# Spectra of infinitesimal generators of composition semigroups on weighted
Bergman spaces induced by doubling weights

Ruishen Qian  School of Mathematics and Statistics, Lingnan Normal University,
Zhanjiang 524048, Guangdong, China<EMAIL_ADDRESS>,  Fanglei Wu
University of Eastern Finland, P.O.Box 111, 80101 Joensuu, Finland
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Hasi Wulan  Department of
Mathematics, Shantou University, Shantou, 515063, People’s Republic of China
<EMAIL_ADDRESS>
###### Abstract.

Suppose $(C_{t})_{t\geq 0}$ is the composition semigroup induced by a one-
parameter semigroup $(\varphi_{t})_{t\geq 0}$ of analytic self-maps of the
unit disk. The main purpose of the paper is to investigate the spectrum of the
infinitesimal generator of $(C_{t})_{t\geq 0}$ acting on the weighted Bergman
space induced by doubling weights, provided $(\varphi_{t})_{t\geq 0}$ is
elliptic. The method applied is a certain spectral mapping theorem and a
characterization of the spectra of certain composition operators. Eventual
norm-continuity of $(C_{t})_{t\geq 0}$ also plays an important role, which can
be depicted in terms of studying the difference of two distinct composition
operators. As a byproduct, we also characterize a certain compact integral
operator that is closely related to the resolvent of the infinitesimal
generator of $(C_{t})_{t\geq 0}$.

###### Key words and phrases:

Spectrum, Composition semigroup, Weighted Bergman space, Doubling weights,
Difference of composition operators, Spectral mapping theorem, Eventually
norm-continuous.

###### 2010 Mathematics Subject Classification:

30H20, 47B33, 47D06, 47A10

The second author is indebted to University of Eastern Finland for the
financial support.

##  1\. Introduction and main results

A family $(\varphi_{t})_{t\geq 0}$ of analytic self-maps of the unit disk
$\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}$ in the complex plane $\mathbb{C}$ is
said to be a semigroup if the following conditions hold.

  * (i)

$\varphi_{0}$ is the identity map of $\mathbb{D}$;

  * (ii)

$\varphi_{t}\circ\varphi_{s}=\varphi_{t+s}$, for $t,s\geq 0$;

  * (iii)

for each $z\in\mathbb{D}$, $\varphi_{t}(z)\rightarrow z$, as $t\rightarrow
0^{+}$.

A semigroup $(\varphi_{t})_{t\geq 0}$ is said to be trivial if each
$\varphi_{t}$ is the identity of $\mathbb{D}$. For any non-trivial semigroup
$(\varphi_{t})_{t\geq 0}$, the infinitesimal generator of
$(\varphi_{t})_{t\geq 0}$ is defined as the function

$G(z)=\lim_{t\to 0^{+}}\frac{\varphi_{t}(z)-z}{t},\quad z\in\mathbb{D}.$

There exist a point $b\in\overline{\mathbb{D}}$ and an analytic function
$P:\mathbb{D}\mapsto\mathbb{C}$ with $\text{Re}\,P\geq 0$ such that

$G(z)=(\overline{b}z-1)(z-b)P(z).$ (1.1)

See [3] for the details. The representation (1.1) is unique and the point $b$
is said to be the Denjoy-Wolff point of $(\varphi_{t})_{t\geq 0}$. In general,
we say that $(\varphi_{t})_{t\geq 0}$ is elliptical if its Denjoy-Wolff point
$b$ belongs to $\mathbb{D}$. In this case, there exists a univalent function
$h$, called Koenigs function, on $\mathbb{D}$ with $h(b)=0$ and
$h^{\prime}(b)=1$ such that

$h(\varphi_{t}(z))=e^{G^{\prime}(b)t}h(z),\,z\in\mathbb{D},\,t\geq 0.$ (1.2)

