# Composition operators on weighted Hardy spaces of polynomial growth

Bingzhe Hou  Bingzhe Hou, School of Mathematics, Jilin University, 130012,
Changchun, P. R. China<EMAIL_ADDRESS>and  Chunlan Jiang  Chunlan Jiang,
Department of Mathematics, Hebei Normal University, 050016, Shijiazhuang, P.
R. China<EMAIL_ADDRESS>
###### Abstract.

In the present paper, we study the composition operators acting on weighted
Hardy spaces of polynomial growth, which are concerned with norms, spectra and
(semi-)Fredholmness. Firstly, we estimate the norms of the composition
operators with symbols of disk automorphisms. Secondly, we discuss the spectra
of the composition operators with symbols of disk automorphisms. In
particular, it is proven of that the spectrum of a composition operator with
symbol of any parabolic disk automorphism is always the unit circle. Thirdly,
we consider the Fredholmness of the composition operator $C_{\varphi}$ with
symbol $\varphi$ which is an analytic self-map on the closed unit disk. We
prove that $C_{\varphi}$ acting on a weighted Hardy space of polynomial growth
has closed range (semi-Fredholmness) if and only if $\varphi$ is a finite
Blaschke product. Furthermore, it is obtained that $C_{\varphi}$ is Fredholm
if and only if $\varphi$ is a disk automorphism.

###### Key words and phrases:

Composition operators, weighted Hardy spaces of polynomial growth, norms,
spectra, (semi-)Fredholm operators.

###### 2010 Mathematics Subject Classification:

Primary 47B33, 47A30, 47A25, 47A53; Secondary 47B38, 46E20.

###### Contents

  1. 1 Introduction
  2. 2 Estimations of norms of the composition operators with symbols of disk automorphisms
  3. 3 Spectra of composition operators with symbols of disk automorphisms
  4. 4 Composition operators with closed ranges and Fredholmness

##  1\. Introduction

Denote by ${\rm Hol}(\mathbb{D})$ and ${\rm Hol}(\overline{\mathbb{D}})$ the
space of all analytic functions on the open unit disk $\mathbb{D}$ and the
closed unit disk $\overline{\mathbb{D}}$, respectively. Denote by ${\rm
Aut}(\mathbb{D})$ the analytic automorphism group on $\mathbb{D}$, which is
the set of all analytic bijections from $\mathbb{D}$ to itself. As well known,
each $\varphi\in{\rm Aut}(\mathbb{D})$, called disk automorphism or Möbius
transformation, could be written as the following form

$\varphi(z)={\rm
e}^{\mathbf{i}\theta}\cdot\frac{z_{0}-z}{1-\overline{z_{0}}z},\ \ \ \text{for
some}\ z_{0}\in\mathbb{D}.$

For any $f\in{\rm Hol}(\mathbb{D})$, denote the Taylor expansion of $f(z)$ by

$f(z)=\sum\limits_{k=0}^{\infty}\widehat{f}(k)z^{k}.$

Let $w=\\{w_{k}\\}_{k=1}^{\infty}$ be a sequence of positive numbers. Write
$\beta=\\{\beta_{k}\\}_{k=0}^{\infty}$,

$\beta_{0}=1,\ \ \text{and}\ \ \beta_{k}=\prod\limits_{j=1}^{k}w_{j},\ \
\text{for}\ k\geq 1.$

The weighted Hardy space $H^{2}_{\beta}$ induced by the weight sequence $w$
(or $\beta$) is defined by

$H^{2}_{\beta}=\\{f(z)=\sum\limits_{k=0}^{\infty}\widehat{f}(k)z^{k};\
\sum\limits_{k=0}^{\infty}|\widehat{f}(k)|^{2}{\beta}_{k}^{2}<\infty\\}.$

Moreover, the weighted Hardy space $H^{2}_{\beta}$ is a complex separable
Hilbert space, on which the inner product is defined by, for any $f,g\in
H^{2}_{\beta}$

$\langle
f,g\rangle_{H^{2}_{\beta}}=\sum\limits_{k=0}^{\infty}{\beta}_{k}^{2}\overline{\widehat{g}(k)}\widehat{f}(k)$

Then, any $f\in H^{2}_{\beta}$ equips the following norm

$\|f(z)\|_{H^{2}_{\beta}}=\sqrt{\langle
f,f\rangle_{H^{2}_{\beta}}}=\sqrt{\sum\limits_{k=0}^{\infty}{\beta}_{k}^{2}|\widehat{f}(k)|^{2}}.$

One can see that $\\{z^{n}\\}_{n=0}^{\infty}$ for each
$n\in\mathbb{\mathbb{N}}$. Moreover, we denote by $\|T\|_{H^{2}_{\beta}}$ the
operator norm of the linear operator $T$ acting on $H^{2}_{\beta}$. In
addition, we denote by $\|f\|$ the norm of the function $f$ in $H^{2}$, and
denote by $\|T\|$ the operator norm of the linear operator $T$ acting on
$H^{2}$.

In the present article, we always assume the weight sequence $w$ satisfies the
regular condition as follows

$\lim\limits_{k\rightarrow\infty}w_{k}=1.$

Then every weighted Hardy space $H^{2}_{\beta}$ is contained in ${\rm
Hol}(\mathbb{D})$.

An analytic self-map $g:\mathbb{D}\rightarrow\mathbb{D}$ introduces a linear
operator $C_{g}:{\rm Hol}(\mathbb{D})\rightarrow{\rm Hol}(\mathbb{D})$,
defined by

$C_{g}(f)=f(g),\ \ \ \text{for any}\ f\in{\rm Hol}(\mathbb{D}).$

Then the operator $C_{g}$ is said to be a composition operator. In this paper,
we are interested the composition operators with the symbols of analytic
functions on the closed unit disk acting on weighted Hardy spaces of
polynomial growth, which are concerned with norms, spectra and
(semi-)Fredholmness.

It is a natural problem to study the boundedness or norms of the composition
operators $C_{g}$ acting on weighted Hardy spaces $H^{2}_{\beta}$. There
numerous authors have contributed to it, see [43], [29], [13], [36], [21],
[4], [22], [17] and [18] for instance. We have known that the composition
operators induced by some analytic self-maps acting on some weighted Hardy
spaces. For instance, the composition operators induced by linear fractional
transformations are continuous on the classical Hardy space $H^{2}$, the
weighted Bergman spaces, the weighted Hardy spaces which are larger than
$H^{2}$, the weighted Dirichlet spaces and so on. However, it is still an open
problem to determine the boundedness of the composition operators $C_{g}$
acting on weighted Hardy spaces $H^{2}_{\beta}$ in general, even for the
composition operators induced by disk automorphisms. Recently, inspired by M.
Gromov’s work in the geometry group theory [20], the authors in a previous
article [24] introduced the notion of weighted Hardy spaces of polynomial
growth and proved that each composition operators induced by a disk
automorphism acting on a weighted Hardy space of polynomial growth is bounded
and the composition operators induced by a disk automorphism acting on a
weighted Hardy space of intermediate growth is possible to be unbounded.

######  Definition 1.1.

Let $w=\\{w_{k}\\}_{k=1}^{\infty}$ be a sequence of positive numbers with
$w_{k}\rightarrow 1$.

  1. (1)

If $\sup\limits_{k}(k+1)|w_{k}-1|<\infty$, we say that the weighted Hardy
space $H^{2}_{\beta}$ is of polynomial growth.

  2. (2)

If $\sup\limits_{k}(k+1)|w_{k}-1|=\infty$, we say that the weighted Hardy
space $H^{2}_{\beta}$ is of intermediate growth.

######  Remark 1.2.

Notice that many functions spaces such as the classical Hardy space, the
weighted Bergman spaces, and the weighted Dirichlet spaces are all the
weighted Hardy spaces of polynomial growth.

In addition, the condition $\sup_{k}(k+1)|w_{k}-1|<\infty$ holds if and only
if there exists a positive number $M$ such that for each $k\in\mathbb{N}$,

$\frac{k+1}{k+M+1}\leq w_{k}\leq\frac{k+M+1}{k+1}.$

Moreover, such weighted Hardy space $H^{2}_{\beta}$ is said to be of
$M$-polynomial growth.

In this paper, we focus on the composition operators acting on weighted Hardy
spaces of polynomial growth. In Section 2, we will estimate the norms of the
composition operators with symbols of disk automorphisms acting on weighted
Hardy spaces of polynomial growth.

The research in the spectra of composition operators began formally with
Nordgren’s paper [38], in which he determined the spectra of invertible
composition operators on the classical Hardy space $H^{2}$. Then, Kamowitz
studied the same problem on $H^{p}$ and the disk algebra in [27] and [28],
respectively. In the following decades, this topic catched widely attention,
we refer to the context by C. Cowen [13] and the references therein. It is
worthy to note that when consider the small space (the weighted Hardy spaces
contained in $H^{2}$ such as Dirichlet spaces and weighted Dirichlet spaces),
the effective techniques are different from the case of classical Hardy space
or Bergman space. Following from P. Hurst [26], E. Gallardo-Gutiérrez and A.
Montes-Rodríguez [16], W. Higdon [23], A. Pons [41], J. Pau and P. Pérez [39],
E. Gallardo-Gutiérrez and R. Schroderus [18], one can see the spectra of the
composition operators induced by disk automorphisms and linear fractional non-
automorphisms acting on the weighted Dirichlet spaces. In Section 3, we
discuss the spectra of the composition operators with symbols of disk
automorphisms acting on weighted Hardy spaces of polynomial growth. In
particular, it is proven of that the spectrum of a composition operator with
symbol of any parabolic disk automorphism is always the unit circle.

Recall that a bounded linear operator $T$ acting on a Hilbert space to itself
is called a semi-Fredholm operator, if $T$ has closed range and at least one
of $\textrm{dim}\ \textrm{Ker}(T)<\infty$ and $\textrm{dim}\
\textrm{Ker}(T^{*})<\infty$ is established. Moreover, $T$ is called a Fredholm
operator, if $T$ has closed range and both $\textrm{dim}\
\textrm{Ker}(T)<\infty$ and $\textrm{dim}\ \textrm{Ker}(T^{*})<\infty$ are
established. Fredholm composition operators on the Hardy spaces of the open
unit disk were characterized in [10], [3] and [5], Fredholm composition
operators on the Bergman space were characterized in [2] and [3], and Fredholm
composition operators on the Dirichlet space were characterized in [11]. And
there have been so many other articles to study the Fredholm operators on the
spaces of holomorphic functions on domains in $\mathbb{C}$ or
$\mathbb{C}^{n}$, see [6], [15], [30], [31], [44] and [45] for instance. In
particular, G. Cao, L. He and K. Zhu [7] in 2019 proved that for an
holomorphic self-map $g$ on a domain $\Omega$ in $\mathbb{C}^{n}$, if the
reproducing kernel $K(w,w)$ of the Hilbert space tends to infinity as $w$
approaches the boundary of $\Omega$, $C_{g}$ is a Fredholm operator if and
only if $C_{g}$ is a Fredholm operator, if and only if $g$ is a disk
automorphism. Recently, G. Cao, L. He and J. Li [8] obtained the same
conclusion without the assumption that $K(w,w)\rightarrow\infty$ as $w$
approaches $\partial\Omega$.

Notice that the composition operators induced by non-trivial analytic self-
maps acting on weighted Hardy spaces are always injective. Then such
composition operator is semi-Fredholm if it has closed range. The closed range
composition operators on various function spaces also has attracted wide
attention. In 1974, J. Cima, J. Thompson and W. Wogen [10] determined the
composition operators on $H^{2}$ with closed range via the boundary behaviour
of the inducing maps. Then, N. Zorboska [44] characterized the closed range
composition operators via the properties of the range of the inducing maps on
the unit disk, which answered a question in [10]. The closed range composition
operators on Bergman space and weighted Bergman spaces were studied by D.
Luecking in [32], [33], and J. Akeroyd and S. Fulmer in [2], in which one can
see that the Navanlinna counting function inducing a reverse Carleson measure
is a necessary and sufficient condition for the closed range composition
operators on the classical Hardy space and weighted Bergman spaces. However,
in the case of the Dirichlet space, this seems to be more difficult. Luecking
proved that the Navanlinna counting function inducing a reverse Carleson
measure is a necessary condition for the closed range composition operators on
the Dirichlet space in [34], but is not sufficient condition [35]. More
recently, G. Cao and L. He [9] gave a necessary and sufficient condition for
the closed range composition operators on the Dirichlet space.

In Section 4, we consider the Fredholmness of the composition operator
$C_{\varphi}$ with symbol of an analytic self-map $\varphi$ on the closed unit
disk. We prove that $C_{\varphi}$ acting on a weighted Hardy space of
polynomial growth has closed range (semi-Fredholmness) if and only if
$\varphi$ is a finite Blaschke product. Furthermore, it is obtained that
$C_{\varphi}$ is Fredholm if and only if $\varphi$ is a disk automorphism.

Up to now, there have been numerous research work on the composition operator
acting on the classical Hardy space, weighted Bergman spaces and weighted
Dirichlet spaces, in which it plays an important role of measure methods
(integral representation, Carleson measure, Navanlinna counting function and
so on). However, when considering weighted Hardy spaces in general, we may
have no appropriate measure to use. The techniques in the present paper
involve the operator theory, base theory and some topology as well as the
composition operators theory, functional theory and complex analysis.

