. ∎

######  Theorem 7.3.

Let $\Theta_{1},\Theta_{2}$ be two pure inner functions and let
$B:K_{\Theta_{1}}\to K_{\Theta_{2}}$ be a bounded linear operator. Then
$B\in\mathcal{MH}(\Theta_{1},\Theta_{2})$ if and only if there exist
$\widehat{B}_{1}:\mathcal{D}_{\Theta_{1}}\to K_{\Theta_{2}}$ and
$\widehat{B}_{2}:\widetilde{\mathcal{D}}_{\Theta_{2}}\to K_{\Theta_{1}}$ such
that

(7.4)
$B-S_{\Theta_{2}}^{*}BS_{\Theta_{1}}^{*}=\widehat{B}_{1}D_{\Theta_{1}}+\widetilde{D}_{\Theta_{2}}\widehat{B}^{*}_{2}.$

###### Proof.

Let
$A=A_{\Phi}^{\Theta_{1},\Theta_{2}}\in\mathcal{MT}(\Theta_{1},\Theta_{2})$. By
Proposition 6.1(e), we have

$C_{\Theta_{2}}A_{\Phi}^{\Theta_{1},\Theta_{2}}{\mathbf{J}}_{1}^{*}=B_{J_{2}\Theta(\bar{z})^{*}\Phi(\bar{z})J_{1}}^{\widetilde{\Theta}_{1},\Theta_{2}}=B_{\Psi}^{\widetilde{\Theta}_{1},\Theta_{2}}$

and so

$A_{\Phi}^{\widetilde{\Theta}_{1},\Theta_{2}}=C_{\Theta_{2}}B_{\Psi}^{\Theta_{1},\Theta_{2}}{\mathbf{J}}_{1}^{*}=C_{\Theta_{2}}B{\mathbf{J}}_{1}^{*}\in\mathcal{MT}(\widetilde{\Theta}_{1},\Theta_{2})$

where $B=B_{\Psi}^{\Theta_{1},\Theta_{2}}.$ Therefore by Theorem 5.2, we have

$\displaystyle
C_{\Theta_{2}}B{\mathbf{J}}_{1}^{*}-S_{\Theta_{2}}C_{\Theta_{2}}B{\mathbf{J}}_{1}^{*}S_{\widetilde{\Theta}_{1}}^{*}=B_{1}{D}_{\widetilde{\Theta}_{1}}+D_{\Theta_{2}}B_{2}^{*}$
$\displaystyle\Rightarrow$ $\displaystyle
B-C_{\Theta_{2}}S_{\Theta_{2}}C_{\Theta_{2}}B{\mathbf{J}}_{1}^{*}S_{\widetilde{\Theta}_{1}}^{*}{\mathbf{J}}_{1}^{*}=C_{\Theta_{2}}B_{1}{D}_{\widetilde{\Theta}_{1}}{\mathbf{J}}_{1}^{*}+C_{\Theta_{2}}D_{\Theta_{2}}B_{2}^{*}{\mathbf{J}}_{1}^{*}$
$\displaystyle\Rightarrow$ $\displaystyle
B-S_{\Theta_{2}}^{*}BS_{\Theta_{1}}^{*}=C_{\Theta_{2}}B_{1}{\mathbf{J}}_{1}^{*}D_{\Theta_{1}}+\widetilde{D}_{\Theta_{2}}C_{\Theta_{2}}B_{2}^{*}{\mathbf{J}}_{1}^{*}$
$\displaystyle\Rightarrow$ $\displaystyle
B-S_{\Theta_{2}}^{*}BS_{\Theta_{1}}^{*}=\widehat{B}_{1}D_{\Theta_{1}}+\widetilde{D}_{\Theta_{2}}\widehat{B}_{2}^{*}$

where
$\widehat{B}_{1}=C_{\Theta_{2}}B_{1}{\mathbf{J}}_{1}^{*}:\mathcal{D}_{\Theta_{1}}\to
K_{\Theta_{2}}$ and
$\widehat{B}_{2}=(C_{\Theta_{2}}B_{2}^{*}{\mathbf{J}}_{1}^{*})^{*}:\widetilde{\mathcal{D}}_{\Theta_{2}}\to
K_{\Theta_{1}}$. ∎