If $b\in\mathbb{T}$, the boundary of $\mathbb{D}$, then the non-tangential
derivative $\varphi_{t}^{\prime}(b)$ always exists and
$\varphi_{t}^{\prime}(b)\in(0,1]$ for every $t>0$. Moreover,
$(\varphi_{t})_{t\geq 0}$ is said to be hyperbolic if
$\varphi_{t}^{\prime}(b)<1$ and parabolic if $\varphi_{t}^{\prime}(b)=1$ for
every $t>0$. For $b\in\mathbb{T}$, there exists a univalent function $h$,
called Koenigs function, on $\mathbb{D}$ with $h(b)=0$ such that

$h(\varphi_{t}(z))=h(z)+t,\,z\in\mathbb{D},\,t\geq 0.$

Each semigroup $(\varphi_{t})_{t\geq 0}$ gives rise to a semigroup
$(C_{t})_{t\geq 0}$ consisting of composition operators on
$\mathcal{H}(\mathbb{D})$, the set of analytic functions on $\mathbb{D}$,
where

$C_{t}(f)=C_{\varphi_{t}}f=f\circ\varphi_{t},\quad
f\in\mathcal{H}(\mathbb{D}).$

The composition semigroup $(C_{t})_{t\geq 0}$ is said to be strongly
continuous on a Banach space $X$ if $C_{t}$ is bounded on $X$ and

$\lim_{t\rightarrow
0^{+}}\|C_{t}x-x\|_{X}=0\quad\mbox{for}~{}~{}\mbox{all}~{}~{}x\in X.$

The problem of studying the strong continuity of composition semigroup can be
traced back to the initial work of E. Berkson and H. Porta [3] in which it was
shown that composition semigroup is strongly continuous on Hardy spaces. The
same results were obtained on weighted Bergman spaces with standard weights
and Dirichlet space, see [34, 35] and references therein. The aforementioned
literature also proved that the (infinitesimal) generator $\Gamma$ of the
composition semigroup $(C_{t})_{t\geq 0}$ on these spaces has the form:

$\Gamma(f)=Gf^{\prime},$ (1.3)

where $G$ is the infinitesimal generator of $(\varphi_{t})_{t\geq 0}$.

It is well-known that the generator of a strongly continuous semigroup is
closed. The classical Hille–Yosida theorem implies that the spectrum of the
(infinitesimal) generator of a strongly continuous semigroup always lies in a
proper left half-plane, but it seems that there is no standard method to find
out the complete distribution of the spectrum for a concrete generator. In
general, the problem of describing the spectra of generators of strongly
continuous composition semigroups is a tough question and remains unsettled in
many cases.

The existing literature offers several descriptions of the point spectrum of
composition semigroup acting on Hardy spaces and Bergman spaces with the
standard weights. To be concrete, given a semigroup $(\varphi_{t})_{t\geq 0}$
with the infinitesimal generator $G$, the Denjoy-Wolff point $b$ and Koenigs
function $h$, the point spectrum of the generator of the corresponding
composition semigroup $(C_{t})_{t\geq 0}$, denote by $P\sigma(\Gamma)$ can be
characterized as follows:

  * (i)

If $b\in\mathbb{D}$, then $P\sigma(\Gamma)=\\{kG^{\prime}(b):h^{k}\in X\\}$;

  * (ii)

If $b\in\mathbb{T}$, then $P\sigma(\Gamma)=\\{\lambda G(0):e^{\lambda h}\in
X\\}$,

where $X$ can be chosen as the Hardy space and the standard weighted Bergman
space. The method used to find out the point spectrum of $\Gamma$ depends on
the argument principle, see [33, 34] for the proof. Using the harmonic
measure, D. Betsakos provided an intuitive characterization of the point
spectrum of the generator $\Gamma$ on the Hardy space and standard weighted
Bergman space in the case of elliptical or hyperbolical semigroup
$(\varphi_{t})_{t\geq 0}$.

