# Complex symmetric weighted composition operators on the space
$\mathcal{H}^{2}_{2}(\mathbb{D})$

Molla Basir Ahamed∗ and Taimur Rahman  Department of Mathematics, Jadavpur
University, Kolkata-700032, West Bengal, India.
<EMAIL_ADDRESS>Department of Mathematics, Jadavpur
University, Kolkata-700032, West Bengal, India.
<EMAIL_ADDRESS>
(Date: Received: xxxxx;
1 Corresponding author
File:- Paper-AT-5-2023)

###### Abstract.

In this paper, we introduce a new norm for $\mathcal{S}^{2}(\mathbb{D})$,
encompassing functions whose first and second derivatives belong to both the
Hardy space $\mathcal{H}^{2}(\mathbb{D})$ and the classical Bergman space
$\mathcal{A}^{2}(\mathbb{D})$. Moreover, we present some basic properties of
the space $\mathcal{H}^{2}_{2}(\mathbb{D})$ and subsequently establish
conditions for symbols $\phi$ and $\Psi$ to provide $W_{\Psi,\phi}$ complex
symmetric, employing a unique conjugation $\mathcal{J}$.

###### Key words and phrases:

Composition Operator, Complex symmetric, Conjugation, Multiplication operator

###### 1991 Mathematics Subject Classification:

47B33, 47B38, 30H10, 30H50, 30J10

##  1\. Introduction

Let $\mathbb{D}$ represent the open unit disk in the complex plane
$\mathbb{C}$, and let $H(\mathbb{D})$ denote the space comprising all
holomorphic functions in this disk. If $f\in H(\mathbb{D})$ with power series
representation $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$, then the Hardy space is
defined by

$\displaystyle\mathcal{H}^{2}(\mathbb{D})=\bigg{\\{}f\in\
H(\mathbb{D}):||f||^{2}_{\mathcal{H}^{2}}=\sum_{n=0}^{\infty}|a_{n}|^{2}<\infty\bigg{\\}}$

and the classical Bergman space is defined by

$\displaystyle\mathcal{A}^{2}(\mathbb{D})=\bigg{\\{}f\in\
H(\mathbb{D}):||f||^{2}_{\mathcal{A}^{2}}=\sum_{n=0}^{\infty}\frac{|a_{n}|^{2}}{n+1}<\infty\bigg{\\}}.$

The composition operators have been the subject of extensive research on these
spaces in [3, 28]. For other aspects of composition operators, we refer to the
articles [13, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 31, 30] and
references therein. There is the space $\mathcal{S}^{2}(\mathbb{D})$ (see [1,
12]) which encompasses holomorphic functions on $\mathbb{D}$, with the
condition that their first derivative lies in the Hardy space
$\mathcal{H}^{2}(\mathbb{D})$ and is defined as follows:

$\displaystyle\mathcal{S}^{2}(\mathbb{D})=\\{f\in\
H(\mathbb{D}):||f||^{2}_{\mathcal{S}^{2}}=|f(0)|^{2}+||f^{\prime}||^{2}_{\mathcal{H}^{2}}<\infty\\}.$

In [8], Gu and Luo defined an equivalent norm on
$\mathcal{S}^{2}(\mathbb{D})$, and denote the space by
$\mathcal{S}^{2}_{1}(\mathbb{D})$.

$\displaystyle\mathcal{S}^{2}_{1}(\mathbb{D})$ $\displaystyle=\bigg{\\{}f\in\
H(\mathbb{D}):||f||^{2}_{\mathcal{S}^{2}_{1}}=||f||^{2}_{\mathcal{H}^{2}}+\frac{3}{2}||f^{\prime}||^{2}_{\mathcal{H}^{2}}+\frac{1}{2}||f^{\prime}||^{2}_{\mathcal{A}^{2}}<\infty\bigg{\\}}$
$\displaystyle=\bigg{\\{}f\in\
H(\mathbb{D}):||f||^{2}_{\mathcal{S}^{2}_{1}}=\sum_{n\geq
0}\frac{(n+1)(n+2)}{2}|a_{n}|^{2}<\infty\bigg{\\}}.$

This space is called the derivative Hardy space (see [8]). Motivated by the
definition of the derivative Hardy space, Gupta and Malhotra [9] introduced an
equivalent norm for the space $\mathcal{S}^{2}(\mathbb{D})$ and referred to it
as $\mathcal{H}^{2}_{1}(\mathbb{D})$. This space is defined as follows:

$\displaystyle\mathcal{H}^{2}_{1}(\mathbb{D})$ $\displaystyle=\bigg{\\{}f\in\
H(\mathbb{D}):||f||^{2}_{\mathcal{H}^{2}_{1}}=7||f||^{2}_{\mathcal{H}^{2}}+||f^{\prime}||^{2}_{\mathcal{H}^{2}}+||f||^{2}_{\mathcal{A}^{2}}+5||f^{\prime}||^{2}_{\mathcal{A}^{2}}<\infty\bigg{\\}}$
$\displaystyle=\bigg{\\{}f\in\
H(\mathbb{D}):||f||^{2}_{\mathcal{H}^{2}_{1}}=7\sum_{n=0}^{\infty}|a_{n}|^{2}+\sum_{n=0}^{\infty}n^{2}|a_{n}|^{2}+\sum_{n=0}^{\infty}\frac{|a_{n}|^{2}}{n+1}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+5\sum_{n=0}^{\infty}n|a_{n}|^{2}<\infty\bigg{\\}}$
$\displaystyle=\bigg{\\{}f\in\
H(\mathbb{D}):||f||^{2}_{\mathcal{H}^{2}_{1}}=\sum_{n\geq
0}\frac{(n+2)^{3}}{n+1}|a_{n}|^{2}<\infty\bigg{\\}}.$

A reproducing kernel Hilbert space, denoted as $\mathcal{H}$, is characterized
by its inner product $\langle.,.\rangle_{\mathcal{H}}$, which has the property
that for every $w\in\mathbb{D}$, there is a function $K_{w}\in\mathcal{H}$,
called the point evaluation kernel, such that $f(w)=\langle
f,K_{w}\rangle_{\mathcal{H}}$ for all $f\in\mathcal{H}$. The Hardy space,
Bergman space, $\mathcal{S}^{2}(\mathbb{D})$ space, and
$\mathcal{H}^{2}_{1}(\mathbb{D})$ space all belong to the class of reproducing
kernel Hilbert spaces, and their respective kernels are defined as follows:

$\displaystyle\begin{cases}&K_{w}(z)=\dfrac{1}{(1-\overline{w}z)}\;\mbox{on}\;\mathcal{H}^{2}(\mathbb{D});\\\
&K_{w}(z)=\dfrac{1}{(1-\overline{w}z)^{2}}\;\mbox{on}\;\mathcal{A}^{2}(\mathbb{D});\\\
&K_{w}(z)=\displaystyle\sum_{n=0}^{\infty}\frac{(\overline{w}z)^{n}}{1+n^{2}},\;\mbox{on}\;\mathcal{S}^{2}(\mathbb{D});\\\
&K_{w}(z)=\displaystyle\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{3}}(\overline{w}z)^{n}\;\mbox{on}\;\mathcal{H}^{2}_{1}(\mathbb{D}).\end{cases}$

With the definition of the derivative Hardy space as our motivation, our aim
is to establish an equivalent norm for the space
$\mathcal{S}^{2}(\mathbb{D})$, which we will refer to as
$\mathcal{H}^{2}_{2}(\mathbb{D})$. The new space
$\mathcal{H}^{2}_{2}(\mathbb{D})$ is defined as follows:

$\displaystyle\mathcal{H}^{2}_{2}(\mathbb{D})$ $\displaystyle=\bigg{\\{}f\in\
H(\mathbb{D}):||f||^{2}_{\mathcal{H}^{2}_{2}}=31||f||^{2}_{\mathcal{H}^{2}}+41||f^{\prime}||^{2}_{\mathcal{H}^{2}}+||f^{\prime\prime}||^{2}_{\mathcal{H}^{2}}+||f||^{2}_{\mathcal{A}^{2}}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+49||f^{\prime}||^{2}_{\mathcal{A}^{2}}+11||f^{\prime\prime}||^{2}_{\mathcal{A}^{2}}<\infty\bigg{\\}}$
$\displaystyle=\bigg{\\{}f\in\
H(\mathbb{D}):||f||^{2}_{\mathcal{H}^{2}_{2}}=31\sum_{n=0}^{\infty}|a|^{2}+41\sum_{n=0}^{\infty}n^{2}|a|^{2}+\sum_{n=0}^{\infty}n^{2}(n-1)^{2}|a|^{2}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\sum_{n=0}^{\infty}\frac{|a|^{2}}{n+1}+49\sum_{n=0}^{\infty}n|a|^{2}+11\sum_{n=0}^{\infty}n^{2}(n-1)|a|^{2}<\infty\bigg{\\}}$
$\displaystyle=\bigg{\\{}f\in\
H(\mathbb{D}):||f||^{2}_{\mathcal{H}^{2}_{1}}=\sum_{n\geq
0}\frac{(n+2)^{5}}{n+1}|a_{n}|^{2}<\infty\bigg{\\}}.$