To avoid confusion, we denote by $(f)^{n}$ or $(f(z))^{n}$ the $n$-th power of
a function $f$, and denote by $f^{n}$ the $n$-th iteration of a function $f$
if the iteration makes sense.

##  2\. Estimations of norms of the composition operators with symbols of disk
automorphisms

In the process of studying the similar representation of analytic functions in
[24], the authors obtained the boundedness of the composition operators with
symbols of disk automorphisms acting on weighted Hardy spaces of polynomial
growth. For an analytic function $g\in{\rm Hol}(\mathbb{D})$, denote by
$M_{g}$ the multiplication operator, which means for any $f\in{\rm
Hol}(\mathbb{D})$, $M_{g}(f(z))=g(z)f(z)$.

######  Theorem 2.1 (Theorem 3.1 in [24]).

Let $H^{2}_{\beta}$ be the weighted Hardy space of polynomial growth induced
by a weight sequence $w=\\{w_{k}\\}_{k=1}^{\infty}$. Then for any
$\varphi\in{\textrm{A}ut(\mathbb{D})}$, $M_{z}\sim M_{\varphi}$, i.e., $M_{z}$
is weakly homogeneous on $H^{2}_{\beta}$. In fact, the composition operator
$C_{\varphi}:H^{2}_{\beta}\rightarrow H^{2}_{\beta}$ is an isomorphism.

Moreover, the composition operators with symbols of finite Blaschke products
and analytic self-maps acting on weighted Hardy spaces of polynomial growth
were studied.

######  Theorem 2.2 (Theorem 3.3 in [24]).

Let $H^{2}_{\beta}$ be the weighted Hardy space of polynomial growth induced
by a weight sequence $w=\\{w_{k}\\}_{k=1}^{\infty}$. If $\psi(z)$ is an
analytic function on $\overline{\mathbb{D}}$ with
$\psi(\mathbb{D})\subseteq\mathbb{D}$, then $C_{\psi}$ is bounded on
$H^{2}_{\beta}$.

######  Theorem 2.3 (Theorem 3.2 in [24]).

Let $H^{2}_{\beta}$ be a weighted Hardy space of polynomial growth, and let
$B(z)$ be a finite Blaschke product with order $m$ on $\mathbb{D}$. Then

$M_{B}\sim\bigoplus_{1}^{m}M_{z}.$

######  Remark 2.4.

Notice that $M_{B}C_{B}=C_{B}M_{z}$. Then, following from the above Theorem,
one can see that the composition operators with symbols of finite Blaschke
products with order $m$ have closed range. Consequently they are semi-
Fredholm, since

$\textrm{dim}\ \textrm{Ker}(T)=0<\infty\ \ \ \text{and}\ \ \ \textrm{dim}\
\textrm{Ker}(T^{*})=\left\\{\begin{array}[]{cc}0,&\text{if}\ m=1\\\
\infty,&\text{if}\ m>1\end{array}\right..$

Now, we have known the boundedness of the composition operators induced by
disk automorphisms acting on weighted Hardy spaces of polynomial growth.
Furthermore, we will give the estimation of the norms of those composition
operators. First of all, let us review some notations and basic statements in
[24], which are related to the bases theory (we refer to [42] and [37]).

Let ${H}$ be a complex separable Hilbert space and let
$\mathfrak{X}=\\{x_{n}\\}_{n=0}^{\infty}$ and
$\mathfrak{Y}=\\{y_{n}\\}_{n=0}^{\infty}$ be two sequences of vectors in
${H}$. Define

$\langle\mathfrak{X},\mathfrak{Y}\rangle_{{H}}=(\langle
x_{j},y_{i}\rangle_{{H}})_{i,j}.$

In particular, the Gram matrix of $\mathfrak{X}$ on ${H}$ is defined by

$\Gamma_{{H}}(\mathfrak{X})=\langle\mathfrak{X},\mathfrak{X}\rangle_{{H}}=(\langle
x_{i},x_{j}\rangle_{{H}})_{i,j}.$

Let $\mathfrak{F}=\\{f_{n}\\}_{n=0}^{\infty}$ be a sequence of vectors in a
weighted Hardy space $H^{2}_{\beta}$. Then $\mathfrak{F}$ induces a linear
operator $X_{\mathfrak{F}}$ on $H^{2}_{\beta}$, defined by

$X_{\mathfrak{F}}(z^{n})=f_{n}(z),\ \ \ \text{for}\ n=0,1,\ldots.$

Furthermore, the operator $X_{\mathfrak{F}}$ has a matrix representation under
the orthogonal base $\\{z^{n}\\}_{n=0}^{\infty}$ as follows

$X_{\mathfrak{F}}=\begin{bmatrix}\widehat{f_{0}}(0)&\widehat{f_{1}}(0)&\widehat{f_{2}}(0)&\cdots&\widehat{f_{k}}(0)&\cdots\\\
\widehat{f_{0}}(1)&\widehat{f_{1}}(1)&\widehat{f_{2}}(1)&\cdots&\widehat{f_{k}}(1)&\cdots\\\
\widehat{f_{0}}(2)&\widehat{f_{1}}(2)&\widehat{f_{2}}(2)&\cdots&\widehat{f_{k}}(2)&\cdots\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
\widehat{f_{0}}(k)&\widehat{f_{1}}(k)&\widehat{f_{2}}(k)&\cdots&\widehat{f_{k}}(k)&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix}.$

We say that the operator $X_{\mathfrak{F}}$ is the transformation operator of
the sequence $\mathfrak{F}$. In particular, if
$\mathfrak{F}=\\{(g)^{n}\\}_{n=0}^{\infty}$ for some $g\in{\rm
Hol}(\mathbb{D})$ with $g(\mathbb{D})\subseteq\mathbb{D}$, then the
transformation operator $X_{\mathfrak{F}}$ is just the composition operator
$C_{g}$.

Define the operator $D_{\beta}:H^{2}_{\beta}\rightarrow H^{2}$ by

$D_{\beta}(z^{k})=\beta_{k}z^{k}\ \ \ \text{for}\ n=0,1,\ldots.$

As well known, $D_{\beta}$ is an isometry isomorphism. Moreover, $D_{\beta}$
and $D^{-1}_{\beta}$ has matrix representations under the orthogonal base
$\\{z^{n}\\}_{n=0}^{\infty}$ as follows

$D_{\beta}=\begin{bmatrix}\beta_{0}&0&\cdots&0&\cdots\\\
0&\beta_{1}&\cdots&0&\cdots\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&\cdots&\beta_{k}&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix},\ \
D^{-1}_{\beta}=D_{\beta^{-1}}=\begin{bmatrix}\frac{1}{\beta_{0}}&0&\cdots&0&\cdots\\\
0&\frac{1}{\beta_{1}}&\cdots&0&\cdots\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&\cdots&\frac{1}{\beta_{k}}&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix}.$

Then, $D_{\beta}X_{\mathfrak{F}}D^{-1}_{\beta}$ is a linear operator on the
classical Hardy space $H^{2}$.

From now on, we always write
$\varphi_{z_{0}}(z)=\frac{z_{0}-z}{1-\overline{z_{0}}z}$,
$z_{0}\in\mathbb{D}\setminus\\{0\\}$ and $B_{0}(z)=z\varphi_{z_{0}}(z)$.
Denote

$\displaystyle\mathfrak{F}=\\{(B_{0}(z))^{n}\\}_{n=0}^{\infty},$
$\displaystyle\mathfrak{F}_{\beta}=\\{\frac{(B_{0}(z))^{n}}{\beta_{n}}\\}_{n=0}^{\infty},$
$\displaystyle\widetilde{\mathfrak{F}}_{1}=\\{(B_{0}(z))^{n},z(B_{0}(z))^{n};n=0,1,\ldots\\},$
$\displaystyle\widetilde{\mathfrak{F}}_{2}=\\{(B_{0}(z))^{n},\varphi_{z_{0}}(z)(B_{0}(z))^{n};n=0,1,\ldots\\},$
$\displaystyle\widetilde{\mathfrak{F}}_{1,\beta}=\\{\frac{(B_{0}(z))^{n}}{\beta_{n}},\frac{z(B_{0}(z))^{n}}{\beta_{n}};n=0,1,\ldots\\},$
$\displaystyle\widetilde{\mathfrak{F}}_{2,\beta}=\\{\frac{(B_{0}(z))^{n}}{\beta_{n}},\frac{\varphi_{z_{0}}(z)(B_{0}(z))^{n}}{\beta_{n}};n=0,1,\ldots\\}.$

It follows from $B_{0}(z)=z\varphi_{z_{0}}(z)$ and
$\varphi_{z_{0}}(\varphi_{z_{0}}(z))=z$ that for every $n=0,1,2,\ldots$,

$\displaystyle
C_{\varphi_{z_{0}}}((B_{0}(z))^{n})=(B_{0}(\varphi_{z_{0}}(z)))^{n}=(B_{0}(z))^{n},$
$\displaystyle
C_{\varphi_{z_{0}}}(z(B_{0}(z))^{n})=\varphi_{z_{0}}(z)(B_{0}(\varphi_{z_{0}}(z)))^{n}=\varphi_{z_{0}}(z)(B_{0}(z))^{n}.$

Then, the composition operator $C_{\varphi_{z_{0}}}$ is just a base
transformation operator from the Riesz base
$\widetilde{\mathfrak{F}}_{1,\beta}$ to the Riesz base
$\widetilde{\mathfrak{F}}_{2,\beta}$. One can see that $C_{B}$ is bounded on
$H^{2}_{\beta}$ and $H^{2}_{\beta^{-1}}$. Consequently,
$D_{\beta}X_{\mathfrak{F}}D^{-1}_{\beta}$ and
$D^{-1}_{\beta}X_{\mathfrak{F}}D_{\beta}$ are bounded on $H^{2}$.

In addition, let

$D_{tm}=\begin{bmatrix}D_{1}&0&\cdots\\\ 0&D_{2}&\cdots\\\
\vdots&\vdots&\ddots\end{bmatrix}=\begin{bmatrix}1&w_{1}^{-1}\overline{z_{0}}&0&0&\cdots\\\
w_{1}z_{0}&1&0&0&\cdots\\\ 0&0&1&w_{2}^{-1}\overline{z_{0}}&\cdots\\\
0&0&w_{2}z_{0}&1&\cdots\\\ \vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix}.$

where

$D_{n}=\begin{bmatrix}1&w_{n}^{-1}\overline{z_{0}}\\\
w_{n}z_{0}&1\end{bmatrix}\ \ \ \ \text{for any}\ n\in\mathbb{N}.$

Now, let us begin to estimate the norms of the composition operators induced
by disk automorphisms acting on weighted Hardy spaces of polynomial growth.

######  Theorem 2.5.

Let $H^{2}_{\beta}$ be the weighted Hardy space of polynomial growth induced
by a weight sequence $w=\\{w_{k}\\}_{k=1}^{\infty}$. Then for any
$\varphi\in{\rm Aut(\mathbb{D})}$.

(2.1)
$\displaystyle(D_{\beta}X^{*}_{\widetilde{\mathfrak{F}}_{1}}D^{-1}_{\beta})(D_{\beta}M^{*}_{\frac{\sqrt{1-|z_{0}|^{2}}}{1-\overline{z_{0}}z}}D^{-1}_{\beta})(D_{\beta}M_{\frac{\sqrt{1-|z_{0}|^{2}}}{1-\overline{z_{0}}z}}D^{-1}_{\beta})(D_{\beta}X_{\widetilde{\mathfrak{F}}_{1}}D^{-1}_{\beta})=D_{tm},$
(2.2)
$\displaystyle(D_{\beta}X^{*}_{\widetilde{\mathfrak{F}}_{2}}D^{-1}_{\beta})(D_{\beta}X_{\widetilde{\mathfrak{F}}_{2}}D^{-1}_{\beta})=D_{tm}.$

Moreover,

$\|C^{-1}_{\varphi}\|_{H^{2}_{\beta}}=\|C_{\varphi^{-1}}\|_{H^{2}_{\beta}}=\|C_{\varphi}\|_{H^{2}_{\beta}}=\|X_{\widetilde{\mathfrak{F}}_{1}}X^{-1}_{\widetilde{\mathfrak{F}}_{2}}\|_{H^{2}_{\beta}}\leq\|X_{\widetilde{\mathfrak{F}}_{1}}\|_{H^{2}_{\beta}}\|X_{\widetilde{\mathfrak{F}}_{2}}\|_{H^{2}_{\beta^{-1}}}\|D^{-1}_{tm}\|.$

###### Proof.

Each element in $\varphi\in{\rm Aut(\mathbb{D})}$ could be written as follows

$\varphi(z)={\rm
e}^{\mathbf{i}\theta}\cdot\frac{z_{0}-z}{1-\overline{z_{0}}z}={\rm
e}^{\mathbf{i}\theta}\cdot\varphi_{z_{0}},\ \ \ \text{for some}\
z_{0}\in\mathbb{D}.$

Since $C_{{\rm e}^{\mathbf{i}\theta}z}$ is an isometry, we have

$\|C_{\varphi}\|_{H^{2}_{\beta}}=\|C_{\frac{z_{0}-z}{1-\overline{z_{0}}z}}C_{{\rm
e}^{\mathbf{i}\theta}z}\|_{H^{2}_{\beta}}=\|C_{\varphi_{z_{0}}}\|_{H^{2}_{\beta}}.$

So, it suffices to consider $\varphi_{z_{0}}(z)$. Following some computation
(the details could be found in the proof of Theorem 3.1 in [24]), one can see
that the equations (2.1) and (2.2) hold.