######  Corollary 7.4.

Let $\Theta_{1},\Theta_{2}$ be two pure inner functions and let
$B:K_{\Theta_{1}}\to K_{\Theta_{2}}$ be a bounded linear operator. Then
$B\in\mathcal{MH}(\Theta_{1},\Theta_{2})$ if and only if there exist
$\widehat{B}_{1}^{{}^{\prime}}:\mathcal{D}_{\Theta_{1}}\to K_{\Theta_{2}}$ and
$\widehat{B}_{2}^{{}^{\prime}}:\mathcal{D}_{\Theta_{2}}\to K_{\Theta_{1}}$
such that

(7.5)
$S_{\Theta_{2}}B-BS_{\Theta_{1}}^{*}=\widehat{B}_{1}^{{}^{\prime}}D_{\Theta_{1}}+D_{\Theta_{2}}\widehat{B}^{{}^{\prime}*}_{2}.$

###### Proof.

By Theorem 7.3, $B\in\mathcal{MH}(\Theta_{1},\Theta_{2})$ if and only if

$B-S_{\Theta_{2}}^{*}BS_{\Theta_{1}}^{*}=\widehat{B}_{1}D_{\Theta_{1}}+\widetilde{D}_{\Theta_{2}}\widehat{B}^{*}_{2},$

which is equivalent to

$S_{\Theta_{2}}B-S_{\Theta_{2}}S_{\Theta_{2}}^{*}BS_{\Theta_{1}}^{*}=S_{\Theta_{2}}\widehat{B}_{1}D_{\Theta_{1}}+S_{\Theta_{2}}\widetilde{D}_{\Theta_{2}}\widehat{B}^{*}_{2}.$

By the defect operator $D_{\Theta_{2}}=I-S_{\Theta_{2}}S_{\Theta_{2}}^{*}$, we
have

$\displaystyle
S_{\Theta_{2}}B-(I-D_{\Theta_{2}})BS_{\Theta_{1}}^{*}=S_{\Theta_{2}}\widehat{B}_{1}D_{\Theta_{1}}+S_{\Theta_{2}}\widetilde{D}_{\Theta_{2}}\widehat{B}^{*}_{2}$
$\displaystyle\Rightarrow$ $\displaystyle
S_{\Theta_{2}}B-BS_{\Theta_{1}}^{*}=S_{\Theta_{2}}\widehat{B}_{1}D_{\Theta_{1}}+S_{\Theta_{2}}\widetilde{D}_{\Theta_{2}}\widehat{B}^{*}_{2}-D_{\Theta_{2}}BS_{\Theta_{1}}^{*}$
$\displaystyle\Rightarrow$ $\displaystyle
S_{\Theta_{2}}B-BS_{\Theta_{1}}^{*}=S_{\Theta_{2}}\widehat{B}_{1}D_{\Theta_{1}}+D_{\Theta_{2}}S_{\Theta_{2}}\widehat{B}^{*}_{2}-D_{\Theta_{2}}BS_{\Theta_{1}}^{*}$
$\displaystyle\Rightarrow$ $\displaystyle
S_{\Theta_{2}}B-BS_{\Theta_{1}}^{*}=S_{\Theta_{2}}\widehat{B}_{1}D_{\Theta_{1}}+D_{\Theta_{2}}(S_{\Theta_{2}}\widehat{B}^{*}_{2}-BS_{\Theta_{1}}^{*})$
$\displaystyle\Rightarrow$ $\displaystyle
S_{\Theta_{2}}B-BS_{\Theta_{1}}^{*}=\widehat{B}_{1}^{{}^{\prime}}D_{\Theta_{1}}+D_{\Theta_{2}}\widehat{B}^{{}^{\prime}*}_{2}$