For $f,g\in\mathcal{H}^{2}_{2}(\mathbb{D})$, with power series representation
$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ and $g(z)=\sum_{n=0}^{\infty}b_{n}z^{n}$,
the inner product on $\mathcal{H}^{2}_{2}(\mathbb{D})$ is defined by

$\displaystyle\langle f,g\rangle_{\mathcal{H}^{2}_{2}}=$ $\displaystyle
31\langle f,g\rangle_{\mathcal{H}^{2}}+41\langle
f^{\prime},g^{\prime}\rangle_{\mathcal{H}^{2}}+\langle
f^{\prime\prime},g^{\prime\prime}\rangle_{\mathcal{H}^{2}}+\langle
f,g\rangle_{\mathcal{A}^{2}}+49\langle
f^{\prime},g^{\prime}\rangle_{\mathcal{A}^{2}}+11\langle
f^{\prime\prime},g^{\prime\prime}\rangle_{\mathcal{A}^{2}}$ $\displaystyle=$
$\displaystyle
31\sum_{n=0}^{\infty}a_{n}\overline{b_{n}}+41\sum_{n=0}^{\infty}n^{2}a_{n}\overline{b_{n}}+\sum_{n=0}^{\infty}n^{2}(n-1)^{2}a_{n}\overline{b_{n}}+\sum_{n=0}^{\infty}\frac{a_{n}\overline{b_{n}}}{n+1}+49\sum_{n=0}^{\infty}na_{n}\overline{b_{n}}$
$\displaystyle+11\sum_{n=0}^{\infty}n^{2}(n-1)a_{n}\overline{b_{n}}$
$\displaystyle=$ $\displaystyle\sum_{n\geq
0}\frac{(n+2)^{5}}{n+1}a_{n}\overline{b_{n}}.$

For $w\in\mathbb{D}$, we define

$\displaystyle
K_{w}(z):=\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}(\overline{w}z)^{n}\;\;\mbox{for}\;\;z\in\mathbb{D}.$

An easy computation shown that

$\displaystyle\langle
f(z),K_{w}(z)\rangle_{\mathcal{H}^{2}_{2}}=\sum_{n=0}^{\infty}\frac{(n+2)^{5}}{n+1}a_{n}\overline{\bigg{(}\frac{(n+1)}{(n+2)^{5}}\overline{w}^{n}\bigg{)}}=\sum_{n=0}^{\infty}a_{n}w^{n}=f(w),z\in\mathbb{D}.$

Hence, $K_{w}$ is the reproducing kernel for the space
$\mathcal{H}^{2}_{2}(\mathbb{D})$.

In this paper, we initiate by introducing some fundamental properties of the
space $\mathcal{H}^{2}_{2}(\mathbb{D})$. We then proceed to investigate the
criteria under which the composition operator
$C_{\phi}:\mathcal{H}^{2}_{2}(\mathbb{D})\to\mathcal{H}^{2}_{2}(\mathbb{D})$,
symbolized by $\phi$ and defined as $C_{\phi}f=f\circ\phi$ where $f$ is
holomorphic on $\mathbb{D}$ and $\phi$ is a holomorphic map from $\mathbb{D}$
to itself, qualifies as a Hilbert-Schmidt operator.

The study of complex symmetric operators on Hilbert spaces was initiated by
Gartia and Putinar [6, 7]. Since then, this area has attracted significant
research attention in complex analysis and operator theory. Complex symmetric
operators include various instances, such as normal operators, Volterra
integration operators, Hankel operators, Toeplitz operators, and more. Let
$\Psi:\mathbb{D}\to\mathbb{C}$ and $\phi:\mathbb{D}\to\mathbb{D}$ be
holomorphic functions. Then the weighted composition operator $W_{\Psi,\phi}$
is defined as

$\displaystyle W_{\Psi,\phi}f=\Psi\cdot f\circ\phi.$

Several authors in [2, 5, 14] studied normal and complex symmetric weighted
composition operators with the special conjugation on the Hardy space
$\mathcal{H}^{2}(\mathbb{D})$. Khoi and Lim [15] studied the complex symmetry
of weighted composition operators on the Hilbert space of analytic functions
$\mathcal{H}_{\gamma}(\mathbb{D})$ over $\mathbb{D}$ with reproducing kernels
$K^{(\gamma)}_{w}=(1-\overline{w}z)^{-\gamma}$, where $\gamma\in\mathbb{N}$.
Hai and Khoi [10, 11] established the conditions under which a weighted
composition operator exhibits complex symmetry in the Fock space (see [10,
11]).

The outline of the paper is as follows. In Section 2, we characterize the
space $\mathcal{H}^{2}_{2}(\mathbb{D})$ by exploring the basis. In the same
spirit, In Section 3, we study multiplication operator on the space
$\mathcal{H}^{2}_{2}(\mathbb{D})$. In the Section 4, of this paper, we explore
the properties of $\phi$ and $\Psi$ concerning the weighted composition
operator $W_{\phi,\Psi}$ in the space $\mathcal{H}^{2}_{2}(\mathbb{D})$, with
the specific aim of confirming its complex symmetry. The proof of the main
results are discussed in each sections.

##  2\. Characterization of the space $\mathcal{H}^{2}_{2}(\mathbb{D})$

In this section, we will provide an account of some essential properties of
the space $\mathcal{H}^{2}_{2}(\mathbb{D})$ that will be drawn upon in later
sections.

######  Proposition 2.1.

The space $\mathcal{H}^{2}_{2}(\mathbb{D})$ satisfies the following:

  1. (a)

$\mathcal{H}^{2}_{2}(\mathbb{D})$ is a normed space.

  2. (b)

polynomials are contained in $\mathcal{H}^{2}_{2}(\mathbb{D})$.

  3. (c)

$\mathcal{H}^{2}_{2}(\mathbb{D})$ is a proper subset of
$\mathcal{H}^{2}(\mathbb{D})$,

The following example serves as proof of the proper inclusion of part (c) in
Proposition 2.1.

######  Example 2.1.

Let
$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}=\sum_{n=0}^{\infty}\sqrt{(n+1)/(n+2)^{5}}z^{n}$.
Then

$\displaystyle||f||_{\mathcal{H}^{2}}=\sum_{n=0}^{\infty}|a_{n}|^{2}=\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}=\zeta(4)-\zeta(5)\approxeq
0.04539<\infty.$

Since

$\displaystyle\sum_{n=0}^{\infty}\frac{(n+2)^{5}}{n+1}|a_{n}|^{2}=\sum_{n=0}^{\infty}\frac{(n+2)^{5}}{n+1}\frac{n+1}{(n+2)^{5}}=\sum_{n=0}^{\infty}1=\infty,$

so $f\notin\mathcal{H}^{2}_{2}(\mathbb{D})$. Hence, $f$ belongs to
$\mathcal{H}^{2}(\mathbb{D})$ but it does not belong to
$\mathcal{H}^{2}_{2}(\mathbb{D})$.

To gain a comprehensive understanding of any space, it is essential to be
familiar with its basis. For instance, when attempting to demonstrate that an
operator defined on a given space is a Hilbert-Schmidt operator, it is
necessary to have an orthonormal basis for that space. Since space
$\mathcal{H}^{2}_{2}(\mathbb{D})$ has been recently constructed, understanding
its basis is crucial for a thorough comprehension of this new space. The
following theorem provides an orthonormal basis for the space
$\mathcal{H}^{2}_{2}(\mathbb{D})$.

######  Theorem 2.1.

Suppose $e_{n}(z)=\sqrt{(n+1)/(n+2)^{5}}z^{n}$, $z\in\mathbb{D}$ and $n$ is
any non-negative integer. Then the set
$\\{e_{n}(z):n\in\mathbb{N}\cup\\{0\\}\\}$ forms an orthonormal basis for
$\mathcal{H}^{2}_{2}(\mathbb{D})$.