Since
$C_{\varphi_{z_{0}}}(\widetilde{\mathfrak{F}}_{1})=\widetilde{\mathfrak{F}}_{1}$,
we have

$C_{\varphi_{z_{0}}}X_{\widetilde{\mathfrak{F}}_{1}}=X_{\widetilde{\mathfrak{F}}_{2}}.$

Together with $\varphi_{z_{0}}^{-1}=\varphi_{z_{0}}$, one can see that

$\|C^{-1}_{\varphi_{z_{0}}}\|_{H^{2}_{\beta}}=\|C_{\varphi_{z_{0}}^{-1}}\|_{H^{2}_{\beta}}=\|C_{\varphi_{z_{0}}}\|_{H^{2}_{\beta}}=\|X_{\widetilde{\mathfrak{F}}_{1}}X^{-1}_{\widetilde{\mathfrak{F}}_{2}}\|_{H^{2}_{\beta}}\leq\|X_{\widetilde{\mathfrak{F}}_{1}}\|_{H^{2}_{\beta}}\|X^{-1}_{\widetilde{\mathfrak{F}}_{2}}\|_{H^{2}_{\beta}}.$

By the equation (2.2),

$\displaystyle\|X^{-1}_{\widetilde{\mathfrak{F}}_{2}}\|_{H^{2}_{\beta}}=$
$\displaystyle\|D_{\beta}X^{-1}_{\widetilde{\mathfrak{F}}_{2}}D^{-1}_{\beta}\|$
$\displaystyle=$
$\displaystyle\|D^{-1}_{tm}(D_{\beta}X^{*}_{\widetilde{\mathfrak{F}}_{2}}D^{-1}_{\beta})\|$
$\displaystyle\leq$
$\displaystyle\|D^{-1}_{tm}\|\|D_{\beta}X^{*}_{\widetilde{\mathfrak{F}}_{2}}D^{-1}_{\beta}\|$
$\displaystyle=$
$\displaystyle\|D^{-1}_{tm}\|\|D^{-1}_{\beta}X_{\widetilde{\mathfrak{F}}_{2}}D_{\beta}\|$
$\displaystyle=$
$\displaystyle\|D^{-1}_{tm}\|\|X_{\widetilde{\mathfrak{F}}_{2}}\|_{H^{2}_{\beta^{-1}}}.$

This finishes the proof. ∎

Then, our task is to estimate
$\|X_{\widetilde{\mathfrak{F}}_{1}}\|_{H^{2}_{\beta}}$,
$\|X_{\widetilde{\mathfrak{F}}_{2}}\|_{H^{2}_{\beta^{-1}}}$ and
$\|D^{-1}_{tm}\|$.

######  Lemma 2.6.

The notations as above. We have

$\|D^{-1}_{tm}\|\leq\frac{\sqrt{1+|z_{0}|^{2}\sup\limits_{n}(w_{n}^{2}+w_{n}^{-2})}}{1-|z_{0}|^{2}}\
\ \ \text{and}\ \
\|D_{tm}\|\leq\sqrt{1+|z_{0}|^{2}\sup\limits_{n}(w_{n}^{2}+w_{n}^{-2})}.$

###### Proof.

For each $n\in\mathbb{N}$,

$D^{*}_{n}D_{n}=\begin{bmatrix}1+|w_{n}z_{0}|^{2}&(w_{n}+w_{n}^{-1})\overline{z_{0}}\\\
(w_{n}+w_{n}^{-1})z_{0}&1+|z_{0}w_{n}^{-1}|^{2}\end{bmatrix}.$

It has two eigenvalues as follows

$\lambda_{1}^{(n)}=\frac{1+(w_{n}^{2}+w_{n}^{-2})|z_{0}|^{2}-\sqrt{(1+(w_{n}^{2}+w_{n}^{-2})|z_{0}|^{2})^{2}-4(1-|z_{0}|^{2})^{2}}}{2},$
$\lambda_{2}^{(n)}=\frac{1+(w_{n}^{2}+w_{n}^{-2})|z_{0}|^{2}+\sqrt{(1+(w_{n}^{2}+w_{n}^{-2})|z_{0}|^{2})^{2}-4(1-|z_{0}|^{2})^{2}}}{2}.$

Consequently,

$\|D_{n}\|=|\lambda_{2}^{(n)}|^{\frac{1}{2}}\leq\sqrt{1+|z_{0}|^{2}(w_{n}^{2}+w_{n}^{-2})},$
$\displaystyle\|D_{n}^{-1}\|=$
$\displaystyle|\lambda_{1}^{(n)}|^{-\frac{1}{2}}$ $\displaystyle=$
$\displaystyle\left|\frac{2}{1+(w_{n}^{2}+w_{n}^{-2})|z_{0}|^{2}-\sqrt{(1+(w_{n}^{2}+w_{n}^{-2})|z_{0}|^{2})^{2}-4(1-|z_{0}|^{2})^{2}}}\right|^{\frac{1}{2}}$
$\displaystyle=$
$\displaystyle\left|\frac{1+(w_{n}^{2}+w_{n}^{-2})|z_{0}|^{2}+\sqrt{(1+(w_{n}^{2}+w_{n}^{-2})|z_{0}|^{2})^{2}-4(1-|z_{0}|^{2})^{2}}}{2(1-|z_{0}|^{2})^{2}}\right|^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\frac{\sqrt{1+|z_{0}|^{2}(w_{n}^{2}+w_{n}^{-2})}}{1-|z_{0}|^{2}}.$

Then,

$\|D_{tm}\|=\sup\limits_{n}\|D_{n}\|\leq\sqrt{1+|z_{0}|^{2}\sup\limits_{n}(w_{n}^{2}+w_{n}^{-2})},$
$\|D^{-1}_{tm}\|=\sup\limits_{n}\|D^{-1}_{n}\|\leq\frac{\sqrt{1+|z_{0}|^{2}\sup\limits_{n}(w_{n}^{2}+w_{n}^{-2})}}{1-|z_{0}|^{2}}.$

∎

######  Lemma 2.7.

The notations as above. We have

$\|X_{{\mathfrak{F}}}\|_{H^{2}_{\beta}}\leq\|X_{\widetilde{\mathfrak{F}}_{1}}\|_{H^{2}_{\beta}}\leq(1+\|M_{z}\|_{H^{2}_{\beta}})\|X_{{\mathfrak{F}}}\|_{H^{2}_{\beta}}$

and

$\|X_{{\mathfrak{F}}}\|_{H^{2}_{\beta^{-1}}}\leq\|X_{\widetilde{\mathfrak{F}}_{2}}\|_{H^{2}_{\beta^{-1}}}\leq(1+\|M_{\varphi_{z_{0}}}\|_{H^{2}_{\beta^{-1}}})\|X_{{\mathfrak{F}}}\|_{H^{2}_{\beta^{-1}}}.$

###### Proof.

One can see that

$\Gamma_{H^{2}_{\beta}}(\widetilde{\mathfrak{F}}_{1,\beta})=\langle\widetilde{\mathfrak{F}}_{1,\beta},\widetilde{\mathfrak{F}}_{1,\beta}\rangle_{H^{2}_{\beta}}=\begin{bmatrix}\Gamma_{11}&\Gamma_{12}\\\
\Gamma_{21}&\Gamma_{22}\\\
\end{bmatrix}=\begin{bmatrix}\langle\mathfrak{F}_{\beta},\mathfrak{F}_{\beta}\rangle_{H^{2}_{\beta}}&\langle\mathfrak{F}_{\beta},M_{z}\mathfrak{F}_{\beta}\rangle_{H^{2}_{\beta}}\\\
\langle
M_{z}\mathfrak{F}_{\beta},\mathfrak{F}_{\beta}\rangle_{H^{2}_{\beta}}&\langle
M_{z}\mathfrak{F}_{\beta},M_{z}\mathfrak{F}_{\beta}\rangle_{H^{2}_{\beta}}\\\
\end{bmatrix},$

where

$\displaystyle\Gamma_{11}=\langle\mathfrak{F}_{\beta},\mathfrak{F}_{\beta}\rangle_{H^{2}_{\beta}}=(D^{-1}_{\beta}X^{*}_{\mathfrak{F}}D_{\beta})(D_{\beta}X_{\mathfrak{F}}D^{-1}_{\beta}),$
$\displaystyle\Gamma_{12}=\langle\mathfrak{F}_{\beta},M_{z}\mathfrak{F}_{\beta}\rangle_{H^{2}_{\beta}}=(D^{-1}_{\beta}X^{*}_{\mathfrak{F}}D_{\beta})(D^{-1}_{\beta}M_{z}^{*}D_{\beta})(D_{\beta}X_{\mathfrak{F}}D^{-1}_{\beta}),$
$\displaystyle\Gamma_{21}=\langle
M_{z}\mathfrak{F}_{\beta},\mathfrak{F}_{\beta}\rangle_{H^{2}_{\beta}}=(D^{-1}_{\beta}X^{*}_{\mathfrak{F}}D_{\beta})(D_{\beta}M_{z}D^{-1}_{\beta})(D_{\beta}X_{\mathfrak{F}}D^{-1}_{\beta}),$
$\displaystyle\Gamma_{22}=\langle
M_{z}\mathfrak{F}_{\beta},M_{z}\mathfrak{F}_{\beta}\rangle_{H^{2}_{\beta}}=(D^{-1}_{\beta}X^{*}_{\mathfrak{F}}D_{\beta})(D^{-1}_{\beta}M_{z}^{*}D_{\beta})(D_{\beta}M_{z}D^{-1}_{\beta})(D_{\beta}X_{\mathfrak{F}}D^{-1}_{\beta}).$

Consequently,

$\displaystyle\|\Gamma_{11}\|=\|D_{\beta}X_{\mathfrak{F}}D^{-1}_{\beta})\|^{2}=\|X_{{\mathfrak{F}}}\|_{H^{2}_{\beta}}^{2},$
$\displaystyle\|\Gamma_{12}\|\leq\|D^{-1}_{\beta}X^{*}_{\mathfrak{F}}D_{\beta}\|\|D^{-1}_{\beta}M_{z}^{*}D_{\beta}\|\|(D_{\beta}X_{\mathfrak{F}}D^{-1}_{\beta}\|=\|M_{z}\|_{H^{2}_{\beta}}\|X_{{\mathfrak{F}}}\|_{H^{2}_{\beta}}^{2},$
$\displaystyle\|\Gamma_{21}\|\leq\|D^{-1}_{\beta}X^{*}_{\mathfrak{F}}D_{\beta}\|\|D_{\beta}M_{z}D^{-1}_{\beta}\|\|D_{\beta}X_{\mathfrak{F}}D^{-1}_{\beta}\|=\|M_{z}\|_{H^{2}_{\beta}}\|X_{{\mathfrak{F}}}\|_{H^{2}_{\beta}}^{2},$
$\displaystyle\|\Gamma_{22}\|\leq\|D^{-1}_{\beta}X^{*}_{\mathfrak{F}}D_{\beta}\|\|D_{\beta}M_{z}D^{-1}_{\beta}\|^{2}\|D_{\beta}X_{\mathfrak{F}}D^{-1}_{\beta}\|=\|M_{z}\|_{H^{2}_{\beta}}^{2}\|X_{{\mathfrak{F}}}\|_{H^{2}_{\beta}}^{2}.$

Then,

$\|X_{{\mathfrak{F}}}\|_{H^{2}_{\beta}}^{2}=\|\Gamma_{11}\|\leq\|\Gamma_{H^{2}_{\beta}}(\widetilde{\mathfrak{F}}_{1,\beta})\|=\|X_{\widetilde{\mathfrak{F}}_{1}}\|_{H^{2}_{\beta}}^{2}$

and

$\displaystyle\|X_{\widetilde{\mathfrak{F}}_{1}}\|_{H^{2}_{\beta}}^{2}=\Gamma_{H^{2}_{\beta}}(\widetilde{\mathfrak{F}}_{1,\beta})\leq$
$\displaystyle\|\Gamma_{11}\|+\|\Gamma_{12}\|+\|\Gamma_{21}\|+\|\Gamma_{22}\|$
$\displaystyle\leq$
$\displaystyle(1+\|M_{z}\|_{H^{2}_{\beta}})^{2}\|X_{{\mathfrak{F}}}\|_{H^{2}_{\beta}}^{2}.$

Therefore,

$\|X_{{\mathfrak{F}}}\|_{H^{2}_{\beta}}\leq\|X_{\widetilde{\mathfrak{F}}_{1}}\|_{H^{2}_{\beta}}\leq(1+\|M_{z}\|_{H^{2}_{\beta}})\|X_{{\mathfrak{F}}}\|_{H^{2}_{\beta}},$

and similarly, we have

$\|X_{{\mathfrak{F}}}\|_{H^{2}_{\beta^{-1}}}\leq\|X_{\widetilde{\mathfrak{F}}_{2}}\|_{H^{2}_{\beta^{-1}}}\leq(1+\|M_{\varphi_{z_{0}}}\|_{H^{2}_{\beta^{-1}}})\|X_{{\mathfrak{F}}}\|_{H^{2}_{\beta^{-1}}}.$

∎

Recall that for $\lambda>0$, the weighted Dirichlet space $D_{\lambda}^{2}$
consists of those analytic functions $f$ on $\mathbb{D}$ with

$\|f\|_{D_{\lambda}^{2}}=\left(|f(0)|^{2}+\int_{\mathbb{D}}|f^{\prime}(z)|^{2}(1-|z|^{2})^{\lambda-1}\textrm{d}A(z)\right)^{\frac{1}{2}}<\infty$

where $\textrm{d}A(z)=\frac{1}{\pi}\textrm{d}x\textrm{d}y$ is the normalized
area measure on $\mathbb{D}$. As well known, the weighted Dirichlet space
$D_{\lambda}^{2}$ could be seemed as a weighted Hardy space
$H^{2}_{\beta^{(\lambda)}}$, where $\beta^{(\lambda)}_{0}=1$ and for
$k=1,2,\ldots$, $\beta_{k}^{(\lambda)}=\prod_{j=1}^{k}w_{j}^{(\lambda)}$ with
$w_{j}^{(\lambda)}=\sqrt{\frac{j+2\lambda+1}{j+1}}$.