where
$\widehat{B}_{1}^{{}^{\prime}}=S_{\Theta_{2}}\widehat{B}_{1}:\mathcal{D}_{\Theta_{1}}\to
K_{\Theta_{2}}$ and
$\widehat{B}_{2}^{{}^{\prime}}=(P_{\mathcal{D}_{2}}(S_{\Theta_{2}}\widehat{B}^{*}_{2}-BS_{\Theta_{1}}^{*}))^{*}:\mathcal{D}_{\Theta_{2}}\to
K_{\Theta_{1}}$. ∎

##  8\. Characterization by modified compressed shift

In the scalar case the compressed shift was introduced in [19, section 10].
The notion of modified compressed shift was introduced in section 5.3 (see
[15]). Characterization of matrix valued asymmetric truncated Toeplitz
operators by using modified compressed shift was presented in [14]. In this
part we use the modified compressed shift to characterize MATHO’s.

######  Theorem 8.1.

Let $\Theta_{1},\Theta_{2}$ be two pure inner functions and let
$B:K_{\Theta_{1}}\to K_{\Theta_{2}}$ be a bounded linear operator. Then
$B\in\mathcal{MH}(\Theta_{1},\Theta_{2})$ if and only if one (and all) of the
following conditions holds.

$B-S_{\Theta_{2},X_{2}}BS_{\Theta_{1},X_{1}}=\mathbb{B}_{1}D_{\Theta_{1}}+D_{\Theta_{2}}\mathbb{B}_{2}^{{}^{\prime}*}$.

$S_{\Theta_{2},X_{2}}^{*}B-BS_{\Theta_{1},X_{1}}=\widetilde{\mathbb{B}}_{1}\widetilde{D}_{\Theta_{1}}+\widetilde{D}_{\Theta_{2}}\widetilde{\mathbb{B}}_{2}^{{}^{\prime}*}$.

$B-S_{\Theta_{2},X_{2}}^{*}BS_{\Theta_{1},X_{1}}^{*}=\mathbb{B}_{3}\widetilde{D}_{\Theta_{1}}+\widetilde{D}_{\Theta_{2}}\mathbb{B}_{4}^{{}^{\prime}*}$.

$S_{\Theta_{2},X_{2}}B-BS_{\Theta_{1},X_{1}}^{*}=\widetilde{\mathbb{B}}_{3}D_{\Theta_{1}}+D_{\Theta_{2}}\widetilde{\mathbb{B}}_{4}^{{}^{\prime}*}$.

###### Proof.

The proof is analogous to the proof of Theorem 7.1. ∎

######  Corollary 8.2.

Let $B:K_{\Theta_{1}}\to K_{\Theta_{2}}$ be a bounded linear operator. Then
$B\in\mathcal{MH}(\Theta_{1},\Theta_{2})$ if and only if it has one (and all)
of the following properties

$\langle BSf,S^{*}g\rangle=\langle Bf,g\rangle~{}\text{for all}~{}f\in
K_{\Theta_{1}},~{}g\in K_{\Theta_{2}}~{}\text{such that
}~{}f\in\widetilde{\mathcal{D}}_{\Theta_{1}}^{\perp}~{}\text{and}~{}g\in\mathcal{D}_{\Theta_{2}}^{\perp}$.

$\langle Bf,Sg\rangle=\langle BSf,g\rangle~{}\text{for all}~{}f\in
K_{\Theta_{1}},~{}g\in K_{\Theta_{2}}~{}\text{such that
}~{}f\in\widetilde{\mathcal{D}}_{\Theta_{1}}^{\perp}~{}\text{and}~{}g\in\widetilde{\mathcal{D}}_{\Theta_{2}}^{\perp}$.