###### Proof of Theorem 2.1.

For any non-negative integers $m$ and $n$, we have

$\displaystyle\langle z^{m},z^{n}\rangle_{\mathcal{H}^{2}_{2}}=31$
$\displaystyle\langle z^{m},z^{n}\rangle_{\mathcal{H}^{2}}+41\langle
mz^{m-1},nz^{n-1}\rangle_{\mathcal{H}^{2}}+\langle
m(m-1)z^{m-2},n(n-1)z^{n-2}\rangle_{\mathcal{H}^{2}}$ (2.1) $\displaystyle+$
$\displaystyle\langle z^{m},z^{n}\rangle_{\mathcal{A}^{2}}+49\langle
mz^{m-1},nz^{n-1}\rangle_{\mathcal{A}^{2}}+11\langle
m(m-1)z^{m-2},n(n-1)z^{n-2}\rangle_{\mathcal{A}^{2}}.$

For $n=m$, we obtain that

$\displaystyle\langle z^{m},z^{n}\rangle_{\mathcal{H}^{2}_{2}}=$
$\displaystyle 31\cdot
1+41n^{2}+n^{2}(n-1)^{2}+\frac{1}{(n+1)}+49n+11n^{2}(n-1)$ $\displaystyle=$
$\displaystyle\frac{n^{5}+10n^{4}+40n^{3}+80n^{2}+80n+32}{(n+1)}$
$\displaystyle=$ $\displaystyle\frac{(n+2)^{5}}{(n+1)}.$

Therefore, it is easy to see that

$\displaystyle\langle
z^{m},z^{n}\rangle_{\mathcal{H}^{2}_{2}}=\begin{cases}\frac{(n+2)^{5}}{(n+1)}\;\;\;\;\;\mbox{if}\;m=n\\\
\;\;0\;\;\;\;\;\;\;\;\;\;\mbox{if}\;m\neq n.\end{cases}$

Hence,
$\\{e_{n}(z):n\in\mathbb{N}\cup\\{0\\}\\}=\\{\sqrt{(n+1)/(n+2)^{5}}z^{n}:n\in\mathbb{N}\cup\\{0\\}\\}$
forms an orthonormal set in $\mathcal{H}^{2}_{2}(\mathbb{D})$.

Let $f\in\mathcal{H}^{2}_{2}(\mathbb{D})$ be such that $f$ is orthogonal to
$e_{n}(z)$ for all $n\geq 0$ and let the power series expansion of $f$ be
given by $f(z)=\sum_{m=0}^{\infty}a_{m}z^{m}$. Then for $n\geq 0$, $\langle
f(z),e_{n}(z)\rangle=\langle\sum_{m=0}^{\infty}a_{m}z^{m},e_{n}(z)\rangle=0$.
This gives $a_{m}=0$ for all $m\geq 0$. Thus, $f\equiv 0$ and hence the set
$\\{e_{n}(z):n\in\mathbb{N}\cup\\{0\\}\\}$ forms an orthonormal basis for
$\mathcal{H}^{2}_{2}(\mathbb{D})$. ∎

##  3\. Multiplication operator on the space $\mathcal{H}^{2}_{2}(\mathbb{D})$

Let $\mathcal{B}$ be a Banach space consisting of holomorphic functions
defined on the open unit disk $\mathbb{D}$.Then for any function
$f\in\mathcal{B}$, the multiplication operator $M_{f}$ is defined as follows:

$\displaystyle M_{f}g=fg$

for every holomorphic function $g$ defined on $\mathbb{D}$. Multiplication
operators have a key role to play in the analysis of weighted composition
operators in holomorphic function spaces. The aim is to find a connection
between the properties of the multiplication operator $M_{f}$ and those of its
corresponding symbol $f$. In this section, we will examine the circumstances
that determine the boundedness of the multiplication operator $M_{f}$ on
$\mathcal{H}^{2}_{2}(\mathbb{D})$.

In a recent study, Gu and Luo (see [8, Proposition 2.1]) demonstrated that for
functions belonging to $\mathcal{S}^{2}_{1}(\mathbb{D})$, it holds that
$||f||{\infty}\leq\sqrt{2}||f||{\mathcal{S}^{2}_{1}}$ where the constant
$\sqrt{2}$ is sharp. Subsequently, Gupta and Malhotra (in Theorem-3.1, as
referenced in [9]) extended this result by establishing that for functions in
$\mathcal{H}^{2}_{1}(\mathbb{D})$, the norm
$||f||{\infty}\leq\sqrt{\pi^{2}/6-\zeta(3)}||f||{\mathcal{H}^{2}_{1}}$, and it
was confirmed that the constant $\sqrt{\pi^{2}/6-\zeta(3)}$ is sharp.

In the following result, we will introduce an analogous counterpart to
Proposition 2.1 mentioned in [8] and Theorem 3.1 discussed in [9].

######  Theorem 3.1.

Suppose $f\in\mathcal{H}^{2}_{2}(\mathbb{D})$. Then
$||f||_{\infty}\leq\sqrt{\zeta(4)-\zeta(5)}||f||_{\mathcal{H}^{2}_{2}}$.
Furthermore, $\sqrt{\zeta(4)-\zeta(5)}$ is the sharp constant($\zeta$ is the
Riemann zeta function and $\zeta(4)\approxeq 1.08232$ and $\zeta(5)\approxeq
1.03693$).

###### Proof of Theorem 3.1.

Suppose $f\in\mathcal{H}^{2}_{2}(\mathbb{D})$ with power series representation
$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$. Then

$\displaystyle|f(z)|=\bigg{|}\sum_{n=0}^{\infty}a_{n}z^{n}\bigg{|}\leq\sum_{n=0}^{\infty}|a_{n}||z|^{n}=\sum_{n=0}^{\infty}\sqrt{\frac{(n+2)^{5}}{n+1}}|a_{n}|\sqrt{\frac{n+1}{(n+2)^{5}}}|z|^{n}.$

By making use of the Cauchy-Schwarz inequality, we obtain that

$\displaystyle|f(z)|\leq$
$\displaystyle\sqrt{\bigg{(}\sum_{n=0}^{\infty}\frac{(n+2)^{5}}{n+1}|a_{n}|^{2}\bigg{)}}\sqrt{\bigg{(}\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}|z|^{2n}\bigg{)}}$
$\displaystyle=$
$\displaystyle\sqrt{\bigg{(}\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}|z|^{2n}\bigg{)}}||f||_{\mathcal{H}^{2}_{2}}$
$\displaystyle\leq$
$\displaystyle\sqrt{\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}}||f||_{\mathcal{H}^{2}_{2}}$
$\displaystyle=$
$\displaystyle\sqrt{(\zeta(4)-\zeta(5))}||f||_{\mathcal{H}^{2}_{2}}.$

Hence,
$||f||_{\infty}\leq\sqrt{\zeta(4)-\zeta(5)}||f||_{\mathcal{H}^{2}_{2}}$.

If we consider the function $f(z)=\sum_{n=0}^{\infty}((n+1)/(n+2)^{5})z^{n}$,
then
$||f||_{\infty}=f(1)=\sum_{n=0}^{\infty}(n+1)/(n+2)^{5}=\zeta(4)-\zeta(5)$ and
$||f||_{\mathcal{H}^{2}_{2}}=\sqrt{\zeta(4)-\zeta(5)}$. Hence, we conclude
that the constant $\sqrt{\zeta(4)-\zeta(5)}$ is sharp. This completes the
proof. ∎

Operator algebras constitute a core subject within functional analysis, a
mathematical discipline concerned with vector spaces endowed with a
topological structure. They offer a structure for comprehending the
characteristics of linear operators, with a special focus on operators that
are either bounded or unbounded. In 1969, the paper [16] presented a proof
that $D_{\alpha}$ is an algebra for $\alpha>1$. Furthermore, it was noted in
[17] that $\mathcal{S}^{2}_{2}(\mathbb{D})$ and its analogous counterparts
$D_{n}$ for integer values of $n\geq 1$ are also algebras, with reference to
Russian literature. The research presented in [1] and [4] established that
both $\mathcal{S}^{2}(\mathbb{D})$ and $\mathcal{S}^{2}_{2}(\mathbb{D})$
possess the characteristics of algebras. Recently, Guo and Luo [8]
demonstrated that $\mathcal{S}^{2}_{1}(\mathbb{D})$ is an algebra, and
subsequently, Gupta and Malhotra [9] established that
$\mathcal{H}^{2}_{1}(\mathbb{D})$ is also an algebra. Here, we will provide
proof demonstrating that $\mathcal{H}^{2}_{2}(\mathbb{D})$ constitutes an
algebra.

######  Proposition 3.1.

The space $\mathcal{H}^{2}_{2}(\mathbb{D})$ is an algebra.