For $N\in\mathbb{N}$, let
$\beta^{(N,1)}=\\{\beta^{(N,1)}_{k}\\}_{k=0}^{\infty}$ and
$\beta^{(N,2)}=\\{\beta^{(N,2)}_{k}\\}_{k=0}^{\infty}$, where

$w_{j}^{(N,1)}=\frac{j+N}{j},\ \
\beta_{k}^{(N,1)}=\prod_{j=1}^{k}w_{j}^{(N,1)}=C_{N+k}^{k}=\frac{(N+k)!}{N!\cdot
k!},$

and

$w_{j}^{(N,2)}=\frac{(j+1)^{N}}{j^{N}},\ \
\beta_{k}^{(N,2)}=\prod_{j=1}^{k}w_{j}^{(N,2)}=(k+1)^{N}.$

Furthermore, denote by $D_{N,1}^{2}$ and $D_{N,2}^{2}$ the weighted Hardy
space $H^{2}_{\beta^{(N,1)}}$ and $H^{2}_{\beta^{(N,2)}}$, respectively. It is
not difficult to see that both $D_{N,1}^{2}$ and $D_{N,2}^{2}$ are the
weighted Dirichlet space $D_{N}^{2}$ with equivalent norms.

######  Lemma 2.8.

Let $H^{2}_{\beta}$ be a weighted Hardy space of $N$-polynomial growth induced
by the weight sequence $\\{\beta_{n}\\}_{n=1}^{\infty}$. Then

$\|M_{z^{n}}\|_{H^{2}_{\beta}}\leq C_{N+k}^{k}=\beta^{(N,1)}_{n},\ \ \
\|M_{\frac{1}{1-\overline{z_{0}}z}}\|_{H^{2}_{\beta}}\leq\frac{1}{(1-|z_{0}|)^{N+1}},$

and

$\|M_{\frac{z_{0}-z}{1-\overline{z_{0}}z}}\|_{H^{2}_{\beta}}\leq\frac{N+2}{(1-|z_{0}|)^{N+1}}.$

###### Proof.

Since $w_{n}\leq\frac{n+1+N}{n+1}<\frac{n+N}{n}=w_{n}^{(N,1)}$ for all
$n=1,2,\ldots$, and $\\{w_{n}^{(N,1)}\\}$ is a decreasing sequence of positive
numbers, it is easy to see that

$\|M_{z^{n}}\|_{H^{2}_{\beta}}\leq\|M_{z^{n}}\|_{D_{N}^{2}}=C_{N+n}^{n}=\beta^{(N)}_{n}.$

Furthermore,

$\|M_{\frac{1}{1-\overline{z_{0}}z}}\|_{H^{2}_{\beta}}=\|\sum\limits_{n=0}^{\infty}z_{0}^{n}M_{z^{n}}\|_{H^{2}_{\beta}}\leq\sum\limits_{n=0}^{\infty}|z_{0}|^{n}\|M_{z^{n}}\|_{H^{2}_{\beta}}\leq\sum\limits_{n=0}^{\infty}|z_{0}|^{n}\beta^{(N,1)}_{n}.$

Then

$\frac{1}{(1-|z_{0}|)^{N+1}}=\sum\limits_{n=0}^{\infty}|z_{0}|^{n}C_{N+n}^{n}=\sum\limits_{n=0}^{\infty}|z_{0}|^{n}\beta^{(N,1)}_{n}\geq\|M_{\frac{1}{1-\overline{z_{0}}z}}\|_{H^{2}_{\beta}}$

and consequently,

$\|M_{\frac{z_{0}-z}{1-\overline{z_{0}}z}}\|_{H^{2}_{\beta}}\leq(|z_{0}|+\|M_{z}\|_{H^{2}_{\beta}})\|M_{\frac{1}{1-\overline{z_{0}}z}}\|_{H^{2}_{\beta}}\leq\frac{N+2}{(1-|z_{0}|)^{N+1}}.$

This finishes the proof. ∎

Define $D_{w},D:H^{2}_{\beta}\rightarrow H^{2}_{\beta}$ by, for any $f(z)\in
H^{2}_{\beta}$

$D_{w}f(z)=\sum\limits_{k=0}^{\infty}w_{k+1}\widehat{f}(k)z^{k}\ \ \text{and}\
\ Df(z)=\sum\limits_{k=0}^{\infty}\frac{k+2}{k+1}\cdot\widehat{f}(k)z^{k}.$

We could write the two operators $D_{w}$ and $D$ in matrix form under the
orthogonal base $\\{z^{k}\\}_{k=0}^{\infty}$ as follows,

$D_{w}=\begin{bmatrix}w_{1}&0&0&\cdots&0&\cdots\\\
0&w_{2}&0&\cdots&0&\cdots\\\ 0&0&w_{3}&\cdots&0&\cdots\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&0&\cdots&w_{k}&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix}\ \ \text{and}\ \
D=\begin{bmatrix}2&0&0&\cdots&0&\cdots\\\ 0&\frac{3}{2}&0&\cdots&0&\cdots\\\
0&0&\frac{4}{3}&\cdots&0&\cdots\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&\cdots\frac{k+1}{k}&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix},$

Let $\widetilde{\beta}_{n}=(n+1)\beta_{n}$ for $n=0,1,\ldots$ and
$\widetilde{w}_{k}=\frac{\beta_{k}}{\beta_{k-1}}=w_{k}\cdot\frac{k+1}{k}$ for
$k=1,2,\ldots$. Moreover, define $D_{\widetilde{w}}:H^{2}_{\beta}\rightarrow
H^{2}_{\beta}$ by, for any $f(z)\in H^{2}_{\beta}$

$D_{w}f(z)=\sum\limits_{k=0}^{\infty}w_{k+1}\widehat{f}(k)z^{k},$

which could be written in a matrix form under the orthogonal base
$\\{z^{k}\\}_{k=0}^{\infty}$ as follows,

$D_{\widetilde{w}}=DD_{w}=\begin{bmatrix}\widetilde{w_{1}}&0&0&\cdots&0&\cdots\\\
0&\widetilde{w_{2}}&0&\cdots&0&\cdots\\\
0&0&\widetilde{w_{3}}&\cdots&0&\cdots\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&0&\cdots&w_{k}&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix}.$

######  Lemma 2.9.

Let $H^{2}_{\beta}$ be the weighted Hardy space induced by a weight sequence
$w=\\{w_{k}\\}_{k=1}^{\infty}$ with $w_{k}\rightarrow 1$. Denote
$\widetilde{\beta}=\\{\widetilde{\beta}_{n}\\}_{n=0}^{\infty}$, where
$\widetilde{\beta}_{n}=(n+1)\beta_{n}$. Then,

$\|C_{B_{0}}\|_{H^{2}_{\widetilde{\beta}}}\leq\|D_{\widetilde{w}}\|\cdot\|M_{B^{\prime}_{0}}\|_{H^{2}_{\beta}}\cdot\|C_{B_{0}}\|_{H^{2}_{\beta}}.$

###### Proof.

Following from the proof of Proposition 2.15 in [24], one can see that

$\langle\frac{(B_{0})^{i+1}}{\widetilde{\beta_{i}}},\frac{(B_{0})^{j+1}}{\widetilde{\beta_{j}}}\rangle_{H^{2}_{\widetilde{\beta}}}=\langle
DD_{w}M_{B^{\prime}_{0}}\left(\frac{(B_{0})^{i}}{\beta_{i}}\right),DD_{w}M_{B^{\prime}_{0}}\left(\frac{(B_{0})^{j}}{\beta_{j}}\right)\rangle_{H^{2}_{\beta}},$

and then

$\Gamma_{H^{2}_{\widetilde{\beta}}}\left(\left\\{\frac{(B_{0})^{n+1}}{\widetilde{\beta}_{n}}\right\\}_{n=0}^{\infty}\right)=\Gamma_{H^{2}_{\beta}}\left(\left\\{DD_{w}M_{B^{\prime}_{0}}\left(\frac{(B_{0})^{n}}{\beta_{n}}\right)\right\\}_{n=0}^{\infty}\right).$

Notice that

$\Gamma_{H^{2}_{\widetilde{\beta}}}\left(\left\\{\frac{(B_{0})^{n}}{\widetilde{\beta}_{n}}\right\\}_{n=0}^{\infty}\right)=\begin{bmatrix}1&0\\\
0&\Gamma_{H^{2}_{\widetilde{\beta}}}(\\{\frac{B_{0}^{n}}{\widetilde{\beta}_{n}}\\}_{n=1}^{\infty})\end{bmatrix}=\begin{bmatrix}1&0\\\
0&D^{-1}_{\widetilde{w}}\Gamma_{H^{2}_{\widetilde{\beta}}}(\\{\frac{B_{0}^{n+1}}{\widetilde{\beta}_{n}}\\}_{n=0}^{\infty})D^{-1}_{\widetilde{w}}.\end{bmatrix}$

Then,

$\displaystyle\|X_{\mathfrak{F}}\|^{2}_{H^{2}_{\widetilde{\beta}}}=$
$\displaystyle\left\|\Gamma_{H^{2}_{\widetilde{\beta}}}\left(\left\\{\frac{(B_{0})^{n}}{\widetilde{\beta}_{n}}\right\\}_{n=0}^{\infty}\right)\right\|$
$\displaystyle=$
$\displaystyle\max\left\\{1,\left\|D^{-1}_{\widetilde{w}}\Gamma_{H^{2}_{\widetilde{\beta}}}\left(\left\\{\frac{(B_{0})^{n+1}}{\widetilde{\beta}_{n}}\right\\}_{n=0}^{\infty}\right)D^{-1}_{\widetilde{w}}\right\|\right\\}.$

Since

$\displaystyle\left\|D^{-1}_{\widetilde{w}}\Gamma_{H^{2}_{\widetilde{\beta}}}\left(\left\\{\frac{(B_{0})^{n+1}}{\widetilde{\beta}_{n}}\right\\}_{n=0}^{\infty}\right)D^{-1}_{\widetilde{w}}\right\|\leq$
$\displaystyle\left\|\Gamma_{H^{2}_{\widetilde{\beta}}}\left(\left\\{\frac{(B_{0})^{n+1}}{\widetilde{\beta}_{n}}\right\\}_{n=0}^{\infty}\right)\right\|$
$\displaystyle=$
$\displaystyle\left\|\Gamma_{H^{2}_{\beta}}\left(\left\\{DD_{w}M_{B^{\prime}_{0}}\left(\frac{(B_{0})^{n}}{\beta_{n}}\right)\right\\}_{n=0}^{\infty}\right)\right\|$
$\displaystyle=$
$\displaystyle\|D_{\widetilde{w}}M_{B^{\prime}_{0}}X_{\mathfrak{F}}\|^{2}_{H^{2}_{\beta}}$
$\displaystyle\leq$
$\displaystyle(\|D_{\widetilde{w}}\|_{H^{2}_{\beta}}\|M_{B^{\prime}_{0}}\|_{H^{2}_{\beta}}\|X_{\mathfrak{F}}\|_{H^{2}_{\beta}})^{2}$
$\displaystyle=$
$\displaystyle(\|D_{\widetilde{w}}\|\|M_{B^{\prime}_{0}}\|_{H^{2}_{\beta}}\|C_{B_{0}}\|_{H^{2}_{\beta}})^{2},$

we have

$\displaystyle\|C_{B_{0}}\|_{H^{2}_{\widetilde{\beta}}}=$
$\displaystyle\|X_{\mathfrak{F}}\|_{H^{2}_{\widetilde{\beta}}}$
$\displaystyle\leq$
$\displaystyle\max\\{1,\|D_{\widetilde{w}}\|\|M_{B^{\prime}_{0}}\|_{H^{2}_{\beta}}\|C_{B_{0}}\|_{H^{2}_{\beta}}\\}$
$\displaystyle=$
$\displaystyle\|D_{\widetilde{w}}\|\|M_{B^{\prime}_{0}}\|_{H^{2}_{\beta}}\|C_{B_{0}}\|_{H^{2}_{\beta}}.$

∎

The following result of C. Cowen [12] is also useful in our estimation.