$\langle BS^{*}f,Sg\rangle=\langle Bf,g\rangle~{}\text{for all}~{}f\in
K_{\Theta_{1}},~{}g\in K_{\Theta_{2}}~{}\text{such that
}~{}f\in\mathcal{D}_{\Theta_{1}}^{\perp}~{}\text{and}~{}g\in\widetilde{\mathcal{D}}_{\Theta_{2}}^{\perp}$.

$\langle Bf,S^{*}g\rangle=\langle BS^{*}f,g\rangle~{}\text{for all}~{}f\in
K_{\Theta_{1}},~{}g\in K_{\Theta_{2}}~{}\text{such that
}~{}f\in\mathcal{D}_{\Theta_{1}}^{\perp}~{}\text{and}~{}g\in\mathcal{D}_{\Theta_{2}}^{\perp}$.

###### Proof.

(a) Let $f\in\widetilde{\mathcal{D}}_{\Theta_{1}}^{\perp}$ and
$g\in\mathcal{D}_{\Theta_{2}}^{\perp}$. Then $Sf=S_{\Theta_{1}}f$ and
$S^{*}g=S_{\Theta_{2}}g$ and hence

$\displaystyle\langle BSf,S^{*}g\rangle$ $\displaystyle=\langle
BS_{\Theta_{1}}f,S_{\Theta_{2}}^{*}g\rangle=\langle
S_{\Theta_{2}}BS_{\Theta_{1}}f,g\rangle.$

By Theorem 7.1, we have

$\displaystyle\langle BSf,S^{*}g\rangle$
$\displaystyle=\langle(B-(B_{1}^{{}^{\prime}}\widetilde{D}_{\Theta_{1}}+D_{\Theta_{2}}B_{2}^{{}^{\prime}*}))f,g\rangle$
$\displaystyle=\langle Bf,g\rangle-\langle
B_{1}^{{}^{\prime}}\widetilde{D}_{\Theta_{1}}f,g\rangle-\langle
D_{\Theta_{2}}B_{2}^{{}^{\prime}*}f,g\rangle$

where $B_{1}^{{}^{\prime}}:\widetilde{D}_{\Theta_{1}}\to K_{\Theta_{2}}$ and
$B_{2}^{{}^{\prime}}:\mathcal{D}_{\Theta_{2}}\to K_{\Theta_{1}}$. Since
$f\in\widetilde{\mathcal{D}}_{\Theta_{1}}^{\perp}$. This implies that
$S_{\Theta_{1}}f=zf(z)$ and hence $S_{\Theta_{1}}^{*}S_{\Theta_{1}}f=f$ means
that $\widetilde{D}_{\Theta_{1}}f=(I-S_{\Theta_{1}}^{*}S_{\Theta_{1}})f=0$.
Therefore

$\langle B_{1}^{{}^{\prime}}\widetilde{D}_{\Theta_{1}}f,g\rangle=0.$

Furthermore, since $f\in\widetilde{\mathcal{D}}_{\Theta_{1}}^{\perp}$ implies
that $D_{\Theta_{2}}B_{2}^{{}^{\prime}*}f\in\mathcal{D}_{\Theta_{2}}$ and
$f\in\mathcal{D}_{\Theta_{2}}^{\perp}$ follows that $\langle
D_{\Theta_{2}}B_{2}^{{}^{\prime}*}f,g\rangle=0$. Therefore, we have

$\langle BSf,S^{*}g\rangle=\langle Bf,g\rangle.$

The proofs of (b), (c) and (d) are analogous and is left to the reader. ∎

## References

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[bib,biblist]nameyeardelim,

# Experimental overview of electromagnetic probes in ultra-relativistic
nucleus-nucleus collisions††thanks: Presented at the Quark Matter 2022
conference in Kraków

Klaus Reygers
Physikalisches Institut, Heidelberg University, Germany

###### Abstract

Electromagnetic probes are not affected by hadronization and provide direct
information about the space-time evolution of high-energy nucleus-nucleus
collisions. In particular, the measurement of thermal radiation from the
quark-gluon plasma and the extraction of an effective medium temperature
belong to the key objectives in heavy-ion physics. We provide a brief tour of
current results and an outlook to future measurements.