###### Proof of Proposition 3.1.

Suppose $f\in\mathcal{H}^{2}_{2}(\mathbb{D})$ with power series representation
$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$. By simple computation, we obtain the
following inequality

$\displaystyle\bigg{|}f(z)-\sum_{n=0}^{N}a_{n}z^{n}\bigg{|}\leq$
$\displaystyle\bigg{|}\sum_{n=N+1}^{\infty}\sqrt{\frac{(n+2)^{5}}{n+1}}a_{n}\sqrt{\frac{n+1}{(n+2)^{5}}}z^{n}\bigg{|}$
$\displaystyle\leq$
$\displaystyle\sum_{n=N+1}^{\infty}\sqrt{\frac{(n+2)^{5}}{n+1}}|a_{n}|\sqrt{\frac{n+1}{(n+2)^{5}}}|z|^{n}$
$\displaystyle\leq$
$\displaystyle\sqrt{\sum_{n=N+1}^{\infty}\frac{(n+2)^{5}}{n+1}|a_{n}|^{2}}\sqrt{\sum_{n=N+1}^{\infty}\frac{n+1}{(n+2)^{5}}|z|^{2n}}$
$\displaystyle\leq$
$\displaystyle||f||_{\mathcal{H}^{2}_{2}}\sqrt{\sum_{n=N+1}^{\infty}\frac{n+1}{(n+2)^{5}}}$
$\displaystyle=$
$\displaystyle||f||_{\mathcal{H}^{2}_{2}}\sqrt{\sum_{n=N+1}^{\infty}\frac{1}{(n+2)^{4}}-\sum_{n=N+1}^{\infty}\frac{1}{(n+2)^{5}}}$
$\displaystyle=$
$\displaystyle||f||_{\mathcal{H}^{2}_{2}}\sqrt{\sum_{n=0}^{\infty}\frac{1}{(n+N+3)^{4}}-\sum_{n=0}^{\infty}\frac{1}{(n+N+3)^{5}}}$
$\displaystyle=$
$\displaystyle||f||_{\mathcal{H}^{2}_{2}}\sqrt{\frac{1}{6}Polygamma[3,N+3]-(-\frac{1}{24})Polygamma[4,N+3]}$
$\displaystyle=$
$\displaystyle||f||_{\mathcal{H}^{2}_{2}}\sqrt{\frac{1}{24}(4Polygamma[3,N+3]+Polygamma[4,N+3])}$

(note that $Polygamma[n,z]$ represents the $n^{th}$ derivative, denoted as
$\Psi^{n}(z)$, of the Digamma function $\Psi(z)$. The Digamma function
$\Psi(z)$ serves as the logarithmic derivative of the gamma function
$\Gamma(z)$, and we can express it as
$\Psi(z)=(d/dz)\ln(\Gamma(z))=\Gamma^{\prime}(z)/\Gamma(z)$, where
$\Gamma(z)=\int_{0}^{\infty}t^{t-1}e^{-t}dt$. The power series representation
of the Polygamma function is as follows:

$\displaystyle
Polygamma[m,z]:=\Psi^{m}(z)=(-1)^{m+1}m!\sum_{n=0}^{\infty}1/(n+z)^{m+1}.$

It should be noted that this representation is valid for values of $z$ greater
than zero and for all natural numbers $n$.) By utilizing Mathematica software
to observe that the limit of the expression

$\displaystyle\sqrt{\frac{1}{24}(4Polygamma[3,N+3]+Polygamma[4,N+3])}$

approaches zero as $N$ tends to $\infty$, we can conclude that $f(z)$ is the
uniform limit of its partial sum. Consequently, $f(z)$ maintains a continuous
extension to the boundary of $\mathbb{D}$. The space
$\mathcal{H}^{2}_{2}(\mathbb{D})$ is a subset of the disk algebra
$A(\mathbb{D})$, where $A(\mathbb{D})$ comprises holomorphic functions mapping
from $\mathbb{D}$ to $\mathbb{C}$ that exhibit continuous extensions to the
boundary of $\mathbb{D}$. Let $A^{1}(\mathbb{D})=\\{f\in
A(\mathbb{D}):f^{\prime}(z)\in A(\mathbb{D})\\}$. Then,
$A^{1}(\mathbb{D})\subseteq\mathcal{H}^{2}_{2}(\mathbb{D})\subseteq
A(\mathbb{D})$. Therefore, $\mathcal{H}^{2}_{2}(\mathbb{D})$ is an algebra.
This completes the proof. ∎

In a recent investigation, Gu and Luo (see Theorem 2.2, [8]) established that
for $f$ and $g$ in $\mathcal{S}^{2}_{1}(\mathbb{D})$, the inequality

$\displaystyle||fg||{\mathcal{S}^{2}_{1}}\leq
2\sqrt{2}||f||{\mathcal{S}^{2}_{1}}||g||{\mathcal{S}^{2}_{1}}$

holds. Subsequently, Gupta and Malhotra (see Theorem 3.4, [9]) established
that for $f$ and $g$ belonging to $\mathcal{H}^{2}_{1}(\mathbb{D})$, the
inequality

$\displaystyle||fg||^{2}_{\mathcal{H}^{2}_{1}}\leq
6(\pi^{2}/6-\zeta(3))||f||^{2}_{\mathcal{H}^{2}_{1}}||g||^{2}_{\mathcal{H}^{2}_{1}}$

is valid. By a similar line of reasoning to that employed in Theorem 2.2 from
[8] and Theorem 3.4 from [9], our objective in this section is to establish
that, for $f$ and $g$ in $\mathcal{H}^{2}_{2}(\mathbb{D})$, the inequality

$\displaystyle||fg||_{\mathcal{H}^{2}_{2}}\leq
2\sqrt{2}\sqrt{\zeta(4)-\zeta(5)}||f||_{\mathcal{H}^{2}_{2}}||g||_{\mathcal{H}^{2}_{2}}$

holds true.

We obtain the following result for the class $\mathcal{H}^{2}_{2}(\mathbb{D})$
which is similar to [8, Theorem 2.3] and [9, Theorem 3.5].

######  Theorem 3.2.

If $f,g\in\mathcal{H}^{2}_{2}(\mathbb{D})$, then
$||fg||_{\mathcal{H}^{2}_{2}}\leq
2\sqrt{2}\sqrt{\zeta(4)-\zeta(5)}||f||_{\mathcal{H}^{2}_{2}}||g||_{\mathcal{H}^{2}_{2}}$.
(note that $\zeta(4)-\zeta(5)\approxeq 0.04539$).

###### Proof of Theorem 3.2.

Suppose that $f,g\in\mathcal{H}^{2}_{2}(\mathbb{D})$. Then a straightforward
computation leads to the following

$\displaystyle||fg||^{2}_{\mathcal{H}^{2}_{2}}=$ $\displaystyle
31||fg||^{2}_{\mathcal{H}^{2}}+41||f^{\prime}g+fg^{\prime}||^{2}_{\mathcal{H}^{2}}+||(fg)^{\prime\prime}||^{2}_{\mathcal{H}^{2}}+||fg||^{2}_{\mathcal{A}^{2}}+49||f^{\prime}g+fg^{\prime}||^{2}_{\mathcal{A}^{2}}$
$\displaystyle+11||(fg)^{\prime\prime}||^{2}_{\mathcal{A}^{2}}$
$\displaystyle\leq$ $\displaystyle
31(||f||^{2}_{\infty}||g||^{2}_{\mathcal{H}^{2}}+||f||^{2}_{\mathcal{H}^{2}}||g||^{2}_{\infty})+41(2||f||^{2}_{\infty}||g^{\prime}||^{2}_{\mathcal{H}^{2}}+2||f^{\prime}||^{2}_{\mathcal{H}^{2}}||g||^{2}_{\infty})$
$\displaystyle+(4||f||^{2}_{\infty}||g^{\prime\prime}||^{2}_{\mathcal{H}^{2}}+2||f^{\prime\prime}||^{2}_{\mathcal{H}^{2}}||g||^{2}_{\infty}+8||f||^{2}_{\infty}||g^{\prime}||^{2}_{\mathcal{H}^{2}}+8||f^{\prime}||^{2}_{\mathcal{H}^{2}}||g||^{2}_{\infty})$
$\displaystyle+(||f||^{2}_{\infty}||g||^{2}_{\mathcal{A}^{2}}+||f||^{2}_{\mathcal{A}^{2}}||g||^{2}_{\infty})+49(2||f||^{2}_{\infty}||g^{\prime}||^{2}_{\mathcal{A}^{2}}+2||f^{\prime}||^{2}_{\mathcal{A}^{2}}||g||^{2}_{\infty})$
$\displaystyle+11(4||f||^{2}_{\infty}||g^{\prime\prime}||^{2}_{\mathcal{H}^{2}}+2||f^{\prime\prime}||^{2}_{\mathcal{H}^{2}}||g||^{2}_{\infty}+8||f||^{2}_{\infty}||g^{\prime}||^{2}_{\mathcal{H}^{2}}+8||f^{\prime}||^{2}_{\mathcal{H}^{2}}||g||^{2}_{\infty})$
$\displaystyle=$
$\displaystyle||f||^{2}_{\infty}(31||g||^{2}_{\mathcal{H}^{2}}+41||g^{\prime}||^{2}_{\mathcal{H}^{2}}+||g^{\prime\prime}||^{2}_{\mathcal{H}^{2}}+||g||^{2}_{\mathcal{A}^{2}}+49||g^{\prime}||^{2}_{\mathcal{A}^{2}}+11||g^{\prime\prime}||^{2}_{\mathcal{A}^{2}})$
$\displaystyle+||g||^{2}_{\infty}(31||f||^{2}_{\mathcal{H}^{2}}+41||f^{\prime}||^{2}_{\mathcal{H}^{2}}+||f^{\prime\prime}||^{2}_{\mathcal{H}^{2}}+||f||^{2}_{\mathcal{A}^{2}}+49||f^{\prime}||^{2}_{\mathcal{A}^{2}}+11||f^{\prime\prime}||^{2}_{\mathcal{A}^{2}})$
$\displaystyle+||f||^{2}_{\infty}(49||g^{\prime}||^{2}_{\mathcal{H}^{2}}+3||g^{\prime\prime}||^{2}_{\mathcal{H}^{2}}+137||g^{\prime}||^{2}_{\mathcal{A}^{2}}+33||g^{\prime\prime}||^{2}_{\mathcal{A}^{2}})+||g||^{2}_{\infty}(49||f^{\prime}||^{2}_{\mathcal{H}^{2}}$
$\displaystyle+||f^{\prime\prime}||^{2}_{\mathcal{H}^{2}}+137||f^{\prime}||^{2}_{\mathcal{A}^{2}}+11||f^{\prime\prime}||^{2}_{\mathcal{A}^{2}})$
$\displaystyle\leq$
$\displaystyle||f||^{2}_{\infty}||g||^{2}_{\mathcal{H}^{2}_{2}}+||g||^{2}_{\infty}||f||^{2}_{\mathcal{H}^{2}_{2}}+3||f||^{2}_{\infty}||g||^{2}_{\mathcal{H}^{2}_{2}}+3||g||^{2}_{\infty}||f||^{2}_{\mathcal{H}^{2}_{2}}$
$\displaystyle=$ $\displaystyle
4(||f||^{2}_{\infty}||g||^{2}_{\mathcal{H}^{2}_{2}}+||g||^{2}_{\mathcal{H}^{2}_{2}}||g||^{2}_{\infty})$
$\displaystyle\leq$ $\displaystyle
8(\zeta(4)-\zeta(5))||f||^{2}_{\mathcal{H}^{2}_{2}}||g||^{2}_{\mathcal{H}^{2}_{2}}.$