######  Theorem 2.10 (Theorem 7 in [12]).

Suppose that $H^{2}_{\beta}$ and $H^{2}_{\beta^{\prime}}$ are weighted Hardy
spaces and

$w_{k+1}=\frac{\beta_{k+1}}{\beta_{k}}\geq\frac{\beta^{\prime}_{k+1}}{\beta^{\prime}_{k}}=w^{\prime}_{k+1},\
\ \ \text{for}\ k=0,1,2,\ldots.$

Let $\psi(z)$ be an analytic function on $\mathbb{D}$ with
$\psi(\mathbb{D})\subseteq\mathbb{D}$ and $\psi(0)=0$. If $C_{\psi}$ is
bounded on $H^{2}_{\beta}$, then $C_{\psi}$ is bounded on
$H^{2}_{\beta^{\prime}}$ and
$\|C_{\psi}\|_{H^{2}_{\beta}}\geq\|C_{\psi}\|_{H^{2}_{\beta^{\prime}}}$.

Now, based on all of the previous preliminaries, we can estimate the norms of
the composition operators induced by $B_{0}$ and disk automorphisms acting on
a weighted Hardy space of $N$-polynomial growth.

######  Proposition 2.11.

Let $H^{2}_{\beta}$ be a weighted Hardy space of $N$-polynomial growth. Then

$\|C_{B_{0}}\|_{H^{2}_{\beta}}\leq\frac{K_{2}K_{1}^{N}\prod_{j=1}^{N}(j^{3}+5j^{2}+2j)}{(1-|z_{0}|)^{N(N+1)}},$

where $K_{1}$ and $K_{2}$ are positive constants.

###### Proof.

Obviously,

$B^{\prime}_{0}(z)=\left(z\cdot\frac{z_{0}-z}{1-\overline{z_{0}}z}\right)^{\prime}=\frac{z_{0}-2z+\overline{z_{0}}z^{2}}{(1-\overline{z_{0}}z)^{2}}.$

By Lemma 2.8, for $j=1,2,\ldots,N$,

$\displaystyle\|M_{B^{\prime}_{0}}\|_{D^{2}_{j-1,1}}\leq$
$\displaystyle(|z_{0}|+2\|M_{z}\|_{D^{2}_{j-1,1}}+|z_{0}|\|M_{z^{2}}\|_{D^{2}_{j-1,1}})\|M_{\frac{1}{1-\overline{z_{0}}z}}\|^{2}_{D^{2}_{j-1,1}}$
$\displaystyle\leq$ $\displaystyle\frac{1+2j+j(j+1)/2}{(1-|z_{0}|)^{2j}}$
$\displaystyle=$ $\displaystyle\frac{j^{2}+5j+2}{2(1-|z_{0}|)^{2j}}.$

Moreover, since both $D_{N,1}^{2}$ and $D_{N,2}^{2}$ are the weighted
Dirichlet space $D_{N}^{2}$ with equivalent norm, there exists constants
$K_{1}>0$ and $K_{2}>0$ such that

$\|M_{B^{\prime}_{0}}\|_{D^{2}_{j-1,2}}\leq
K_{1}\|M_{B^{\prime}_{0}}\|_{D^{2}_{j-1,1}}\leq\frac{K_{1}(j^{2}+5j+2)}{2(1-|z_{0}|)^{2j}}$

and

$\|C_{B_{0}}\|_{D^{2}_{N,1}}\leq K_{2}\|C_{B_{0}}\|_{D^{2}_{N,2}}.$

In addition, for $j=1,2,\ldots,N$,

$\|DD_{w^{(j-1,1)}}\|=2j.$

Then, by Lemma 2.9 and $C_{B_{0}}$ being an isometry on the classical Hardy
space,

$\displaystyle\|C_{B_{0}}\|_{D^{2}_{N,2}}\leq$
$\displaystyle\prod\limits_{j=1}^{N}\|DD_{w^{(j-1,1)}}\|\cdot\prod\limits_{j=1}^{N}\|M_{C_{B^{\prime}_{0}}}\|_{D^{2}_{j-1,2}}\cdot\|C_{B_{0}}\|_{D^{2}_{0,2}}$
$\displaystyle\leq$
$\displaystyle\prod\limits_{j=1}^{N}(2j)\cdot\prod\limits_{j=1}^{N}\frac{K_{1}(j^{2}+5j+2)}{2(1-|z_{0}|)^{2j}}\cdot\|C_{B_{0}}\|$
$\displaystyle=$
$\displaystyle\frac{K_{1}^{N}\prod_{j=1}^{N}(j^{3}+5j^{2}+2j)}{(1-|z_{0}|)^{N(N+1)}}.$

Thus, it follows from Theorem 2.10 that

$\|C_{B_{0}}\|_{H^{2}_{\beta}}\leq\|C_{B_{0}}\|_{D^{2}_{N,1}}\leq
K_{2}\|C_{B_{0}}\|_{D^{2}_{N,2}}\leq\frac{K_{2}K_{1}^{N}\prod_{j=1}^{N}(j^{3}+5j^{2}+2j)}{(1-|z_{0}|)^{N(N+1)}}.$

∎

######  Theorem 2.12.

Let $H^{2}_{\beta}$ be a weighted Hardy space of $N$-polynomial growth induced
by the weight sequence $w=\\{w_{k}\\}_{k=1}^{\infty}$. Then for any
$\varphi\in{\rm Aut(\mathbb{D})}$, there exists a positive number $K$ such
that

$\|C_{\varphi}\|_{H^{2}_{\beta}}\leq\frac{K}{(1-|z_{0}|)^{2N^{2}+3N+2}}.$

###### Proof.

By Theorem 2.5,

$\|C_{\varphi}\|_{H^{2}_{\beta}}\leq(1+\|M_{z}\|_{H^{2}_{\beta}})(1+\|M_{\varphi}\|_{H^{2}_{\beta^{-1}}})\|X_{\mathfrak{F}}\|_{H^{2}_{\beta}}\|X_{\mathfrak{F}}\|_{H^{2}_{\beta^{-1}}}\|D^{-1}_{tm}\|,$

it suffices to estimate the items in the right of the above inequality.
Following from Lemma 2.8, one can see

$\|M_{z}\|_{H^{2}_{\beta}}\leq N+1\ \ \ \text{and}\ \ \
\|M_{\frac{z_{0}-z}{1-\overline{z_{0}}z}}\|_{H^{2}_{\beta}}\leq\frac{N+2}{(1-|z_{0}|)^{N+1}}.$

By Proposition 2.11, we have

$\|X_{\mathfrak{F}}\|_{H^{2}_{\beta}}=\|C_{B_{0}}\|_{H^{2}_{\beta}}\leq\frac{K_{2}K_{1}^{N}\prod_{j=1}^{N}(j^{3}+5j^{2}+2j)}{(1-|z_{0}|)^{N(N+1)}}$

and

$\|X_{\mathfrak{F}}\|_{H^{2}_{\beta^{-1}}}=\|C_{B_{0}}\|_{H^{2}_{\beta^{-1}}}\leq\frac{K_{2}K_{1}^{N}\prod_{j=1}^{N}(j^{3}+5j^{2}+2j)}{(1-|z_{0}|)^{N(N+1)}}.$

Notice that

$\sqrt{1+|z_{0}|^{2}\sup\limits_{n}(w_{n}^{2}+w_{n}^{-2})}\leq\sqrt{\sup\limits_{n}(1+w_{n}^{2}+w_{n}^{-2})}\leq\sqrt{\sup\limits_{n}(w_{n}+w_{n}^{-1})^{2}}\leq
N+2.$

Then by Lemma 2.6,

$\|D^{-1}_{tm}\|=\sup\limits_{n}\|D^{-1}_{n}\|\leq\frac{\sqrt{1+|z_{0}|^{2}\sup\limits_{n}(w_{n}^{2}+w_{n}^{-2})}}{1-|z_{0}|^{2}}\leq\frac{N+2}{1-|z_{0}|^{2}}\leq\frac{N+2}{1-|z_{0}|}.$

Therefore, if let

$K=K^{2}_{2}K_{1}^{2N}(N+2)^{2}(N+3)\prod_{j=1}^{N}(j^{3}+5j^{2}+2j)^{2},$

we obtain that

$\displaystyle\|C_{\varphi}\|_{H^{2}_{\beta}}\leq$
$\displaystyle(1+\|M_{z}\|_{H^{2}_{\beta}})(1+\|M_{\varphi}\|_{H^{2}_{\beta^{-1}}})\|X_{\mathfrak{F}}\|_{H^{2}_{\beta}}\|X_{\mathfrak{F}}\|_{H^{2}_{\beta^{-1}}}\|D^{-1}_{tm}\|$
$\displaystyle\leq$
$\displaystyle(N+2)\left(1+\frac{N+2}{(1-|z_{0}|)^{N+1}}\right)\left(\frac{K_{2}K_{1}^{N}\prod_{j=1}^{N}(j^{3}+5j^{2}+2j)}{(1-|z_{0}|)^{N(N+1)}}\right)^{2}\frac{N+2}{1-|z_{0}|}$
$\displaystyle\leq$
$\displaystyle\frac{K^{2}_{2}K_{1}^{2N}(N+2)^{2}(N+3)\prod_{j=1}^{N}(j^{3}+5j^{2}+2j)^{2}}{(1-|z_{0}|)^{2N^{2}+3N+2}}$
$\displaystyle\leq$ $\displaystyle\frac{K}{(1-|z_{0}|)^{2N^{2}+3N+2}}.$

∎

Obviously, if $H^{2}_{\beta}$ is a weighted Hardy space of $M$-polynomial
growth for some $M>0$, then for any positive integer $N$ more than $M$,
$H^{2}_{\beta}$ is also a weighted Hardy space of $N$-polynomial growth. Thus,
we could estimate the norms of the composition operators induced by disk
automorphisms acting on any weighted Hardy space of polynomial growth.

##  3\. Spectra of composition operators with symbols of disk automorphisms

In this section, we study the spectra of composition operators with symbols of
disk automorphisms acting on weighted Hardy spaces of polynomial growth. Every
element of $\textrm{Aut}(\mathbb{D})$ has two fixed points (counting
multiplicity) in $\mathbb{C}$, and thus can be classified by the location of
the fixed points.

######  Definition 3.1.

For an element $\varphi$ in $\textrm{Aut}\mathbb{D}$,

  1. (1)

$\varphi$ is said to be elliptic, if it has one fixed point in $\mathbb{D}$
and one in the complement of $\overline{\mathbb{D}}$;

  2. (2)

$\varphi$ is said to be parabolic, if it has just one fixed point on the unit
circle $\partial\mathbb{D}$ (of multiplicity $2$);

  3. (3)

$\varphi$ is said to be hyperbolic, if it has two distinct fixed points on
$\partial\mathbb{D}$.

Two disk automorphisms $\varphi$ and $\psi$ are called conformally equivalent,
if there exists a disk automorphism $\tau$ such that

$\psi=\tau\circ\varphi\circ\tau^{-1}.$

The main advantage of conformal equivalence is in the placement of the fixed
points. Every elliptic disk automorphism is conformally equivalent to one
whose fixed point in $\mathbb{D}$ is $0$.

######  Lemma 3.2.

Let $\varphi$ be an elliptic disk automorphism with fixed point $\alpha$ in
$\mathbb{D}$. Then $\varphi$ is conformally equivalent to $\psi(z)=\lambda z$,
where $\lambda=\varphi^{\prime}(\alpha)$.

This result is well-known, and one can find a proof in [1] (Lemma 2.1 in it).

Every parabolic disk automorphism is conformally equivalent to one whose fixed
point (of multiplicity $2$) is $1$.

######  Lemma 3.3 (Lemma 4.1.2 in [40]).

Let $\varphi$ be a parabolic disk automorphism. Then $\varphi$ is conformally
equivalent to either

$\psi_{1}(z)=\frac{(1+\mathbf{i})z-1}{z+\mathbf{i}-1}\ \ \ \text{or}\ \ \
\psi_{2}(z)=\frac{(1-\mathbf{i})z-1}{z-\mathbf{i}-1}.$

######  Remark 3.4.

After a straight computation, one can see

$\psi_{1}(z)=\psi_{2}^{-1}(z),$

and for any positive integer $n$,

$\psi_{1}^{n}(z)=\underset{n}{\underbrace{\psi_{1}\circ\cdots\circ\psi_{1}}}(z)=\frac{(n+\mathbf{i})z-n}{nz+\mathbf{i}-n},$

and

$\psi_{2}^{n}(z)=\underset{n}{\underbrace{\psi_{2}\circ\cdots\circ\psi_{2}}}(z)=\frac{(n-\mathbf{i})z-n}{nz-\mathbf{i}-n}.$

Moreover, for any positive integer $n$, $\psi_{1}^{n}(z)$ is conformally
equivalent to $\psi_{1}(z)$, and $\psi_{2}^{n}(z)$ is conformally equivalent
to $\psi_{2}(z)$. In fact, let

$\tau_{n}(z)=\frac{(n+1)z+(n-1)}{(n-1)z+(n+1)},$

then

$\tau_{n}\circ\psi_{1}\circ\tau_{n}^{-1}=\psi_{1}^{n}\ \ \ \text{and}\ \ \
\tau_{n}\circ\psi_{2}\circ\tau_{n}^{-1}=\psi_{2}^{n}.$

Every hyperbolic disk automorphism is conformally equivalent to one whose
fixed points in $\partial\mathbb{D}$ are $1$ and $-1$.