We are mainly interested in the question of characterizing the spectrum of the
generator of composition semigroup acting on weighted Bergman spaces induced
by a class of weights admitting a certain doubling property. For a non-
negative function $\omega\in L^{1}(\mathbb{D})$ such that
$\omega(z)=\omega(|z|)$ for all $z\in\mathbb{D}$ is called a radial weight.
For $0<p<\infty$ and such an $\omega$, the Lebesgue space $L^{p}_{\omega}$
consists of complex-valued measurable functions $f$ on $\mathbb{D}$ such that

$\|f\|_{L^{p}_{\omega}}^{p}=\int_{\mathbb{D}}|f(z)|^{p}\omega(z)\,dA(z)<\infty,$

where $dA(z)=\frac{dx\,dy}{\pi}$ is the normalized area measure on
$\mathbb{D}$. The corresponding weighted Bergman space is
$A^{p}_{\omega}=L^{p}_{\omega}\cap\mathcal{H}(\mathbb{D})$. As usual, we write
$A^{p}_{\alpha}$ for the classical weighted Bergman spaces induced by the
standard weight $\omega(z)=(\alpha+1)(1-|z|^{2})^{\alpha}$ with
$-1<\alpha<\infty$. Throughout this paper we assume
$\widehat{\omega}(z)=\int_{|z|}^{1}\omega(s)\,ds>0$ for all $z\in\mathbb{D}$,
for otherwise $A^{p}_{\omega}=\mathcal{H}(\mathbb{D})$. A radial weight
$\omega$ belongs to the class $\widehat{\mathcal{D}}$ if there exists a
constant $C=C(\omega)\geq 1$ such that the tail integral $\widehat{\omega}$
satisfies the doubling condition $\widehat{\omega}(r)\leq
C\widehat{\omega}(\frac{1+r}{2})$ for all $0\leq r<1$. Moreover, if there
exist $K=K(\omega)>1$ and $C=C(\omega)>1$ such that $\widehat{\omega}(r)\geq
C\widehat{\omega}\left(1-\frac{1-r}{K}\right)$ for all $0\leq r<1$, then we
write $\omega\in\widecheck{\mathcal{D}}$. In other words,
$\omega\in\widecheck{\mathcal{D}}$ if there exist $K=K(\omega)>1$ and
$C^{\prime}=C^{\prime}(\omega)>0$ such that

$\widehat{\omega}(r)\leq
C^{\prime}\int_{r}^{1-\frac{1-r}{K}}\omega(t)\,dt,\quad 0\leq r<1.$ (1.4)

The intersection $\widehat{\mathcal{D}}\cap\widecheck{\mathcal{D}}$ is denoted
by $\mathcal{D}$. It is well-known that for a radial weight $\omega$ the norm
convergence in $A^{p}_{\omega}$ implies the uniform convergence in compact
subsets of $\mathbb{D}$. In particular, the point evaluation functionals are
bounded on $A^{p}_{\omega}$. If further $\omega\in\widehat{\mathcal{D}}$, then
a classical Hardy-Littlewood inequality yields

$|f(z)|\lesssim\frac{\|f\|_{A^{p}_{\omega}}}{((1-|z|)\widehat{\omega}(z))^{1/p}},\quad
z\in\mathbb{D},~{}~{}f\in A^{p}_{\omega}.$ (1.5)

See, for example [26, p.35]. Therefore, for a fixed $z\in\mathbb{D}$, the
point evaluation functional $\delta_{z}$ satisfies
$\|\delta_{z}\|\lesssim\frac{1}{((1-|z|)\widehat{\omega}(z))^{1/p}}$. In fact,
the other way around in the above estimate is correct as well. We just note
that there exists a sufficiently large $\gamma=\gamma(\omega)>0$ such that for
any $a\in\widehat{\mathcal{D}}$ the function
$f_{a,p,\gamma}(z)=\left(\frac{1-|a|^{2}}{1-\overline{a}z}\right)^{(\gamma+1)/p}$
belongs to $A^{p}_{\omega}$ with
$\|f_{a,p,\gamma}\|^{p}_{A^{p}_{\omega}}\asymp\widehat{\omega}(z)(1-|z|).$
Then the definition of $\delta_{z}$ simply gives the other direction.
Therefore, we actually have

$\|\delta_{z}\|\asymp\frac{1}{((1-|z|)\widehat{\omega}(z))^{1/p}},\quad
z\in\mathbb{D}.$ (1.6)

Apart from this we refer to [25, 26] and references therein for basic
properties of these weighted Bergman spaces.