Consequently, the desired inequality $||fg||_{\mathcal{H}^{2}_{2}}\leq
2\sqrt{2}\sqrt{(\zeta(4)-\zeta(5))}||f||_{\mathcal{H}^{2}_{2}}||g||_{\mathcal{H}^{2}_{2}}$
is established. This completes the proof. ∎

######  Theorem 3.3.

If $f\in H(\mathbb{D})$, then $M_{f}$ is a bounded linear operator on
$\mathcal{H}^{2}_{2}(\mathbb{D})$ if and only if
$f\in\mathcal{H}^{2}_{2}(\mathbb{D})$. Moreover,

$\displaystyle\max\\{||f||_{\infty},||f||_{\mathcal{H}^{2}_{2}}\\}\leq||M_{f}||\leq
2\sqrt{2}\sqrt{(\zeta(4)-\zeta(5))}||f||_{\mathcal{H}^{2}_{2}}.$

###### Proof of Theorem 3.3.

For all $w\in\mathbb{D}$,
$||M^{*}_{f}K_{w}||=||\overline{f(w)}K_{w}||=|f(w)|||K_{w}||$. Therefore,
$|f(w)|\leq||M^{*}_{f}||=||M_{f}||$ for all $w\in\mathbb{D}$ and hence
$||f||_{\infty}\leq||M_{f}||$. Also,
$||M_{f}||\geq||M_{f}1||=||f||_{\mathcal{H}^{2}_{2}}$. Consequently,
$\max\\{||f||_{\infty},||f||_{\mathcal{H}^{2}_{2}}\\}\leq||M_{f}||$. It
follows from Theorem 3.2 that for all $g\in\mathcal{H}^{2}_{2}(\mathbb{D})$,
we have

$\displaystyle||M_{f}g||=||fg||\leq
2\sqrt{2}\sqrt{(\zeta(4)-\zeta(5))}||f||_{\mathcal{H}^{2}_{2}}||g||_{\mathcal{H}^{2}_{2}}.$

By setting $g$ to be the identity function, we obtain that

$\displaystyle||M_{f}||\leq
2\sqrt{2}\sqrt{(\zeta(4)-\zeta(5))}||f||_{\mathcal{H}^{2}_{2}}.$

This leads to the following conclusion

$\displaystyle\max\\{||f||_{\infty},||f||_{\mathcal{H}^{2}_{2}}\\}\leq||M_{f}||\leq
2\sqrt{2}\sqrt{(\zeta(4)-\zeta(5))}||f||_{\mathcal{H}^{2}_{2}}.$

This completes the proof. ∎

##  4\. Complex symmetry of weighted composition operators on
$\mathcal{H}^{2}_{2}(\mathbb{D})$

Let $\mathcal{B}(\mathcal{H})$ be the algebra of all bounded linear operators
on a separable complex Hilbert space $\mathcal{H}$. A mapping
$T:\mathcal{H}\to\mathcal{H}$ is said to be anti-linear if

$\displaystyle T(ax+by)=\overline{a}T(x)+\overline{b}T(y),$

for all $x,y\in\mathcal{H}$ and for all $a,b\in\mathbb{C}$. A conjugation on
$\mathcal{H}$ is an antilinear operator
$\mathcal{C}:\mathcal{H}\to\mathcal{H}$ which satisfies
$\langle\mathcal{C}x,\mathcal{C}y\rangle_{\mathcal{H}}=\langle
y,x\rangle_{\mathcal{H}}$ for all $x,y\in\mathcal{H}$ and
$\mathcal{C}^{2}=I_{\mathcal{H}}$, where $I_{\mathcal{H}}$ is the identity
operator on ${\mathcal{H}}$. An operator $T\in\mathcal{B}({\mathcal{H}})$ is
said to be complex symmetric if there exists a conjugation $\mathcal{C}$ on
$\mathcal{H}$ such that $T=\mathcal{C}T^{*}\mathcal{C}$, or equivalently,
$T\mathcal{C}=\mathcal{C}T^{*}$. In this case, we say that $T$ is complex
symmetric with conjugation $\mathcal{C}$.

######  Definition 4.1.

An operator $T$ is said to be a Hilbert-Schmidt operator in a Hilbert space
$\mathcal{H}$ if there exists an orthonormal basis $\\{e_{n}\\}_{n\geq 0}$ for
which the series $\sum_{n=0}^{\infty}||Te_{n}||^{2}$ converges to a finite
value.

In [9], Gupta and Malhotra have established that if $\phi$ belongs to the
space $\mathcal{H}^{2}_{1}(\mathbb{D})$ and satisfies $||\phi||_{\infty}<1$,
the operator $C_{\phi}$ is a Hilbert-Schmidt operator. In the next result, it
is shown that the composition operator $C_{\phi}$ on
$\mathcal{H}^{2}_{2}(\mathbb{D})$ is a Hilbert-Schmidt operator, and it is
compact when $\phi$ belongs to $\mathcal{H}^{2}_{2}(\mathbb{D})$ and satisfies
the condition $||\phi||_{\infty}<1$. The proof for the following result
follows a similar line of reasoning as that of [9, Theorem 4.2].

######  Theorem 4.1.

Let $\phi\in\mathcal{H}^{2}_{2}(\mathbb{D})$ be such that
$||\phi||_{\infty}<1$. Then the composition operator $C_{\phi}$ is a Hilbert-
Schmidt operator on $\mathcal{H}^{2}_{2}(\mathbb{D})$.