######  Lemma 3.5 (Theorem 6 in [38]).

Let $\varphi$ be a hyperbolic disk automorphism. Then, $\varphi$ is
conformally equivalent to

$\psi(z)=\frac{z+r}{1+rz},$

where $r=|\varphi(0)|$.

######  Lemma 3.6.

Let $H^{2}_{\beta}$ be a weighted Hardy space of polynomial growth. If
$\varphi$ and $\psi$ are conformally equivalent disk automorphisms, then the
induced composition operators $C_{\varphi}$ and $C_{\psi}$ acting on
$H^{2}_{\beta}$ are similar.

###### Proof.

Since $\varphi$ and $\psi$ are conformally equivalent, there exists a disk
automorphism $\tau$ such that

$\psi=\tau\circ\varphi\circ\tau^{-1}.$

By Theorem 2.5, $C_{\tau}$ is an invertible bounded operator on
$H^{2}_{\beta}$. For any $f\in H^{2}_{\beta}$, we have

$C_{\tau}^{-1}\circ C_{\varphi}\circ C_{\tau}(f)=C_{\tau^{-1}}\circ
C_{\varphi}\circ
C_{\tau}(f)=C_{\tau\circ\varphi\circ\tau^{-1}}(f)=C_{\psi}(f),$

then $C_{\varphi}$ and $C_{\psi}$ acting on $H^{2}_{\beta}$ are similar. ∎

######  Theorem 3.7.

Let $H^{2}_{\beta}$ be a weighted Hardy space of polynomial growth. If
$\varphi$ is an elliptic disk automorphism with fixed point $\alpha$ in
$\mathbb{D}$, then $C_{\varphi}$ acting on $H^{2}_{\beta}$ is a diagonal
operator and hence it is a normal operator. Moreover, the spectrum of
$C_{\varphi}$ acting on $H^{2}_{\beta}$ is as follows

$\sigma(C_{\varphi})=\overline{\\{(\varphi^{\prime}(\alpha))^{n};\
n=0,1,2,\ldots\\}}.$

###### Proof.

By Lemma 3.2, $\varphi$ is conformally equivalent to $\psi(z)=\lambda z$,
where $\lambda=\varphi^{\prime}(\alpha)$. Since the composition operator
induced by a disk automorphism is an invertible bounded operator on
$H^{2}_{\beta}$, we have $C_{\varphi}$ is similar to $C_{\psi}$ acting on
$H^{2}_{\beta}$. Notice that the composition operator $C_{\psi}$ is a diagonal
operator, more precisely, it has the following matrix presentation under the
orthogonal base $\\{\frac{z^{n}}{\beta_{n}}\\}_{k=1}^{\infty}$

$C_{\psi}=\begin{bmatrix}1&0&0&\cdots&0&\cdots\\\
0&\lambda&0&\cdots&0&\cdots\\\ 0&0&\lambda^{2}&\cdots&0&\cdots\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&0&\cdots\lambda^{n}&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix},$

Then $C_{\varphi}$ acting on $H^{2}_{\beta}$ is a diagonal operator and hence
it is a normal operator. Moreover,

$\sigma(C_{\varphi})=\sigma(C_{\psi})=\overline{\\{(\varphi^{\prime}(\alpha))^{n};\
n=0,1,2,\\}}.$

∎

######  Theorem 3.8.

Let $H^{2}_{\beta}$ be a weighted Hardy space of polynomial growth. If
$\varphi$ is a parabolic disk automorphism, then the spectrum of $C_{\varphi}$
acting on $H^{2}_{\beta}$ is the unit circle.

###### Proof.

Suppose that $H^{2}_{\beta}$ is a weighted Hardy space of $N$-polynomial
growth. By Lemma 3.3, Remark 3.4 and Lemma 3.6, it suffices to consider the
spectra of

$\psi(z)=\frac{(1+\mathbf{i})z-1}{z+\mathbf{i}-1}\ \ \ \text{and}\ \ \
\psi^{-1}(z)=\frac{(1-\mathbf{i})z-1}{z-\mathbf{i}-1}.$

We will prove the spectrum of $C_{\psi}$ acting on $H^{2}_{\beta}$ is the unit
circle divided into three steps.

$\bf(1)$ The spectrum of $C_{\psi}$ acting on $H^{2}_{\beta}$ is contained in
the unit circle.

By Theorem 2.12, we have for any positive integer $n$, there exists a positive
number $K$ such that

$\|C^{n}_{\psi}\|_{H^{2}_{\beta}}=\|C_{\psi^{n}}\|_{H^{2}_{\beta}}\leq\frac{K}{(1-|\frac{n}{n+\mathbf{i}}|)^{2N^{2}+3N+2}},$

and

$\|C^{-n}_{\psi}\|_{H^{2}_{\beta}}=\|C_{\psi^{-n}}\|_{H^{2}_{\beta}}\leq\frac{K}{(1-|\frac{n}{n-\mathbf{i}}|)^{2N^{2}+3N+2}}.$

Consequently

$\rho(C_{\psi})=\lim\limits_{n\rightarrow+\infty}\|C^{n}_{\psi}\|^{\frac{1}{n}}_{H^{2}_{\beta}}\leq
1,$

and

$\rho(C^{-1}_{\psi})=\lim\limits_{n\rightarrow+\infty}\|C^{-n}_{\psi}\|^{\frac{1}{n}}_{H^{2}_{\beta}}\leq
1.$

Therefore,

$\sigma(C_{\psi})\subseteq\partial\mathbb{D}.$

$\bf(2)$ The spectrum of $C_{\psi}$ acting on $H^{2}_{\beta}$ is either the
whole unit circle or the single point set $\\{1\\}$.

By Remark 3.4 and Lemma 3.6, for any positive integer $n$, $C_{\psi}$ is
similar to $C^{n}_{\psi}$ on $H^{2}_{\beta}$ and consequently,

$\sigma(C_{\psi})=\sigma(C^{n}_{\psi}),$

that means if $\lambda\in\sigma(C_{\psi})$, then
$\lambda^{n}\in\sigma(C_{\psi})$.

Now, suppose that the spectrum of $C_{\psi}$ on $H^{2}_{\beta}$ is not the
whole unit circle. Denote $\lambda=\mathrm{e}^{t2\pi\mathbf{i}}$,
$t\in\mathbb{R}$. Then, if $t$ is a irrational number, $\lambda$ is not in
$\sigma(C_{\psi})$. Because if $\lambda$ is in $\sigma(C_{\psi})$, then

$\sigma(C_{\psi})\supseteq\overline{\\{\lambda^{n};\
n\in\mathbb{N}\\}}=\partial\mathbb{D}.$

If $\lambda\in\sigma(C_{\psi})$ and $t=\frac{q}{p}$ is a rational number,
where $p$ and $q$ are two coprime positive integers, then

$\\{1,\mathrm{e}^{\frac{1}{p}\cdot
2\pi\mathbf{i}},\mathrm{e}^{\frac{2}{p}\cdot
2\pi\mathbf{i}},\ldots,\mathrm{e}^{\frac{p-1}{p}\cdot
2\pi\mathbf{i}}\\}\subseteq\sigma(C_{\psi}).$

Since $\sigma(C_{\psi})$ is a closed proper subset of $\partial\mathbb{D}$
which contains no irrational points, $\sigma(C_{\psi})$ is a finite set and
consequently we could write $\sigma(C_{\psi})$ as follows

$\sigma(C_{\psi})=\\{1,\mathrm{e}^{\frac{1}{p_{1}}\cdot
2\pi\mathbf{i}},\ldots,\mathrm{e}^{\frac{p_{1}-1}{p_{1}}\cdot
2\pi\mathbf{i}},\ldots,\mathrm{e}^{\frac{1}{p_{M}}\cdot
2\pi\mathbf{i}},\ldots,\mathrm{e}^{\frac{p_{M}-1}{p_{M}}\cdot
2\pi\mathbf{i}}\\},$

where $M$ is positive integer. Furthermore,

$\sigma(C_{\psi})=\sigma(C^{p_{1}p_{2}\cdots
p_{M}}_{\psi})=\\{\lambda^{p_{1}p_{2}\cdots
p_{M}};\lambda\in\sigma(C_{\psi})\\}=\\{1\\}.$

Therefore, if $\sigma(C_{\psi})\subsetneqq\partial\mathbb{D}$, we have
$\sigma(C_{\psi})=\\{1\\}$.

$\bf(3)$ The spectrum of $C_{\psi}$ acting on $H^{2}_{\beta}$ is just the unit
circle.

Suppose that $\sigma(C_{\psi})=\\{1\\}$. We also have
$\sigma(C_{\psi^{-1}})=\sigma(C^{-1}_{\psi})=\\{1\\}$, and then for any
$\epsilon>0$, when $m\in\mathbb{N}$ is large enough,

$\|C_{\psi^{-m}}\|_{H^{2}_{\beta}}=\|C^{-m}_{\psi}\|_{H^{2}_{\beta}}\leq(1+\epsilon)^{m}.$

In particular,

$\|\psi^{-2m}(z)\|_{H^{2}_{\beta}}=\|C_{\psi^{-2m}}(z)\|_{H^{2}_{\beta}}\leq(1+\epsilon)^{2m}.$

Consider the operator $(C_{\psi}-\mathbf{I})^{4m}$, where $\mathbf{I}$ is the
identity operator. One can see

$(C_{\psi}-\mathbf{I})^{4m}(\psi^{-2m})=\sum\limits_{n=0}^{4m}(-1)^{n}C_{4m}^{n}\cdot
C_{\psi^{n}}(\psi^{-2m})=\sum\limits_{n=0}^{4m}(-1)^{n}C_{4m}^{n}\cdot\psi^{n-2m}.$

Notice that for any $k\in\mathbb{N}$,

$\psi^{k}(0)=\frac{k}{k-\mathbf{i}}=\frac{k^{2}}{k^{2}+1}+\mathbf{i}\frac{k}{k^{2}+1}\
\ \text{and}\ \
\psi^{-k}(0)=\frac{k}{k+\mathbf{i}}=\frac{k^{2}}{k^{2}+1}-\mathbf{i}\frac{k}{k^{2}+1}.$

Then,

$\displaystyle\mathrm{Re}((C_{\psi}-\mathbf{I})^{4m}\psi^{-2m}(0))=$
$\displaystyle\mathrm{Re}(\sum\limits_{n=0}^{4m}(-1)^{n}C_{4m}^{n}\cdot\psi^{n-2m}(0))$
$\displaystyle=$
$\displaystyle\sum\limits_{n=0}^{4m}(-1)^{n}C_{4m}^{n}\cdot\frac{(n-2m)^{2}}{(n-2m)^{2}+1}.$

Furthermore, we have

$\displaystyle-\mathrm{Re}((C_{\psi}-\mathbf{I})^{4m}\psi^{-2m}(0))=(1-1)^{4m}-\mathrm{Re}((C_{\psi}-\mathbf{I})^{4m}\psi^{-2m}(0))$
$\displaystyle=$
$\displaystyle\sum\limits_{n=0}^{4m}(-1)^{n}C_{4m}^{n}-\sum\limits_{n=0}^{4m}(-1)^{n}C_{4m}^{n}\cdot\frac{(n-2m)^{2}}{(n-2m)^{2}+1}$
$\displaystyle=$
$\displaystyle\sum\limits_{n=0}^{4m}(-1)^{n}C_{4m}^{n}\cdot\frac{1}{(n-2m)^{2}+1}.$

One can see that

$\sum\limits_{n=2m-1}^{2m+1}(-1)^{n}C_{4m}^{n}\cdot\frac{1}{(n-2m)^{2}+1}=-\frac{1}{2}C_{4m}^{2m-1}+C_{4m}^{2m}-\frac{1}{2}C_{4m}^{2m+1}>0,$

and for $k=1,2,\ldots,m-1$,

$C_{4m}^{2k}\cdot\frac{1}{(2k-2m)^{2}+1}-C_{4m}^{2k-1}\cdot\frac{1}{(2k-1-2m)^{2}+1}>0,$

and hence

$\displaystyle\sum\limits_{n=1}^{2m-2}\left((-1)^{n}C_{4m}^{n}\cdot\frac{1}{(n-2m)^{2}+1}\right)$
$\displaystyle=$
$\displaystyle\sum\limits_{k=1}^{m-1}\left(-C_{4m}^{2k-1}\cdot\frac{1}{(2k-1-2m)^{2}+1}+C_{4m}^{2k}\cdot\frac{1}{(2k-2m)^{2}+1}\right)$
$\displaystyle>$ $\displaystyle 0.$

In addition, for $n=0,1,\ldots,4m$,

$(-1)^{n}C_{4m}^{n}\cdot\frac{1}{(n-2m)^{2}+1}=(-1)^{4m-n}C_{4m}^{4m-n}\cdot\frac{1}{((4m-n)-2m)^{2}+1},$

and hence

$\sum\limits_{n=2m+1}^{4m-1}\left((-1)^{n}C_{4m}^{n}\cdot\frac{1}{(n-2m)^{2}+1}\right)=\sum\limits_{n=1}^{2m-2}\left((-1)^{n}C_{4m}^{n}\cdot\frac{1}{(n-2m)^{2}+1}\right)>0.$