A possible and promising method to depict the spectrum of a strongly
continuous composition semigroup is the so-called spectral mapping theorem
which states the relationship between the spectrum of every $C_{t}$ and the
spectrum of $\Gamma$. Recall that a composition semigroup $(C_{t})_{t\geq 0}$
and its generator $\Gamma$ satisfy the weakly spectral mapping theorem (SMT)
if the following identity is valid:

$\sigma(C_{t})\backslash\\{0\\}=\overline{e^{t\sigma(\Gamma)}}:=\overline{\\{e^{t\lambda}:\lambda\in\sigma(\Gamma)\\}}\quad\mbox{for}~{}~{}t\geq
0,$ (1.7)

where $\sigma(\cdot)$ is the spectrum of a certain linear operator. The
identity (1.7) naturally holds if $(C_{t})_{t\geq 0}$ is acting on a Hilbert
space, and hence it is valid for $A^{2}_{\omega}$. However, in general case,
(1.7) holds only if $\sigma$ is replaced by point spectrum $P\sigma$ or
residue spectrum $R\sigma$. In fact, it is only the approximate point spectrum
$A\sigma$ that is responsible for the failure of (SMT). In this case, we only
have the inclusion relation in the right direction of (1.7). Nevertheless,
(SMT) can be guaranteed if $(C_{t})_{t\geq 0}$ happens to be eventually norm
continuous. See [13] for more information about (SMT).

In this paper, we will investigate the spectrum of the generator of the
eventually norm continuous composition semigroup $(C_{t})_{t\geq 0}$ induced
by an elliptical semigroup $(\varphi_{t})_{t\geq 0}$ on the weighted Bergman
space $A^{p}_{\omega}$ with $\omega\in\widehat{\mathcal{D}}$. We say that a
strongly continuous composition semigroup $(C_{t})_{t\geq 0}$ is eventually
norm continuous on a Banach space $X$ if there exists a $t_{0}>0$ such that
the composition $C_{t}$ is norm continuous for all $t>t_{0}$, i.e.

$\lim_{s\to t}\|C_{s}-C_{t}\|_{X}=0.$

From now on, denote by $\|T\|_{X}$, $r_{X}(T)$ and $\sigma_{X}(T)$ the
operator norm, spectral radius, and the spectrum of $T$ on $X$. Denote by
$\|T\|_{e,X}$ and $r_{e,X}(T)$ the essential norm and essential spectral
radius of the linear operator $T$ on the Banach space $X$. Our first main
result can be stated as follows.

######  Theorem 1.

Let $1\leq p<\infty$ and $\omega\in\widehat{\mathcal{D}}$. Suppose
$(\varphi_{t})_{t\geq 0}$ is a semigroup of analytic self-maps of $\mathbb{D}$
with Denjoy-Wolff point $b\in\mathbb{D}$, infinitesimal generator $G$, and
associated Koenigs function $h$. Denote by $\Gamma$ the infinitesimal
generator of the corresponding composition semigroup $(C_{t})_{t\geq 0}$. Then
the following statements hold:

  * (i)

If $(\varphi_{t})_{t\geq 0}$ consists of automorphisms of $\mathbb{D}$, then

$\sigma(\Gamma)=P\sigma(\Gamma)=\\{kG^{\prime}(b):h^{k}\in
A^{p}_{\omega},k=0,1,2,3\ldots\\};$

  * (ii)

If there is no element of $(\varphi_{t})_{t\geq 0}$ is automorphic of
$\mathbb{D}$ and $(C_{t})_{t\geq 0}$ is eventually norm continuous, then for
any $t>0$, the spectrum $\sigma(\Gamma)$ satisfies the following equality:

$\\{e^{t\lambda}:\lambda\in\sigma(\Gamma)\\}\cup\\{0\\}=\\{\lambda\in\mathbb{C}:|\lambda|\leq
r_{e,A^{p}_{\omega}}(C_{\varphi_{t}})\\}\cup\\{e^{kG^{\prime}(b)t}:k=0,1,2,3\ldots\\}.$

We make several observations on the above theorem. First of all, the most
basic problem is to show that $(C_{t})_{t\geq 0}$ is strongly continuous on
$A^{p}_{\omega}$ provided $\omega\in\widehat{\mathcal{D}}$. Otherwise, we can
not proceed to consider the further problem of investigating the spectrum of
the generator $\Gamma$ of $(C_{t})_{t\geq 0}$. However, we will show that
$(C_{t})_{t\geq 0}$ is strongly continuous on $A^{p}_{\omega}$ and, without
any surprise, its generator has the same format as the case of Hardy and
standard Bergman spaces. Second, the method used to depict the point spectrum
of the $\Gamma$ is the argument principle, which also allows us to get
$P\sigma(\Gamma)$ when the Denjoy-Wolff point $b$ is on the boundary. Indeed,
if $b\in\mathbb{T}$, then we can get $P\sigma(\Gamma)=\\{\lambda
G(0):e^{\lambda h(z)}\in A^{p}_{\omega}\\}.$ Further, the reason why Theorem 1
is divided into two cases is that any element of the semigroup
$(\varphi_{t})_{t\geq 0}$ is simultaneously automorphic or not, according to
the well-known result in [1]. Finally, the interested readers may also realize
that the second statement of the Theorem 1 requires the precondition that
$(C_{t})_{t\geq 0}$ is eventually continuous on $A^{p}_{\omega}$, but the
assumption can be dropped when $p=2$ since the (SMT) naturally holds.

As we aforementioned, the key approach is spectrum mapping theorem. Since each
element of the semigroup $(\varphi_{t})_{t\geq 0}$ is univalent on
$\mathbb{D}$, to use (SMT), we need first characterize the spectrum of the
composition $C_{\varphi}$, provided $\varphi$ is univalent and has a fixed
point in $\mathbb{D}$. The approach we apply here is inherited from the
argument of Cowen and MacCluer [9] in which the case of unweighted Bergman
spaces was dealt with. The key method applied there is using so-called
iteration sequence, which will also be employed in the proof of Theorem 1.
Another important ingredient of the proof is the useful identity concerning
essential spectral radii $r_{e,A^{p}_{\omega}}(C_{\varphi})$ for distinct
$1\leq p<\infty$, which can be achieved through a shortcut instead of using
the old method that requires many laborious calculations.

Now, another natural question is when the composition semigroup
$(C_{t})_{t\geq 0}$ is eventually continuous on $A^{p}_{\omega}$ provided
$\omega\in\widehat{\mathcal{D}}$. The most straightforward way to characterize
the eventual continuity of $(C_{t})_{t\geq 0}$ is estimating the norm of the
operator $C_{\varphi}-C_{\psi}$, the difference of $C_{\varphi}$ and
$C_{\psi}$, when $\varphi$ and $\psi$ are two distinct self-maps on
$\mathbb{D}$. In the paper, as the second main result, we can get the
characterization if the weight involved is confined to belong to the class
$\mathcal{D}$. To state the result, we define
$\varphi_{a}(z)=\frac{a-z}{1-\overline{a}z}$ for all $a,z\in\mathbb{D}$. The
pseudohyperbolic distance between two points $a$ and $z$ in $\mathbb{D}$ is
$\delta(a,z)=|\varphi_{a}(z)|$. For $a\in\mathbb{D}$ and $0<r<1$, the
pseudohyperbolic disc of center $a$ and of radius $r$ is
$\Delta(a,r)=\\{z\in\mathbb{D}:\delta(a,z)<r\\}$. It is well known that
$\Delta(a,r)$ is an Euclidean disk centered at $(1-r^{2})a/(1-r^{2}|a|^{2})$
and of radius $(1-|a|^{2})r/(1-r^{2}|a|^{2})$.