###### Proof of Theorem 4.1.

In order to establish that the composition operator $C_{\phi}$ is a Hilbert-
Schmidt operator, it is necessary to demonstrate that the series
$\sum_{n=0}^{\infty}||C_{\phi}e_{n}(z)||^{2}$ has a finite sum. An easy
computation shows that

$\displaystyle\sum_{n=0}^{\infty}||C_{\phi}e_{n}(z)||^{2}=\sum_{n=0}^{\infty}\bigg{|}\bigg{|}C_{\phi}\bigg{(}\frac{z^{n}}{||z^{n}||}\bigg{)}\bigg{|}\bigg{|}^{2}=1+\sum_{n=1}^{\infty}\frac{||\phi^{n}||^{2}_{\mathcal{H}^{2}_{2}}}{||z^{n}||^{2}_{\mathcal{H}^{2}_{2}}}=1+\sum_{n=1}^{\infty}\frac{(n+1)||\phi^{n}||^{2}_{\mathcal{H}^{2}_{2}}}{(n+2)^{5}}.$

and

$\displaystyle||\phi^{n}||^{2}_{\mathcal{H}^{2}_{2}}=$ $\displaystyle
31||\phi\phi^{n-1}||^{2}_{\mathcal{H}^{2}}+41||(\phi^{n})^{\prime}||^{2}_{\mathcal{H}^{2}}+||(\phi^{n})^{\prime\prime}||^{2}_{\mathcal{H}^{2}}+||\phi\phi^{n-1}||^{2}_{\mathcal{A}^{2}}+49||(\phi^{n})^{\prime}||^{2}_{\mathcal{A}^{2}}$
$\displaystyle+11||(\phi^{n})^{\prime\prime}||^{2}_{\mathcal{A}^{2}}$
$\displaystyle=$ $\displaystyle
31||\phi\phi^{n-1}||^{2}_{\mathcal{H}^{2}}+41||n\phi^{n-1}\phi^{\prime}||^{2}_{\mathcal{H}^{2}}+||n(n-1)\phi^{n-2}(\phi^{\prime})^{2}+n\phi^{n-1}\phi^{\prime\prime}||^{2}_{\mathcal{H}^{2}}$
$\displaystyle+||\phi\phi^{n-1}||^{2}_{\mathcal{A}^{2}}+49||n\phi^{n-1}\phi^{\prime}||^{2}_{\mathcal{A}^{2}}+11||n(n-1)\phi^{n-2}(\phi^{\prime})^{2}+n\phi^{n-1}\phi^{\prime\prime}||^{2}_{\mathcal{A}^{2}}$
$\displaystyle\leq$ $\displaystyle
31||\phi^{n-1}||^{2}_{\infty}||\phi||^{2}_{\mathcal{H}^{2}}+41n^{2}||\phi^{n-1}||^{2}_{\infty}||\phi^{\prime}||^{2}_{\mathcal{H}^{2}}+2[n^{2}(n-1)^{2}||\phi^{n-1}||^{2}_{\infty}||\phi^{\prime}||^{2}_{\mathcal{H}^{2}}$
$\displaystyle+n^{2}||\phi^{n-1}||^{2}_{\infty}||\phi^{\prime\prime}||^{2}_{\mathcal{H}^{2}}]+||\phi^{n-1}||^{2}_{\infty}||\phi||^{2}_{\mathcal{A}^{2}}+49n^{2}||\phi^{n-1}||^{2}_{\infty}||\phi^{\prime}||^{2}_{\mathcal{A}^{2}}$
$\displaystyle+22[n^{2}(n-1)^{2}||\phi^{n-1}||^{2}_{\infty}||\phi^{\prime}||^{2}_{\mathcal{A}^{2}}+n^{2}||\phi^{n-1}||^{2}_{\infty}||\phi^{\prime\prime}||^{2}_{\mathcal{A}^{2}}]$
$\displaystyle\leq$
$\displaystyle||\phi^{n-1}||^{2}_{\infty}[31||\phi||^{2}_{\mathcal{H}^{2}}+41n^{2}||\phi^{\prime}||^{2}_{\mathcal{H}^{2}}+n^{2}||\phi^{\prime\prime}||^{2}_{\mathcal{H}^{2}}+||\phi||^{2}_{\mathcal{A}^{2}}+49n^{2}||\phi^{\prime}||^{2}_{\mathcal{A}^{2}}$
$\displaystyle+11n^{2}||\phi^{\prime\prime}||^{2}_{\mathcal{A}^{2}}]+||\phi^{n-1}||^{2}_{\infty}[2n^{2}(n-1)^{2}\phi^{\prime}||^{2}_{\mathcal{H}^{2}}+n^{2}||\phi^{\prime\prime}||^{2}_{\mathcal{H}^{2}}$
$\displaystyle+22n^{2}(n-1)^{2}||\phi^{\prime}||^{2}_{\mathcal{A}^{2}}+11n^{2}||\phi^{\prime\prime}||^{2}_{\mathcal{A}^{2}}]$
$\displaystyle\leq$ $\displaystyle
n^{2}||\phi^{n-1}||^{2}_{\infty}[31||\phi||^{2}_{\mathcal{H}^{2}}+41||\phi^{\prime}||^{2}_{\mathcal{H}^{2}}+||\phi^{\prime\prime}||^{2}_{\mathcal{H}^{2}}+||\phi||^{2}_{\mathcal{A}^{2}}+49||\phi^{\prime}||^{2}_{\mathcal{A}^{2}}+11||\phi^{\prime\prime}||^{2}_{\mathcal{A}^{2}}]$
$\displaystyle+n^{4}||\phi^{n-1}||^{2}_{\infty}[2\phi^{\prime}||^{2}_{\mathcal{H}^{2}}+||\phi^{\prime\prime}||^{2}_{\mathcal{H}^{2}}+22||\phi^{\prime}||^{2}_{\mathcal{A}^{2}}+11||\phi^{\prime\prime}||^{2}_{\mathcal{A}^{2}}]$
$\displaystyle\leq$ $\displaystyle
2n^{4}||\phi||^{2(n-1)}_{\infty}||\phi||^{2}_{\mathcal{H}^{2}_{2}}.$

Thus, it follows that

$\displaystyle\sum_{n=0}^{\infty}\bigg{|}\bigg{|}C_{\phi}\bigg{(}\frac{z^{n}}{||z^{n}||}\bigg{)}\bigg{|}\bigg{|}^{2}$
$\displaystyle\leq
1+\sum_{n=1}^{\infty}\frac{2n^{4}(n+1)||\phi||^{2(n-1)}_{\infty}||\phi||^{2}_{\mathcal{H}^{2}_{2}}}{(n+2)^{5}}$
$\displaystyle\leq
1+\sum_{n=1}^{\infty}\frac{2n^{4}(n+1)||\phi||^{2(n-1)}_{\infty}||\phi||^{2}_{\mathcal{H}^{2}_{2}}}{(n+2)^{5}}$
$\displaystyle\leq
1+2||\phi||^{2}_{\mathcal{H}^{2}_{2}}\sum_{n=1}^{\infty}||\phi||^{2(n-1)}_{\infty}$
$\displaystyle=1+2||\phi||^{2}_{\mathcal{H}^{2}_{2}}(1-||\phi||^{2}_{\infty})^{-1}<\infty.$

Hence, $C_{\phi}$ is a Hilbert-Schmidt operator. ∎

Our next result is for complex symmetric with conjugation $\mathcal{J}$ of the
composition operator $C_{\phi}$.

######  Theorem 4.2.

Let $\phi$ be a self holomorphic map on $\mathbb{D}$ such that the composition
operator
$C_{\phi}:\mathcal{H}^{2}_{2}(\mathbb{D})\to\mathcal{H}^{2}_{2}(\mathbb{D})$
is complex symmetric with conjugation $\mathcal{J}$. Then $\phi(z)=az$, where
$a=\phi^{\prime}(0),\phi(0)=0$. Conversely, if $\phi(z)=az$ maps the unit disk
into itself where $a\in\mathbb{D}$, then the composition operator
$C_{\phi}:\mathcal{H}^{2}_{2}(\mathbb{D})\to\mathcal{H}^{2}_{2}(\mathbb{D})$
is complex symmetric with conjugation $\mathcal{J}$.

###### Proof of Theorem 4.2.

Suppose $C_{\phi}$ is complex symmetric with conjugation $\mathcal{J}$. Since
the linear span of reproducing kernels is dense in any reproducing kernel
Hilbert space, for all $w,z\in\mathbb{D}$, we have

$\displaystyle(C_{\phi}\mathcal{J})K_{w}(z)=(\mathcal{J}C^{*}_{\phi})k_{w}(z),$

that is, $K_{\overline{w}(\phi(z))}=K_{\overline{\phi(w)}(z)}$. Thus, for all
$w,z\in\mathbb{D}$,

$\displaystyle\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}(\phi(z)w)^{n}=\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}(z\phi(w))^{n}.$
(4.1)

Taking the derivative with respect to $z$ on the both sides in (4.1), we
obtain

$\displaystyle\sum_{n=0}^{\infty}\frac{n(n+1)}{(n+2)^{5}}(\phi(z)^{n-1}\phi^{\prime}(z)w^{n}=\sum_{n=0}^{\infty}\frac{n(n+1)}{(n+2)^{5}}z^{n-1}\phi(w)^{n}.$
(4.2)

On substituting $z=0$ in (4.2), we obtain

$\displaystyle\phi^{\prime}(0)\sum_{n=0}^{\infty}\frac{n(n+1)}{(n+2)^{5}}(\phi(0)^{n-1}w^{n}=\frac{2}{243}\phi(w)\;\;\mbox{for
all}\;\;w\in\mathbb{D}.$ (4.3)

Again putting $w=0$ in (4.3), it follows that $\phi(0)=0$. Hence, from (4.3)
we get $\phi(z)=az$ where $a=\phi^{\prime}(0)$.