Then,

$\displaystyle-\mathrm{Re}((C_{\psi}-\mathbf{I})^{4m}\psi^{-2m}(0))$
$\displaystyle=$
$\displaystyle\frac{1}{(2m)^{2}+1}+\left(\sum\limits_{n=1}^{2m-2}+\sum\limits_{n=2m-1}^{2m+1}+\sum\limits_{n=2m+1}^{4m-1}\right)\left((-1)^{n}C_{4m}^{n}\cdot\frac{1}{(n-2m)^{2}+1}\right)$
$\displaystyle+\frac{1}{(2m)^{2}+1}$ $\displaystyle\geq$
$\displaystyle\frac{2}{4m^{2}+1}.$

Consequently,

$\displaystyle\left\|(C_{\psi}-\mathbf{I})^{4m}\frac{\psi^{-2m}}{\|\psi^{-2m}\|_{H^{2}_{\beta}}}\right\|_{H^{2}_{\beta}}$
$\displaystyle\geq$
$\displaystyle\frac{1}{\|\psi^{-2m}\|_{H^{2}_{\beta}}}\cdot|(C_{\psi}-\mathbf{I})^{4m}\psi^{-2m}(0)|$
$\displaystyle\geq$
$\displaystyle\frac{1}{\|\psi^{-2m}\|_{H^{2}_{\beta}}}\cdot|-\mathrm{Re}((C_{\psi}-\mathbf{I})^{4m}\psi^{-2m}(0))|$
$\displaystyle\geq$
$\displaystyle\frac{1}{(1+\epsilon)^{2m}}\cdot\frac{2}{4m^{2}+1}.$

Therefore,

$\displaystyle\rho(C_{\psi}-\mathbf{I})=$
$\displaystyle\lim\limits_{m\rightarrow\infty}\|(C_{\psi}-\mathbf{I})^{4m}\|_{H^{2}_{\beta}}^{\frac{1}{4m}}$
$\displaystyle\geq$
$\displaystyle\lim\limits_{m\rightarrow\infty}\left\|(C_{\psi}-\mathbf{I})^{4m}\frac{\psi^{-2m}}{\|\psi^{-2m}\|_{H^{2}_{\beta}}}\right\|_{H^{2}_{\beta}}^{\frac{1}{4m}}$
$\displaystyle\geq$
$\displaystyle\lim\limits_{m\rightarrow\infty}\left(\frac{1}{(1+\epsilon)^{2m}}\cdot\frac{2}{4m^{2}+1}\right)^{\frac{1}{4m}}$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{1+\epsilon}}$ $\displaystyle>$
$\displaystyle 0,$

which is a contradiction to $\sigma(C_{\psi}-\mathbf{I})=\\{0\\}$.

Now we have obtained that the spectrum of $C_{\psi}$ acting on $H^{2}_{\beta}$
is the unit circle. Furthermore,

$\sigma(C_{\psi^{-1}})=\sigma(C^{-1}_{\psi})=\\{\frac{1}{z};\
z\in\sigma(C_{\psi})\\}=\partial\mathbb{D}.$

Thus, for any parabolic disk automorphism $\varphi$, the spectrum of
$C_{\varphi}$ acting on $H^{2}_{\beta}$ is the unit circle.

∎

######  Theorem 3.9.

Let $H^{2}_{\beta}$ be a weighted Hardy space of polynomial growth induced by
the weight sequence $\beta=\\{\beta_{k}\\}_{k=1}^{\infty}$. If $\varphi$ is a
hyperbolic disk automorphism, then the spectrum of $C_{\varphi}$ acting on
$H^{2}_{\beta}$ is contained in an annulus. More precisely,

$\sigma(C_{\varphi})\subseteq\\{z\in\mathbb{C};\left(\frac{1-|\varphi(0)|}{1+|\varphi(0)|}\right)^{2N^{2}+3N+2}\leq|z|\leq\left(\frac{1+|\varphi(0)|}{1-|\varphi(0)|}\right)^{2N^{2}+3N+2}\\}.$

###### Proof.

Suppose that $H^{2}_{\beta}$ is a weighted Hardy space of $N$-polynomial
growth. By Lemma 3.5, it suffices to consider the spectrum of

$\psi(z)=\frac{z+r}{1+rz},$

where $r=|\varphi(0)|$. After a straight computation, one can see

$\psi^{n}(z)=\underset{n}{\underbrace{\psi\circ\cdots\circ\psi}}(z)=\frac{1+\frac{(1+r)^{n}-(1-r)^{n}}{(1+r)^{n}+(1-r)^{n}}}{1+\frac{(1+r)^{n}-(1-r)^{n}}{(1+r)^{n}+(1-r)^{n}}z},$

Then, by Theorem 2.12, there exists a positive number $K$ such that

$\|C^{n}_{\psi}\|_{H^{2}_{\beta}}=\|C_{\psi^{n}}\|_{H^{2}_{\beta}}\leq\frac{K}{\left(1-\frac{(1+r)^{n}-(1-r)^{n}}{(1+r)^{n}+(1-r)^{n}}\right)^{2N^{2}+3N+2}},$

and consequently

$\rho(C_{\psi})=\lim\limits_{n\rightarrow+\infty}\|C^{n}_{\psi}\|^{\frac{1}{n}}_{H^{2}_{\beta}}\leq\left(\frac{1+r}{1-r}\right)^{2N^{2}+3N+2}.$

Similarly, we also obtain

$\rho(C^{-1}_{\psi})\leq\left(\frac{1+r}{1-r}\right)^{2N^{2}+3N+2}.$

Therefore,

$\sigma(C_{\varphi})\subseteq\left\\{z\in\mathbb{C};\left(\frac{1-|\varphi(0)|}{1+|\varphi(0)|}\right)^{2N^{2}+3N+2}\leq|z|\leq\left(\frac{1+|\varphi(0)|}{1-|\varphi(0)|}\right)^{2N^{2}+3N+2}\right\\}.$

∎

##  4\. Composition operators with closed ranges and Fredholmness

In this section, we completely characterize the closed range and Fredholm
composition operators with symbols of analytic self-maps on the closed unit
disk acting on weighted Hardy spaces of polynomial growth.

######  Proposition 4.1.

Let $H^{2}_{\beta}$ be a weighted Hardy space of polynomial growth. Let
$\varphi:\overline{\mathbb{D}}\rightarrow\overline{\mathbb{D}}$ be a
nontrivial analytic map. If the composition operator $C_{\varphi}$ acting on
$H^{2}_{\beta}$ has closed range, then
$\mathbb{T}\subseteq\varphi(\overline{\mathbb{D}})$.

###### Proof.

Suppose that $H^{2}_{\beta}$ is a weighted Hardy space of $N$-polynomial
growth. If the composition operator $C_{\varphi}$ acting on $H^{2}_{\beta}$
has closed range, then there exists a positive number $c$ such that

(4.1) $\|C_{\varphi}(h)\|_{H^{2}_{\beta}}\geq c\|h\|_{H^{2}_{\beta}},\ \ \ \ \
\text{for any}\ h\in H^{2}_{\beta}.$

Suppose that $\varphi(\overline{\mathbb{D}})$ dose not contain the unit circle
$\mathbb{T}$. Then there exists a point $\xi\in\mathbb{T}$ and a positive
number $\epsilon<\frac{1}{2}$ such that

$B(\xi,\epsilon)\bigcap\varphi(\overline{\mathbb{D}})=\emptyset,$

where $B(\xi,\epsilon)=\\{z\in\mathbb{C};\ |z-\xi|<\epsilon\\}$. Furthermore,
we could select a positive number $t\in(0,1)$ such that the arc $J$ which lies
in the unit disk, passes through the point $t\xi$ and is orthogonal to the
unit circle, is contained in $B(\xi,\epsilon)$.

Let $\eta(z)=\frac{1+z}{1-z}$ be the map from the unit disk $\mathbb{D}$ to
the right half-plane $\mathbb{H}_{r}$, and let
$\varsigma(z)=\frac{(1-t)^{2}}{(1+t)^{2}}\cdot z$ be the map from
$\mathbb{H}_{r}$ on to itself. Define $\tau:\mathbb{D}\rightarrow\mathbb{D}$
by

$\tau=\eta^{-1}\circ\varsigma\circ\eta.$

Then $\eta$ is a disk automorphism with fixed point $-\xi$ and maps $J$ to
$-J$, where $-J=\\{-z;z\in J\\}$. Moreover, $\eta$ maps the region bounded by
$J$ and the arc in the unit circle passing through $-\xi$ to the region
bounded by $-J$ and the arc in the unit circle passing through $-\xi$ (see
Figure 1).

Figure 1. Disk automorphism $\tau$

Let $g(z)=z+\xi$. Consider the analytic function
$f:\mathbb{D}\rightarrow\mathbb{D}$ defined by

$f(z)=g\circ\tau(z).$

Then $f\circ\varphi$ maps the unit disk into the disk $B(0,\epsilon)$ (more
precisely, $f(\mathbb{D})$ is contained in the yellow region as following
Figure 2.).

Figure 2. Analytic function $f\circ\varphi$

Now set $f_{k}(z)=(f(z))^{k}$, for $k=1,2,\ldots$. We write

$F=C_{\varphi}(f)=f\circ\varphi,\ \ \ \ F_{k}(z)=(F(z))^{k}$

and

$C_{\varphi}(f_{k}(z))=F_{k}=\sum\limits_{n=0}^{\infty}\widehat{F_{k}}(n)z^{n}.$

Suppose that $\varphi$ is analytic on a disk $B(0,R)$, for some certain $R>1$.
Then each $F_{k}$ is also analytic on $B(0,R)$.

Since the maximum modulus of $f_{k}(z)$ on $\mathbb{D}$ is less than
$\epsilon^{k}$, it follows from Cauchy inequality that

$|\widehat{F_{k}}(n)|\leq\frac{\epsilon^{k}}{R^{n}},$

and consequently as $k\rightarrow\infty$,

$\|C_{\varphi}(f_{k})\|_{H^{2}_{\beta}}=\sqrt{\sum\limits_{n=0}^{\infty}|\widehat{F_{k}}(n)|^{2}\beta_{n}^{2}}\leq\sqrt{\sum\limits_{n=0}^{\infty}\frac{\epsilon^{2k}\beta_{n}^{2}}{R^{2n}}}=\epsilon^{k}\sqrt{\sum\limits_{n=0}^{\infty}\frac{\beta_{n}^{2}}{R^{2n}}}\rightarrow
0.$

Notice that for every $k$,

$\|(g(z))^{k}\|_{H^{2}_{\beta}}\geq|(g(0))^{k}|=|\xi^{k}|=1.$

In addition, we have known that $C_{\tau}$ is a bounded invertible operator
acting on $H^{2}_{\beta}$ by Theorem 2.1, i.e., there exist positive numbers
$a$ and $A$ such that

$a\|h\|_{H^{2}_{\beta}}\leq\|C_{\tau}(h)\|_{H^{2}_{\beta}}\leq
A\|h\|_{H^{2}_{\beta}},\ \ \ \ \ \text{for any}\ h\in H^{2}_{\beta}.$

Then for all $k$,

$\|f_{k}\|_{H^{2}_{\beta}}=\|C_{\tau}((g(z))^{k})\|_{H^{2}_{\beta}}\geq
a\|(g(z))^{k}\|_{H^{2}_{\beta}}\geq a>0.$

This is a contradiction to the inequality (4.1). ∎

To continue our study, some results related to $0$-dimensional spaces are
required (we refer to [25]).

######  Definition 4.2 (Definition II 1 in [25] p.10).

A separable metric space $X$ has dimension $0$ at a point $p$ if $p$ has
arbitrarily small neighborhood with empty boundaries. A nonempty space $X$ has
$0$-dimension if it has dimension $0$ at each of its points.

######  Theorem 4.3.

All spaces considered are separable metric. Then

  1. (1)

The property of being $0$-dimensional is a topological invariant, i.e., if the
space $X$ is homeomorphic to a $0$-dimensional space, then $X$ is also
$0$-dimensional. ((A) in [25] p.10)

  2. (2)

A nonempty subset of a $0$-dimensional space is $0$-dimensional. (Theorem II 1
in [25] p.13)

  3. (3)

A space which is the sum of countable $0$-dimensional closed subsets is itself
$0$-dimensional. (Theorem II 2 in [25] p.18)

  4. (4)

Any set of real numbers containing no interval is $0$-dimensional. (Example II
4 in [25] p.11)

  5. (5)

A connected $0$-dimensional consists of only one point. ((C) in [25] p.15)

######  Theorem 4.4.

Let $H^{2}_{\beta}$ be a weighted Hardy space of polynomial growth induced by
the weight sequence $\beta=\\{\beta_{k}\\}_{k=1}^{\infty}$. Let
$\varphi:\overline{\mathbb{D}}\rightarrow\overline{\mathbb{D}}$ be a
nontrivial analytic map. Then the followings are equivalent.

  1. (1)

The composition operator $C_{\varphi}$ is semi-Fredholm.

  2. (2)

The composition operator $C_{\varphi}$ has closed range.

  3. (3)

$\mathbb{T}\subseteq\varphi(\overline{\mathbb{D}})$.

  4. (4)

$\varphi(\mathbb{T})$ contains an arc in $\mathbb{T}$.

  5. (5)

$\varphi$ is a finite Blaschke product.