######  Theorem 2.

Let $1\leq p<\infty$, $\omega\in\mathcal{D}$. Suppose $(\varphi_{t})_{t\geq
0}$ is a semigroup of analytic self-maps of $\mathbb{D}$ and $(C_{t})_{t\geq
0}$ is the corresponding composition semigroup on $A^{p}_{\omega}$. Then
$(C_{t})_{t\geq 0}$ is an eventually norm-continuous operator semigroup on
$A^{p}_{\omega}$ if and only if there exist $\gamma_{0}=\gamma_{0}(\omega)>0$
and $t_{0}>0$ such that for any $\gamma>\gamma_{0}$ and $t,s>t_{0}$

$\begin{split}&\lim_{s\to
t}\sup_{a\in\mathbb{D}}\frac{(1-|a|)^{\gamma}}{\widehat{\omega}(a)}\int_{\mathbb{D}}\left(\frac{\delta^{p}(\varphi_{t}(z),\varphi_{s}(z))}{|1-\bar{a}\varphi_{t}(z)|^{\gamma+1}}+\frac{\delta^{p}(\varphi_{t}(z),\varphi_{s}(z))}{|1-\bar{a}\varphi_{s}(z)|^{\gamma+1}}\right)\omega(z)\,dA(z)=0.\end{split}$
(1.8)

The quantity on the left of (1.8) without limit is an estimate of
$\|C_{\varphi_{t}}-C_{\varphi_{s}}\|^{p}$. We will actually prove that the
similar estimate remains hold for general $C_{\varphi}-C_{\psi}$ on
$A^{p}_{\omega}$ with $\omega\in\mathcal{D}$ and any different analytic self-
maps $\varphi$ and $\psi$ on $\mathbb{D}$. Although $C_{\varphi}-C_{\psi}$ is
automatically bounded due to the subordination principle, the above estimate
of the norm $C_{\varphi}-C_{\psi}$ requires so-called Carleson measure. Recall
that for a space of analytic functions $X$, a positive measure $\mu$ is said
to be $p$-Carleson measure for $X$ if the identity operator $I_{d}:X\to
L^{p}_{\mu}$ is bounded. It follows from [21, Theorem 2] that a positive $\mu$
is a $p$-Carleson measure for $A^{p}_{\omega}$ with $\omega\in\mathcal{D}$ if
and only if there exists $r=r(\omega)\in(0,1)$ such that

$\sup_{z\in\mathbb{D}}\frac{\mu(\Delta(z,r))}{\omega(\Delta(z,r))}<\infty.$
(1.9)

The reason why the same method cannot be carried out in the case
$A^{p}_{\omega}$ with $\omega\in\widehat{\mathcal{D}}$ is that the same
characterization (1.9) is not valid anymore for those $A^{p}_{\omega}$, see
[25, Theoem 3.3] for example. Indeed, we will show that the same norm estimate
of $C_{\varphi}-C_{\psi}$ for the case $\omega\in\mathcal{D}$ is an upper
bound for the norm $C_{\varphi}-C_{\psi}$ for the case
$\omega\in\widehat{\mathcal{D}}$ but we do not know if it is also a below
bound or not. To get the general characterization for the eventual continuity
of $(C_{t})_{t\geq 0}$ on $A^{p}_{\omega}$ with
$\omega\in\widehat{\mathcal{D}}$, we stress here that an intriguing attempt is
probably to combine the Carleson measure and the fact that every element of
$(\varphi_{t})_{t\geq 0}$ is univalent.

As explained above, we see that (1.8) is a sufficient condition such that
$(C_{t})_{t\geq 0}$ is eventually norm continuous on $A^{p}_{\omega}$ with
$\omega\in\widehat{\mathcal{D}}$, which immediately implies that the class of
$(\varphi_{t})_{t\geq 0}$ such that the corresponding $(C_{t})_{t\geq 0}$ is
norm continuous on those $A^{p}_{\omega}$ is vast. To be more concrete, for a
semigroup $(\varphi_{t})_{t\geq 0}$, if the exists a $t_{0}>0$ such that
$\|\varphi_{t_{0}}\|_{H^{\infty}}<1$, then the corresponding $(C_{t})_{t\geq
0}$ is eventually norm continuous on $A^{p}_{\omega}$, provided
$\omega\in\widehat{\mathcal{D}}$.