Conversely, suppose $\phi(z)=az$ is a self holomorphic map on $\mathbb{D}$
where $a\in\mathbb{D}$. It can be easily verified that for all
$w,z\in\mathbb{D}$

$\displaystyle
K_{\overline{w}(\phi(z))}=\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}=K_{\overline{\phi(w)}}(z).$

This proves that $C_{\phi}$ is complex symmetric with conjugation
$\mathcal{J}$. ∎

We obtain the following result showing that if $W_{\Psi,\phi}$ is complex
symmetric with conjugation $\mathcal{J}$, then explicit form of $\phi(z)$ and
$\Psi(z)$ can be obtained.

######  Theorem 4.3.

Let $\Psi:\mathbb{D}\to\mathbb{C}$ and $\phi:\mathbb{D}\to\mathbb{D}$ be
holomorphic maps such that the weighted composition operator
$W_{\Psi,\phi}:\mathcal{H}^{2}_{2}(\mathbb{D})\to\mathcal{H}^{2}_{2}(\mathbb{D})$
is complex symmetric with conjugation $\mathcal{J}$. Then for all
$z\in\mathbb{D}$,

$\displaystyle\phi(z)=a_{0}+a_{1}\frac{q(z)}{p(z)},\;\Psi(z)=a_{2}p(z),$

where $a_{0}=\phi(0),a_{1}=\phi^{\prime}(0),a_{2}=3888\Psi(0),$

$\begin{cases}p(z)=\displaystyle\frac{2}{243}\sum_{n=0}^{\infty}((n+1)/(n+2)^{5})a^{n}_{0}z^{n}\\\
q(z)=\displaystyle\frac{1}{32}\sum_{n=1}^{\infty}(n(n+1)/(n+2)^{5})a^{n-1}_{0}z^{n}.\end{cases}$

Conversely, let $a_{0},a_{1}\in\mathbb{D}$ and $a_{2}\in\mathbb{C}$. If
$\phi(z)=a_{0}+a_{1}(q(z)/p(z))$ and $\Psi(z)=a_{2}p(z)$ are holomorphic maps
on $\mathbb{D}$, where

$\displaystyle\begin{cases}p(z)=\displaystyle\frac{2}{243}\sum_{n=0}^{\infty}((n+1)/(n+2)^{5})a^{n}_{0}z^{n};\\\
q(z)=\displaystyle\frac{1}{32}\sum_{n=1}^{\infty}(n(n+1)/(n+2)^{5})a^{n-1}_{0}z^{n}\end{cases}$
(4.4)

then the weighted composition operator
$W_{\Psi,\phi}:\mathcal{H}^{2}_{2}(\mathbb{D})\to\mathcal{H}^{2}_{2}(\mathbb{D})$
is complex symmetric with conjugation $\mathcal{J}$ only if $a_{0}=0$ or
$a_{1}=0$ or both are zero.

###### Proof of Theprem 4.3.

First we suppose that $W_{\Psi,\phi}$ is complex symmetric with conjugation
$\mathcal{J}$. Then
$(W_{\Psi,\phi}\mathcal{J})K_{w}(z)=(\mathcal{J}W^{*}_{\Psi,\phi})K_{w}(z)$
for all $w,z\in\mathbb{D}$ and it follows that
$\Psi(z)K_{\overline{w}}(\phi(z))=\Psi(w)K_{\overline{\phi(w)}}(z)$. This
implies that

$\displaystyle\Psi(z)\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}(\phi(z)w)^{n}=\Psi(w)\sum_{n=0}^{\infty}\frac{n+1}{n+2)^{5}}(z\phi(w))^{n}.$
(4.5)

Suppose that

$\displaystyle
g_{z}(w)=\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}(\phi(z)w)^{n}\;\;\mbox{and}\;\;h_{z}(w)=\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}(z\phi(w))^{n}$

for all $w,z\in\mathbb{D}$. Hence, it follows that

$\displaystyle\Psi(z)g_{z}(w)=\Psi(w)h_{z}(w)\;\;\mbox{for
all}\;\;w,z\in\mathbb{D}.$ (4.6)

Setting $w=0$ in (4.6), we obtain that $\psi(z)g_{z}(0)=\Psi(0)h_{z}(w)$ and
hence

$\displaystyle\Psi(z)=\Psi(0)\frac{h_{z}(0)}{g_{z}(0)}\;\;\mbox{for
all}\;\;w,z\in\mathbb{D}.$ (4.7)

By using (4.7), we obtain from (4.6) that

$\displaystyle\Psi(0)\frac{h_{z}(0)}{g_{z}(0)}g_{z}(w)=\Psi(0)\frac{h_{w}(0)}{g_{w}(0)}h_{z}(w).$
(4.8)

It can be easily shown that $g_{z}(w)=h_{w}(z)$ for all $w,z\in\mathbb{D}$.
Using this in (4.8), we have
$(h_{z}(0)/g_{z}(0))g_{z}(w)=(g_{0}(w)/h_{0}(w))h_{z}(w)$ for all
$w,z\in\mathbb{D}$. Therefore, we see that

$\displaystyle\bigg{(}\frac{h_{z}(0)}{g_{z}(0)}\bigg{)}g_{z}(w)h_{0}(w)=g_{0}(w)h_{z}(w).$
(4.9)

Taking the derivative of (4.9) with respect to $w$, we obtain

$\displaystyle\bigg{(}\frac{h_{z}(0)}{g_{z}(0)}\bigg{)}(g^{\prime}_{z}(w)h_{0}(w)+g_{z}(w)h^{\prime}_{0}(w))=g^{\prime}_{0}(w)h_{z}(w)+g_{0}(w)h^{\prime}_{z}(w).$
(4.10)

Putting $w=0$ in (4.10), we obtain

$\displaystyle
h_{z}(0)(g^{\prime}_{z}(0)h_{0}(0)+g_{z}(0)h^{\prime}_{0}(0))=g_{z}(0)(g^{\prime}_{0}(0)h_{z}(0)+g_{0}(0)h^{\prime}_{z}(0))$
(4.11)

By using the defining formula of $g_{z}(w)$ and $h_{z}(w)$, we obtain

$\displaystyle g_{z}(0)$ $\displaystyle=g_{0}(0)=h_{0}(0)=\frac{1}{32};$
$\displaystyle g^{\prime}_{z}(0)$ $\displaystyle=\frac{2}{243}\phi(z);$
$\displaystyle g^{\prime}_{0}(0)$ $\displaystyle=\frac{2}{243}\phi(0);$
$\displaystyle h_{z}(0)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}(z\phi(0))^{n};$
$\displaystyle h^{\prime}_{z}(0)$
$\displaystyle=\sum_{n=1}^{\infty}\frac{n(n+1)}{(n+2)^{5}}(\phi(0))^{n-1}\phi^{\prime}(0)z^{n};$
$\displaystyle h^{\prime}_{0}(0)$ $\displaystyle=0.$

Denoting $a_{0}=\phi(0)$ and $a_{1}=\phi^{\prime}(0)$, and then substituting
the above expressions into equation (4.11), we obtain

$\displaystyle\bigg{(}\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}a^{n}_{0}z^{n}\bigg{)}\bigg{(}\frac{2}{243}\phi(z)\bigg{)}$
$\displaystyle=\frac{2}{243}a_{0}\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}a^{n}_{0}z^{n}+\frac{1}{32}a_{1}\sum_{n=1}^{\infty}\frac{n(n+1)}{(n+2)^{5}}a^{n-1}_{0}z^{n}.$
$\displaystyle\Rightarrow\phi(z)$
$\displaystyle=a_{0}+a_{1}\frac{q(z)}{p(z)},$

where

$\displaystyle
p(z)=\frac{2}{243}\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}a^{n}_{0}z^{n}\;\;\mbox{and}\;\;q(z)=\frac{1}{32}\sum_{n=1}^{\infty}\frac{n(n+1)}{(n+2)^{5}}a^{n-1}_{0}z^{n}.$

Take $a_{2}=3888\Psi(0)$. Consequently, it follows that
$\phi(z)=a_{0}+a_{1}(q(z)/p(z))$ and $\Psi(z)=a_{2}p(z)$ for all
$z\in\mathbb{D}$.