##  1 Introduction

Once electroweak probes ($\gamma$, $\gamma^{*}$, W, Z) are produced in a high-
energy nucleus-nucleus collision, they leave the surrounding medium with
virtually no further interactions. The direct information these probes provide
about the medium is therefore different from the information from light and
heavy quark observables which are affected by hadronization. Electroweak
bosons provide information about the entire evolution of a nucleus-nucleus
collision. Parton distribution functions in nuclei are probed by measuring
$\gamma$, $\gamma^{*}$, W, Z produced in initial hard scattering processes.
Moreover, parton energy loss in the medium can be studied by measuring quark
jets recoiling from a photon or a Z boson. Real and virtual photons produced
in the preequilibrium phase, the quark-gluon plasma (QGP), and the hadron gas
provide information about the space-time evolution and the temperature of the
medium. In addition, the broadening of the $\rho$ resonance in nucleus-nucleus
collisions accessible via the decay $\rho\to e^{+}e^{-}$ is sensitive to the
restoration of chiral symmetry [1]. This paper focuses on thermal radiation
from the hot medium created in high-energy nucleus-nucleus collision.

Thermal photons from the QGP are expected to be a significant component of the
direct-photon yield in the transverse momentum range $1\lesssim p_{T}\lesssim
3~{}\mathrm{GeV}/c$ [2, 3]. The inverse slope parameter $T_{\mathrm{eff}}$ is
sensitive to the effective medium temperature averaged over the space-time
evolution of the collision. Due to the rapid radial expansion of the medium,
the measured photon spectrum is blueshifted and the inverse slope parameter is
expected to be larger than the temperature of the emitting medium. Therefore,
an inverse slope parameter above the critical temperature
$T_{\mathrm{pc}}\approx 156~{}\mathrm{MeV}$ cannot be directly interpreted as
evidence for the formation of a QGP. With increasing $p_{T}$, the measured
direct-photon yield becomes more sensitive to earlier and hotter stages of the
collisions [4]. Photons from the preequilibrium stage could become important
for $p_{T}\gtrsim 2-3~{}\mathrm{GeV}/c$ [5].

Dileptons (virtual photons) probe different aspects of a heavy-ion collisions
depending on the considered mass range. Dileptons with invariant masses around
and below the $\rho$ meson mass are particularly sensitive to modifications of
the $\rho$ meson due to the surrounding medium. These modifications are
experimentally accessible as the $\rho$ decays in the medium due to its short
lifetime of only $1.3~{}\mathrm{fm}/c$. In particular the presence of baryons
in the medium is expected to give rise to $\rho$ melting, i.e., to a strong
increase of its width. The $\rho$ melting is connected to the restoration of
chiral symmetry in a hot and dense medium. The intermediate mass region (IMR,
$1\lesssim m_{ee}\lesssim 3\,\mathrm{GeV}/c^{2}$) is sensitive to thermal
radiation from the QGP. An effective medium temperature not affected by
blueshift can be obtained by fitting the dielectron yield in this range with
$dN/m_{ee}\propto(m_{ee}T)^{3/2}\exp(-m_{ee}/T)$. With increasing mass, the
contribution from dielectrons from the preequilibrium stage becomes more
relevant which needs to be considered in the extraction of a temperature. To
date, the only measurement of a medium temperature in the mass range above
$1\,\mathrm{GeV}/c^{2}$ was made by the NA60 experiment at the CERN SPS in In-
In collisions collisions. A fit in the range $1.2\lesssim m_{ee}\lesssim
2.0\,\mathrm{GeV}/c^{2}$ gave an effective temperature of $T=205\pm
12\,\mathrm{MeV}$ [6].