Another important regular property of $(C_{t})_{t\geq 0}$ is eventual
compactness, which says that there exists a $t_{0}>0$ such that for any
$t>t_{0}$, $C_{t}$ is compact. It is well known that eventual compactness is
stronger than eventual continuity. Indeed, [13, Theorem 5.12] indicates that
the eventual compactness of a strongly continuous operator semigroup is
equivalent to the eventual continuity and the compactness of the resolvent
(operator) of its infinitesimal generator. If the resolvent of the
infinitesimal generator happens to be compact, then the spectrum of the
generator only contains the point spectrum. Recall that the resolvent
$R(\lambda,\Gamma)$ of $\Gamma$, the generator of composition semigroup
$(C_{t})_{t\geq 0}$ on the Banach space $X$ is defined by
$(\lambda-\Gamma)^{-1}$ for any
$\lambda\in\rho(\Gamma)=\mathbb{C}\setminus\sigma_{X}(\Gamma)$, the resolvent
set of $\Gamma$. In general, it can be represented as

$R(\lambda,\Gamma)(f)=\int_{0}^{\infty}e^{-\lambda t}C_{t}(f)\,dt,\quad f\in
X.$

In particular, if the Denjoy-Wolff point of the induced semigroup
$(\varphi_{t})_{t\geq 0}$ point is 0, then the resolvent of $\Gamma$ has a
neatly concrete representation:

$R(\lambda,\Gamma)f(z)=-\frac{1}{G^{\prime}(0)}\frac{1}{(h(z))^{-\frac{\lambda}{G^{\prime}(0)}}}\int_{0}^{z}f(\zeta)(h(\zeta))^{-\frac{\lambda}{G^{\prime}(0)}-1}h^{\prime}(\zeta)d\zeta,~{}~{}f\in
X,$

where $G$ is the infinitesimal generator of $(\varphi_{t})_{t\geq 0}$ and $h$
is the corresponding Koenigs function.

Now, we return to the weighted Bergan space $A^{p}_{\omega}$ with
$\omega\in\widehat{\mathcal{D}}$. We will show that $-G^{\prime}(0)$ belongs
$\rho(\Gamma)$, which means that special case $R(-G^{\prime}(0),\Gamma)$ is
bounded. It is well known that $h$ is a certain spiral-like function, which is
certainly univalent. However, the linear operator

$R_{h}(f)(z)=\frac{1}{h(z)}\int_{0}^{z}f(\zeta)h^{\prime}(\zeta)\,d\zeta\quad
f\in\mathcal{H}(\mathbb{D})$

is well-defined for general univalent function $h$ with $h(0)=0$. This enables
us to consider the boundedness and compactness on $A^{p}_{\omega}$ with
$\omega\in\widehat{\mathcal{D}}$ for the full range $0<p<\infty$ and for
general univalent function $h$ with $h(0)=0$. As a byproduct, we have the
following result:

######  Theorem 3.

Let $0<p<\infty$ and $\omega\in\widehat{\mathcal{D}}$. Suppose $h$ is a
univalent function on $\mathbb{D}$ satisfying $h(0)=0$. Then the following
statements are equivalent:

  * (i)

$R_{h}$ is compact on $A^{p}_{\omega}$;

  * (ii)

$\log\frac{h(z)}{z}\in\mathcal{B}_{0}$;

  * (iii)

$\log\frac{h(z)}{z}\in\mathord{\rm VMOA}$.