Conversely, suppose that $\phi(z)=a_{0}+a_{1}(q(z)/p(z))$ and
$\Psi(z)=a_{2}p(z)$ where $a_{0},a_{1}\in\mathbb{D},a_{2}\in\mathbb{C}$ and
$p(z)$ and $q(z)$ are as in (4.4). Let $w$ and $z$ be elements of
$\mathbb{D}$. For $W_{\Psi,\phi}$ to have complex symmetry, the following must
hold:

$\displaystyle\Psi(z)K_{\overline{w}}(\phi(z))=\Psi(w)K_{\overline{\phi(w)}}(z),$
(4.12)

that is

$\displaystyle\bigg{(}$
$\displaystyle\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}a^{n}_{0}z^{n}\bigg{)}\bigg{(}\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}\bigg{(}a_{0}w+a_{1}\frac{q(z)}{p(z)}w\bigg{)}^{n}\bigg{)}$
(4.13)
$\displaystyle=\bigg{(}\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}a^{n}_{0}w^{n}\bigg{)}\bigg{(}\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}\bigg{(}a_{0}z+a_{1}\frac{q(w)}{p(w)}z\bigg{)}^{n}\bigg{)}.$

In the entirety of the unit disk $\mathbb{D}$, the function $q(z)/p(z)$ is
holomorphic, and it satisfies the condition $q(0)=0$. Consequently, we can
express $q(z)/p(z)$ as an infinite series:

$\displaystyle\frac{q(z)}{p(z)}=\sum_{i=1}^{\infty}c_{i}a^{i-1}_{0}z^{i},$

where $c_{1}=1$, and for $i=2,3,\ldots$, the coefficients $c_{i}$ are positive
real numbers less than one. Thus, (4.13) is equivalent to

$\displaystyle\bigg{(}$
$\displaystyle\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}a^{n}_{0}z^{n}\bigg{)}\bigg{(}\sum_{m=0}^{\infty}\frac{m+1}{(m+2)^{5}}\bigg{(}\sum_{k=0}^{m}\binom{m}{k}(a_{0}w)^{k}\bigg{(}\sum_{i=1}^{\infty}a_{1}c_{i}a^{i-1}_{0}z^{i}w\bigg{)}^{m-k}\bigg{)}\bigg{)}$
$\displaystyle=\bigg{(}\sum_{n=0}^{\infty}\frac{n+1}{(n+2)^{5}}a^{n}_{0}w^{n}\bigg{)}$
$\displaystyle\times\bigg{(}\sum_{m=0}^{\infty}\frac{m+1}{(m+2)^{5}}\bigg{(}\sum_{k=0}^{m}\binom{m}{k}(a_{0}z)^{k}\bigg{(}\sum_{i=1}^{\infty}a_{1}c_{i}a^{i-1}_{0}w^{i}z\bigg{)}^{m-k}\bigg{)}\bigg{)}.$

On comparing the coefficients of $z^{3}w$, on L.H.S, we obtain

$\displaystyle\frac{2}{3^{5}}\bigg{(}\frac{4}{5^{5}}a^{4}_{0}+\frac{3}{4^{5}}a^{2}_{0}a_{1}c_{1}+\frac{2}{3^{5}}a^{2}_{0}a_{1}c_{2}+\frac{1}{2^{5}}a^{2}_{0}a_{1}c_{3}\bigg{)},$
(4.14)

whereas on R.H.S, we obtain

$\displaystyle\frac{4}{5^{5}}\bigg{(}\frac{2}{3^{5}}a^{4}_{0}+\frac{3}{2^{5}}a^{2}_{0}a_{1}c_{1}\bigg{)}.$
(4.15)

Consequently, we find that the expressions of (4.14) and (4.15) will be equal
only if $a_{0}$ is zero or $a_{1}$ is zero or both are zero. With this the
proof is completed. ∎

Compliance of Ethical Standards

Conflict of interest The authors declare that there is no conflict of interest
regarding the publication of this paper.

Data availability statement Data sharing not applicable to this article as no
datasets were generated or analysed during the current study.

## References

  * [1] R. F. Allen, K. C. Heller, and M. A. Pons, Multiplication operators on $\mathcal{S}^{2}(\mathbb{D})$, Acta. Sci. Math.(Szeged),81 (2015), 575–587.
  * [2] P. S. Bourdon and S. K. Narayan, Normal weighted composition operators on the Hardy space $H^{2}(\mathbb{U})$, J. Math. Anal. Appl., 367 (2010), 278-286.
  * [3] C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, (1995).
  * [4] Z. Cuckovic and B. Paudyal,Invariant subspaces of the shift plus complex Volterra operator, J. Math. Anal. Appl. 426, 1174-1181. https://doi.org/10.48550/arXiv.2101.04911.
  * [5] S. Garcia and C. Hammond, Which weighted composition operators are complex symmetric? Oper. Theory Adv. Appl. 236(2013), 171–179.
  * [6] S. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358, (2006), 1285–1315.
  * [7] S. Garcia and M. Putinar, Complex symmetric operators and applications II, Trans. Amer. Math. Soc. 359(2007), 3913–3931.
  * [8] C. Gu and S. Luo, Composition and multiplication operators on the derivative Hardy space $\mathcal{S}^{2}(\mathbb{D})$, Complex Var. Elliptic Equ., 63 (2018), 599–624.
  * [9] A. Gupta and A. Malhotra, Complex symmetric weighted composition operator on the space $\mathcal{H}^{2}_{2}(\mathbb{D})$, Complex Var. Elliptic Equ., 65:9 (2020),1488=1500.
  * [10] P. V. Hai and L. H. Khoi, complex symmetry of weighted composition operators on the Fock space, J. Math. Anal. Appl, 433 (2016), 1757-1771.
  * [11] P. V. Hai and L. H. Khoi, complex symmetry of weighted composition operators on the Fock space in several variables, Complex Var. Elliptic Equ., 63 (2018), 391-405.
  * [12] K. C. Heller, Composition operators on $\mathcal{S}^{2}(\mathbb{D})$ [PhD thesis], Pro Quest LLC. Ann Arbor (MI), University of Virginia, (2010).
  * [13] W. M. Higdon, The spectra of composition operators from linear fractional maps acting upon the Dirichlet space, J. Funct. Anal. 220 (1) (2005), 55–75.
  * [14] S. Jung, Y. Kim, E. Ko, and J. E. Lee, Complex symmetric weighted composition operators on ${H}^{2}(\mathbb{D})$, J. Func. Anal. 267(2014) 323–351.
  * [15] L. Khoi and R. Lim, Complex symmetric weighted composition operators on $\mathcal{H}_{\gamma}(\mathbb{D})$, J. Math. Anal. Appl. 464(2018), 101-118.
  * [16] R. P. Kopp, A subcollection of algebras in a collection of Banach space, Pacific J. math. 30 (1969), 433-435.
  * [17] B. I. Koreblum, Invariant subspaces of the shift operator in a weighted Hilbert space, Mat Sb. 89(131) (1972), 110-137.
  * [18] J. Liu, S. Ponnusamy, and H. Xie, Complex symmetric weighted composition-differentiation operators, Linear and Multilinear Algebra, 71(2023), 737-755.
  * [19] C. Lo and A. W.-k Loh, Complex symmetric generalized weighted composition operators on Hilbert spaces of analytic functions, J. Math. Anal. Appl. 523(2023) 127141.
  * [20] J. S. Manhas and R. Zhao, New estimates of essential norms of weighted composition operators between Bloch type spaces, J. Math. Anal. Appl. 389 (2012), 32–47.
  * [21] M. J. Mart$\acute{i}$n and D. Vukoti$\acute{c}$, Adjoints of composition operators on Hilbert spaces of analytic functions, J. Funct. Anal. 238 (2006), 298-312.
  * [22] M. J. Mart$\acute{i}$n and D. Vukoti$\acute{c}$, Isometries of the Dirichlet space among the composition operators, Proc. Amer. Math. Soc. 134 (6) (2006), 1701–1705.
  * [23] J. N. McDonald, Adjoints of a class of composition operators, Proc. Amer. Math. Soc. 131 (2003), 601-606.
  * [24] T. Mengestie and W. Seyoum, Spectral properties of composition operators on Fock-type spaces, Quaest. Math. 44 (2021), 335–350.
  * [25] T. Mengestie and W. Seyoum, Supercyclicity and resolvent condition for weighted composition operators, Comput. Methods Funct. Theory, 22 (2022), 157-168. https://doi.org/10.1007/s40315-021-00380-x
  * [26] S. Noor and O. Severiano, Complex symmetry and cyclicity of composition operators on $H^{2}(\mathbb{C}_{+})$, Proc. Amer. Math. Soc. 148(2020), 2469-2476.
  * [27] S. Ohno, Products of composition and differentiation between Hardy spaces, Bull. Austral. Math. Soc. 73(2006), 235-243.
  * [28] J. H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York, (1993).
  * [29] S. Ueki, Weighted composition operators on the Fock space, Proc. Amer. Math. Soc. 135(5) (2007), 1405–1410.
  * [30] H. Wulan, D. Zheng, and K. Zhu, Composition operators on BMOA and the Bloch space, Proc. Amer. Math. Soc. 137 (2009), 3861–3868.
  * [31] X. Yao, Complex symmetric composition operators on a Hilbert space of Dirichlet series, J. Math. Anal. Appl. 452(2017), 1413-1419.