##  2 Low-$p_{T}$ direct photons

Direct photon measurements are expected to eventually play a key role in our
understanding of the evolution of the QGP in nucleus-nucleus collisions.
However, it is currently a challenge for models to simultaneously describe the
direct-photons yield and the azimuthal anisotropy ($v_{2}$) of direct photons.
This is known as the direct-photon puzzle [7, 5]. The puzzle became apparent
at the Quark Matter 2011 conference with the observation by the PHENIX
collaboration that the $v_{2}$ of direct photons was similar to the $v_{2}$ of
pions whereas hydrodynamic models predicted a much smaller direct-photon
$v_{2}$. In addition, hydrodynamic models underestimated the direct photon
yield at low $p_{T}$ by about a factor 2–3 [7]. The radial flow velocity
profiles created in the hydrodynamic evolution of the medium need time to
build up. The large direct-photon $v_{2}$ therefore could mean that contrary
to the general expectation, direct photons in the range $1\lesssim
p_{T}\lesssim 3~{}\mathrm{GeV}/c$ are predominantly produced at the later
stages of the medium evolution, perhaps during the transition from the QGP to
a gas of hadrons. This would make direct photons a less useful probe of the
early hot QGP phase. The direct-photon puzzle currently is a puzzle which
concerns the PHENIX spectra and $v_{2}$ measurements. At the LHC, ALICE
direct-photon spectra and $v_{2}$ measurements show similar trends as the
PHENIX data, however, the uncertainties are unfortunately too large to claim a
significant difference between data and theory.

Figure 1: Left panel: Nonprompt (“$\text{Au--
Au}-N_{\mathrm{coll}}\times\text{pp}$”) direct-photon spectrum in central
Au–Au collisions at $\sqrt{s_{\mathrm{NN}}}=200~{}\mathrm{GeV}$ [8]. The new
data from the high-statistics 2014 Au+Au run are shown as closed circles.
Right panel: Inverse slope parameters $T_{\mathrm{eff}}$ in Au–Au collisions
at $\sqrt{s_{\mathrm{NN}}}=200~{}\mathrm{GeV}$ for different centralities
characterized by the charged-particle multiplicity density
$\left.dN_{\mathrm{ch}}/d\eta\right|_{\eta\approx 0}$.

At this conference, PHENIX presented final direct-photon spectra in Au–Au
collisions at $\sqrt{s_{\mathrm{NN}}}=200\,\mathrm{GeV}$ from the high-
statistics 2014 data set [8]. A significant excess above the direct-photon
spectrum in pp collisions at $\sqrt{s}=200\,\mathrm{GeV}$ scaled by the number
of nucleon-nucleon collisions ($N_{\mathrm{coll}}$) is found. The difference
between in direct-photon spectra in Au–Au collisions and the scaled pp
spectrum, denoted by PHENIX as “nonprompt” direct-photon spectrum, is shown in
Fig. 1. PHENIX extracted the inverse slope parameter $T_{\mathrm{eff}}$ in two
fit ranges for various centralities (see Fig. 1, right panel). The inverse
slope parameter is independent of the centrality but increases with increasing
$p_{T}$, in line with a larger early-time contribution at higher $p_{T}$.

To examine at which stage of a nucleus-nucleus collision direct photons are
predominantly produced, it is instructive to study the dependence of the
direct-photon yield on centrality and center-of-mass energy. PHENIX found a
universal scaling according to which the direct-photon yield integrated above
$p_{T}=1\,\mathrm{GeV}/c$ is a function of the charged-particle multiplicity
alone (for different centralities, center-of-mass energies, and collision
systems) and is given by
$dN_{\gamma}^{\mathrm{dir}}/dy\propto(\left.dN_{\mathrm{ch}}/d\eta\right|_{\eta\approx
0})^{\alpha}$, see Fig. 2. A fit to the 2014 data gives $\alpha=1.11\pm
0.02\,(\mathrm{stat}){}^{+0.09}_{-0.08}\,(\mathrm{syst})$. Interestingly, this
value is smaller than the power $\alpha\approx 1.6$ expected for thermal
photons in [4]. Photons from different stages of a collision are actually
expected to exhibit a different scaling behavior. In [4] it is predicted that
the hadron gas, the QGP, and hard scattering contributions scale as
$\alpha_{\mathrm{HG}}\approx 1.23$, $\alpha_{\mathrm{QGP}}\approx 1.83$, and
$\alpha_{\mathrm{pQCD}}\approx 1.25$, respectively. Different stages
contribute differently to different $p_{T}$ intervals and one might expect a
strong $p_{T}$ dependence of $\alpha$. PHENIX, however, finds that
$\alpha(p_{T})$ is consistent with being independent of $p_{T}$.