###### Proof.

Obviously, $(1)$ implies $(2)$ and $(3)$ implies $(4)$. By Proposition 4.1,
one can see that $(2)$ implies $(3)$. By Theorem 2.3, one can see that $(5)$
implies $(1)$, more precisely, the composition operator $C_{\varphi}$ with a
symbol of finite Blaschke product with order $m$ is an injective operator with
closed range, and has codimension $1$ if $m=1$, and has infinite codimension
if $m>1$. So it suffices to prove $(4)$ implies $(5)$.

Suppose that $\varphi(\mathbb{T})$ contains an arc in $\mathbb{T}$. Since
$\varphi$ is analytic on $\overline{\mathbb{D}}$, we could write $\varphi=BF$,
where $B$ is a finite Blaschke product and $F$ is an outer function which is
also analytic on $\overline{\mathbb{D}}$ (see [19] for exmple). Notice that
$|B(\xi)|=1$ for all $\xi\in\mathbb{T}$. Then $F(\mathbb{T})$ contains a
closed arc, denoted by $J_{1}$, in $\mathbb{T}$. Obviously, $F^{-1}(J_{1})$ is
a closed subset in $\mathbb{T}$.

Now we claim that $F^{-1}(J_{1})$ is a closed subset with nonempty interior in
$\mathbb{T}$. Suppose that the closed set $F^{-1}(J_{1})$ has empty interior
in $\mathbb{T}$. Then by Theorem 4.3, $F^{-1}(J_{1})$ is homeomorphic to a set
of real numbers containing no interval and hence is $0$-dimensional. Since
$F(z)$ is analytic on $\overline{\mathbb{D}}$, $F$ is analytic on a disc
$B(0,R)$ for some $R>1$. To avoid confusion, let $F_{R}$ be the analytic
function on $B(0,R)$ such that $F_{R}(z)=F(z)$ for any
$z\in\overline{\mathbb{D}}$. Notice that there is at most finite zero points
of $F^{\prime}_{R}(z)$(or written as $F^{\prime}(z)$) on $\mathbb{T}$. Then we
could choose a sub-arc $J_{2}$ of $J_{1}$ such that $F^{\prime}(\lambda)\neq
0$ for any $\lambda\in F^{-1}(J_{2})$. Furthermore, one can select a point
$\xi_{0}\in J_{2}$ such that

$\textrm{Card}\\{F^{-1}(\xi_{0})\\}=\textrm{Card}\\{\lambda\in\mathbb{T};~{}F(\lambda)=\xi_{0}\\}=\max\limits_{\xi\in
J_{2}}~{}\textrm{Card}\\{\lambda\in\mathbb{T};~{}F(\lambda)=\xi\\}.$

Denote $F^{-1}(\xi_{0})=\\{\lambda_{j},j=1,2,\ldots,n_{0}.\\}$, where
$n_{0}=\textrm{Card}\\{\lambda\in\mathbb{T};~{}F(\lambda)=\xi_{0}\\}$.
Following from $F^{\prime}_{R}(\lambda_{j})\neq 0$ for $j=1,2,\ldots,n_{0}$,
there exists a neighborhood $V$ of $\xi_{0}$ and a neighborhood $U_{j}$ of
$\lambda_{j}$ for all $j$ such that the restriction of $F_{R}$ on $U_{j}$ is
an analytic homeomorphism to $V$. Denote $J_{3}$ be a closed arc in
$J_{2}\bigcap V$, and moreover denote $A_{j}=F^{-1}(J_{3})\bigcap U_{j}$ for
$j=1,2,\ldots,n_{0}$. By Theorem 4.3, each $A_{j}$ is $0$-dimensional and
consequently

$J_{3}=F_{R}(F^{-1}(J_{3}))=F_{R}(\bigcup\limits_{j=1}^{n_{0}}A_{j})=\bigcup\limits_{j=1}^{n_{0}}F_{R}(A_{j})$

is also $0$-dimensional. However, by $(5)$ in Theorem 4.3, the arc $J_{3}$ is
not $0$-dimensional. This is a contradiction.

So, $F^{-1}(J_{1})$ is a closed subset with nonempty interior in $\mathbb{T}$.
Furthermore, there exists an arc $J$ in $F^{-1}(J_{1})\subseteq\mathbb{T}$ and
then $F(J)=\widetilde{J}$ is a sub-arc in $J_{1}\subseteq\mathbb{T}$.

In addition, $F(\mathbb{D})$ is contained in $\mathbb{D}$ and does not contain
zero. Then by the Schwartz reflection principle, one can obtain an analytic
function $\widetilde{F}$ on the domain
$\Omega=J\bigcup(\mathbb{C}\setminus\mathbb{T})$, which is an analytic
continuation of $F$ on $\mathbb{D}$.

Since $F$ is analytic on $\overline{\mathbb{D}}$ and coincide with
$\widetilde{F}$ on $\mathbb{D}$, it follows the uniqueness of analytic
functions that $F$ can be analytically extended to be an entire function on
$\mathbb{C}$. Moreover, one can see that $F(\overline{\mathbb{D}})$ also does
not contain zero. If $0\in F(\overline{\mathbb{D}})$, there exists a sequence
of points $z_{n}\in\mathbb{D}$ tending to a point $\xi\in\mathbb{T}$ such that
$F(z_{n})\rightarrow 0$. By the Schwartz reflection principle, we have
$\widetilde{F}(\frac{1}{\overline{z_{n}}})\rightarrow\infty$. However, this
can not occur because of
$\frac{1}{\overline{z_{n}}}\rightarrow\xi\in\mathbb{T}$. Then, $F$ can be
analytically extended to be a bounded entire function, and consequently $F$
must be a constant function with module $1$. Therefore, $\varphi$ is just a
finite Blaschke product. ∎

######  Theorem 4.5.

Let $H^{2}_{\beta}$ be a weighted Hardy space of polynomial growth. Let
$\varphi:\overline{\mathbb{D}}\rightarrow\overline{\mathbb{D}}$ be an analytic
map. Then the composition operator $C_{\varphi}$ acting on $H^{2}_{\beta}$ is
Fredholm if and only if $\varphi$ is a disk automorphism.

###### Proof.

Notice that for a finite Blaschke product $B$ with order $m$, by Theorem 2.3
and Remark 2.4, the composition operator $C_{B}$ has finite codimension if and
only if the order $m$ equals $1$. Then, by Theorem 4.4, the composition
operator $C_{\varphi}$ is Fredholm if and only if $\varphi$ is a disk
automorphism. ∎

Acknowledgement

The second author was supported by National Natural Science Foundation of
China (Grant No. 11831006, 11920101001 and 11771117). The authors thank
Professor Guangfu Cao and Professor Maofa Wang for many valuable discussions
on this paper.

## References

  * [1] R. Allen, T. Le, M. Pons, Spectrum of a composition operator with automorphic symbol, Involve: a Journal of Mathematics, 9(5) (2016): 813-829.
  * [2] J. R. Akeroyd, S. R. Fulmer, Closed-range composition operators on weighted Bergman spaces, Integr. Equ. Oper. Theory, 72 (2012): 103-114.
  * [3] P. S. Bourdon, Fredholm multiplication and composition operators on the Hardy space, Integr. Equ. Oper. Theory, 13 (1990): 607-610.
  * [4] P. S. Bourdon, E. E. Fry, C. Hammond, C. H. Spofford, Norms of linear-fractional composition operators, T. Am. Math. Soc., 356 (2004): 2459-2480.
  * [5] G. F. Cao, S. H. Sun, On composition operators on Hp-spaces in several variables, Acta Math. Sin., 13 (1997): 281-288.
  * [6] G. F. Cao, Composition and Toeplitz operators on general domains, Tohoku Math. J., 57 (2005): 11-22.
  * [7] G. F. Cao, L. He, K. H. Zhu, Fredholm composition operators and proper holomorphic mappings, Bull. London Math. Soc., 51 (2019): 1104-1112.
  * [8] G. F. Cao, L. He, J. Li, Evaluation functions and composition operators on Banach spaces of holomorphic functions, preprint, available at arXiv: 2211.12236.
  * [9] G. F. Cao, L. He, Composition operators with closed ranges on Dirichlet spaces, preprint.
  * [10] J. A. Cima, J. Thompson, W. R. Wogen, On some properties of composition operators, Indiana Univ. Math. J., 24 (1974): 215-220.
  * [11] J. A. Cima, A theorem on composition operators, Lecture Notes in Mathematics, 604, Springer-Verlag, (1976): 21-24.
  * [12] C. C. Cowen, An application of Hadamard multiplication to operators on weighted Hardy spaces, Linear Alg. Appl. 133 (1990): 21-32.
  * [13] C. C. Cowen, B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.
  * [14] E. Fricain, J. Mashreghi, On a characterization of finite Blaschke products, Complex Var. Elliptic Equ. 59(3) (2014): 362-368.
  * [15] P. Galindo, T. Gamelin and M. Lindstrom, Fredholm Composition operators on algebras of analytic functions on Banach spaces, J. Funct. Anal., 258 (2010): 1504-1512.
  * [16] E. A. Gallardo-Gutiérrez, A. Montes-Rodríguez, The role of the spectrum in the cyclic behavior of composition operators, Mem. Amer. Math. Soc. 167 (791) (2004), x+81.
  * [17] E. A, Gallardo-Gutiérrez, J. R. Partington, Norms of composition operators on weighted Hardy spaces, Israel J. Math. 196(1) (2013): 273-283.
  * [18] E. A, Gallardo-Gutiérrez, R. Schroderus, The spectra of linear fractional composition operators on weighted Dirichlet spaces, J. Funct. Anal., 271 (2016): 720-745.
  * [19] J. B. Garnett, Bounded analytic functions. Academic Press, New York, 1981.
  * [20] M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math-Paris, 53(1) (1981): 53-78.
  * [21] C. Hammond, On the norm of a composition operator with linear fractional symbol, Acta Sci. Math., 69 (2003): 813-829.
  * [22] C. Hammond, L. J. Patton, Norm inequalities for composition operators on Hardy and weighted Bergman spaces, in Topics in Operator Theory. Volume 1. Operators, Matrices and Analytic Functions, Operator Theory: Advances and Applications, Vol. 202, Birkhäuser Verlag, Basel, 2010, pp. 265-272.
  * [23] 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.
  * [24] B. Hou, C. Jiang, Analytic automorphism group and similar representation of analytic functions, preprint, available at arXiv: 2209.03852.
  * [25] W. Hurewicz, H. Wallman, Dimension Theory, Princeton Math. Series, vol. 4, Princeton University Press, 1941.
  * [26] P. R. Hurst, Relating composition operators on different weighted Hardy spaces, Arch. Math. (Basel), 68(6) (1997): 503-513.
  * [27] H. Kamowitz, The spectra of composition operators on Hp, J. Funct. Anal., 18 (1975): 132-150.
  * [28] H. Kamowitz, The spectra of a class of operators on the disc algebra, Indiana Univ. Math. J., 27(4) (1978): 581-610.
  * [29] T. L. Kriete, B. D. MacCluer, Composition operators on large weighted Bergman spaces, Indiana U. Math. J., 41 (1992): 755-788.
  * [30] A. Kumar, Fredholm composition operators, Proc. Amer. Math. Soc., 79 (1980): 233-236.
  * [31] B. MacCluer, Fredholm composition operators, Proc. Amer. Math. Soc., 125 (1997), 163-166.
  * [32] D. H. Luecking, Inequalities on Bergman spaces, Illinois J. Math., 25 (1981): 1-11.
  * [33] D. H. Luecking, Closed ranged restriction operators on weighted Bergman spaces, Pacific J. Math., 110(1) (1984): 145-160.
  * [34] D. H. Luecking, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives, Amer. J. Math., 107 (1985): 85-111.
  * [35] D. H. Luecking, Bounded composition operators with closed range on the Dirichlet space, Proc. Amer. Math. Soc. 128(4) (1999): 1109-1116.
  * [36] B. D. MacCluer, X. Zeng, N. Zorboska, Composition operators on small weighted Hardy spaces, Illinois J. Math., 40 (1996): 662-677.
  * [37] N. K. Nikol’skiǐ, Treatise on the shift operator. Springer Berlin Heidelberg, 1986.
  * [38] E. A. Nordgren, Composition operators, Canad. J. Math. 20 (1968): 442-449.
  * [39] J. Pau, P. A. Pérez, Composition operators acting on weighted Dirichlet spaces, J. Math. Anal. Appl., 401(2) (2013): 682-694.
  * [40] M. A. Pons, Composition Operators on Besov and Dirichlet Type Spaces, Ph.D. thesis, University of Virginia, 2007.
  * [41] M. A. Pons, The spectrum of a composition operator and Calderón’s complex interpolation, in "Topics in Operator Theory", Volume 1. Operators, Matrices and Analytic Functions, in: Oper. Theory Adv. Appl., vol. 202, Birkhäuser Verlag, Basel, 2010, pp. 451-467.
  * [42] I. Singer, Bases in Banach Space I, Springer-Verlag, New York, 1970.
  * [43] N. Zorboska, Composition operators on $\mathcal{S}_{a}$ spaces, Indiana U. Math. J., 39 (1990): 847-857.
  * [44] N. Zorboska, Composition operators with closed range, T. Am. Math. Soc., 344 (1994): 791-801.
  * [45] N. Zorboska, Composition operators on weighted Dirichlet spaces, P. Am. Math. Soc., 126 (1998): 2013-2023.