Figure 2: Left panel: Direct photon yield $dN_{\gamma}^{\mathrm{dir}}/dy$ in
the range $1<p_{T}<5\,\mathrm{GeV}/c$ for collisions with different charged-
particle multiplicity densities
$\left.dN_{\mathrm{ch}}/d\eta\right|_{\eta\approx 0}$. Right panel: Scaling
power $\alpha$ describing the charged-particle multiplicity dependence of the
direct-photon yield in different $p_{T}$ intervals according to
$dN_{\gamma}^{\mathrm{dir}}/dy\propto(\left.dN_{\mathrm{ch}}/d\eta\right|_{\eta\approx
0})^{\alpha}$.

ALICE showed new preliminary direct-photon spectra in Pb–Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=5.02\,\mathrm{TeV}$ [9]. The results of two
independent methods, the photon conversion method and the virtual photons
method, were found to be in agreement, see Fig. 3. Photons from initial hard
parton-parton scatterings as obtained from a scaled next-to-leading-order
perturbative QCD (pQCD) calculation are sufficient to explain the data. A
calculation with additional contributions from thermal and preequilibrium
photons also agrees with the data. To establish a signal of thermal photons in
nucleus-nucleus collisions at the LHC thus requires a further reduction of the
experimental uncertainties.

Figure 3: Direct-photon excess ratio
$R_{\gamma}=\gamma_{\mathrm{total}}/\gamma_{\mathrm{decay}}\equiv
1+\gamma_{\mathrm{direct}}/\gamma_{\mathrm{decay}}$ in central Pb–Pb
collisions at $\sqrt{s_{\mathrm{NN}}}=5.02~{}\mathrm{TeV}$. A scaled pQCD
calculation (gray dashed line) and a model with additional photon production
sources including thermal radiation fom the QGP (orange dashed line) are
compared to the data.

With the calculation of [5] and the new direct-photon spectra presented at
this conference, the differences between data and theory in terms of the
yields do not appear very significant anymore, see Fig. 4. At low
$p_{T}\lesssim 3~{}\mathrm{GeV}/c$ where thermal and preequilibrium photons
are expected, one finds a good agreement of the model calculation with the LHC
measurements. At RHIC, the model predictions agrees with the STAR data [10] at
low $p_{T}$. The calculation is systematically below the PHENIX data, however,
within the experimental systematic uncertainties there is no significant
deviation. While the puzzle appears almost resolved concerning the yields, it
remains a challenge to explain the large azimuthal anisotropy ($v_{2}$)
measured by PHENIX.

Figure 4: Comparison of direct-photon transverse momentum spectra measured at
RHIC and the LHC to a model which includes thermal and preequilibrium photons
in addition to pQCD photons.

##  3 Dileptons

ALICE presented a preliminary dielectron mass spectrum in central Pb–Pb
collisions at $\sqrt{s_{\mathrm{NN}}}=5.02~{}\mathrm{TeV}$ at this conference,
see Fig. 5 [11]. To establish a signal of thermal radiation in the low mass
region (LMR, $m_{ee}\lesssim 1\,\mathrm{GeV}/c^{2}$) and in the intermediate
mass region (IMR, $1\lesssim m_{ee}\lesssim 3\,\mathrm{GeV}/c^{2}$) at LHC
energies, a detailed understanding of $e^{+}e^{-}$ pairs resulting from
correlated semileptonic decays of open charm and beauty hadrons is crucial
($c\bar{c}\to e^{+}e^{-}$, $b\bar{b}\to e^{+}e^{-}$). In Fig. 5 two versions
of the cocktail, with and without nuclear modification of the $c\bar{c}\to
e^{+}e^{-}$ and $b\bar{b}\to e^{+}e^{-}$ contribution are compared to the
data. In both cases no clear enhancement above the cocktail is observed.
However, predictions that include thermal QGP radiation and a medium-modified
$\rho$ also appear to be consistent with the data. Separation of the prompt
thermal signal from the non-prompt background of semileptonic decays based on
the distance of closest approach of the electron and positron tracks to the
primary interaction point will play a crucial role in the extraction of a
thermal signal.

Figure 5: Dielectron invariant mass distribution in central Pb–Pb collisions
at $\sqrt{s_{\mathrm{NN}}}=5.02~{}\mathrm{TeV}$ after subtraction of
uncorrelated background. The data are compared to the expected yield
(“cocktail”) from the decay of hadrons (light flavor hadrons and $J/\psi$) and
from $e^{+}e^{-}$ pairs resulting from correlated semileptonic open charm and
open beauty hadron decays. The two lower panels show the comparison to two
versions of the cocktail, with and without nuclear modifications of the
$c\bar{c}\to e^{+}e^{-}$ and $b\bar{b}\to e^{+}e^{-}$ contribution.

##  4 Future measurements

One of the main current goals in the study of electromagnetic probes is to
obtain a global picture of high-energy nucleus-nucleus collisions in which
both electromagnetic observables as well as light and heavy quark observables
can be naturally explained. This entails the determination of a unique signal
of thermal radiation as well as the extraction of an effective medium
temperature. An effective medium temperature above the critical temperature
$T_{\mathrm{pc}}\approx 156\,\mathrm{MeV}$ would provide additional evidence
for the formation of a QGP. To improve our understand of the QCD phase
diagram, future measurements of electromagnetic probes at vanish and high
baryo-chemical potential $\mu_{\mathrm{B}}$ play a crucial role, see Tab. 1.
These include, e.g., the study of the caloric curve ($T_{\mathrm{eff}}$ as a
fct. of $\sqrt{s_{\mathrm{NN}}}$) to find evidence for a first-order phase
transition at high $\mu_{B}$. In addition, real and virtual photons at very
low $p_{T}$ (and masses) could provide information about the electrical
conductivity of the QGP. Moreover, the long-standing puzzle of the measured
excess of ultra-soft photons above predictions based on Low’s theorem [12]
should finally be resolved.

|  High $\mu_{\mathrm{B}}$ |  Vanishing $\mu_{\mathrm{B}}$
---|---|---
Thermal radiation
($\gamma$, $\gamma^{*}$ spectra) |  STAR, NA60+, CBM, NICA |  PHENIX, STAR, ALICE 2, ALICE 3
Caloric curve
($T_{\mathrm{eff}}$ in IMR vs. $\sqrt{s_{\mathrm{NN}}}$) |  STAR, NA60+, CBM, NICA |
Mechanism of chiral symmetry resoration, $\rho$-$a_{1}$ mixing |  CBM, NA60+ |  STAR, ALICE 3
Dilepton + direct photon $v_{n}$ |  CBM, HADES, NA60+ |  STAR, (ALICE 2), ALICE 3
Time dep. of $T_{\mathrm{eff}}$,
preequilibrium contrib. |  |  ALICE 3
Ultra-soft photons
(test of Low’s theorem) |  |  ALICE 3
Electrical conductivity |  HADES |  STAR, ALICE 3
Table 1: Possible future measurements of electromagnetic observables at high
and vanishing baryochemical potential $\mu_{\mathrm{B}}$ along with the
corresponding experiments [13, 14].

## References

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