The proof of the theorem absorbs from the method provided in [36], where it
was used to characterize the compactness of the resolvent of the infinitesimal
generator on Hardy spaces. The key point of this approach is to show that
$R_{h}$ has ‘similar’ properties as another novel linear operator $Q_{h}$,
defined by

$Q_{h}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\zeta)\,d\zeta+\frac{1}{z}\int_{0}^{z}f(\zeta)\zeta\left(\log\frac{h(\zeta)}{\zeta}\right)^{\prime}\,d\zeta,\quad
f\in\mathcal{H}(\mathbb{D}).$

Since $h$ is univalent with $h(0)=0$, an application of the growth theorem
(see [11, Theorem 2.6]) of univalent function implies that
$\log\frac{h(z)}{z}$ automatically belongs to the Bloch space $\mathcal{B}$,
which is equivalent to belongs to $\mathord{\rm BMOA}$ by [30]. Then it
follows from [25, Proposition 6.1 and Theorem 6.8] that $R_{h}$ is
automatically bounded on $A^{p}_{\omega}$ with
$\omega\in\widehat{\mathcal{D}}$. Likewise, the compactness part can be
deduced similarly.

The paper is organized as follows. In Section 2, we first characterize the
spectra of some composition operators on $A^{p}_{\omega}$, then using this
characterization, we prove Theorem 1. In Section 3, we characterize the
eventual norm-continuity of $(C_{t})_{t\geq 0}$ in terms of estimating the
norm of the difference of two distinct composition operators on
$A^{p}_{\omega}$. Section 4 contains the proof of Theorem 3 as well as several
other discussions about the operator $R_{h}$.

We finish the introduction with a couple of words about the notation used in
this paper. Throughout the paper, the letter $C=C(\cdot)$ will denote an
absolute constant whose value depends on the parameters indicated in the
parenthesis and may change from one occurrence to another. If there exists a
constant $C=C(\cdot)>0$ such that $a\leq Cb$, then we write either $a\lesssim
b$ or $b\gtrsim a$. In particular, if $a\lesssim b$ and $a\gtrsim b$, then we
denote $a\asymp b$ and say that $a$ and $b$ are comparable.

##  2\. Spectra of composition operators on $A^{p}_{\omega}$

Suppose $\omega\in\widehat{\mathcal{D}}$ all
$z\in\mathbb{D}\backslash\\{0\\}$. Let

$\omega^{\star}(z)=\int_{|z|}^{1}s\omega(s)\log\frac{s}{|z|}\,ds.$

From [25, Lemma 2.1], we know

$\omega^{\star}(z)\asymp\widehat{\omega}(z)(1-|z|),\quad|z|>\frac{1}{2}.$
(2.1)

In this section, we focus on the spectrum of a certain composition operator on
weighted Bergman spaces with doubling weights. To be precise, the composition
operator we consider here is confined to be univalent but not an automorphism
with the fixed point in the interior of $\mathbb{D}$. The approach applied
here is inherited from [23] in which the spectrum of the same composition
operator on unweighted Bergman spaces was investigated. It pointed out that a
certain closed subspace of weighted Bergman spaces is supposed to be an
efficient tool. Therefore, for a weight $\omega$ and a fixed $m\in\mathbb{N}$,
we define $A^{p}_{\omega,m}=\\{f\in A^{p}_{\omega}:f(z)=z^{m}g(z),g\in
A^{p}_{\omega}\\}$. In other words, $A^{p}_{\omega,m}$ consists of all $f\in
A^{p}_{\omega}$ that have a zero of at least order $m$ order at 0.

We need the following growth estimate of functions in $A_{\omega,m}^{p}$.

######  Lemma 4.

Let $0<p<\infty$ and $\omega\in\widehat{\mathcal{D}}$. For any
$m\in\mathbb{N}$ there exists an $r_{m}$ such that

$|f(z)|\lesssim\frac{m|z|^{m}}{((1-|z|)\widehat{\omega}(r_{m})\widehat{\omega}(z))^{1/p}}\|f\|_{A^{p}_{\omega}},\quad
f\in A_{\omega,m}^{p},~{}~{}z\in\mathbb{D}.$

###### Proof.

When $m\geq 2$, take $r_{m}=(\frac{1}{m})^{\frac{1}{m}}$. Then the definition
of $A_{\omega,m}^{p}$ indicates that for any $f\in A_{\omega,m}^{p}$, there
exists a $g\in A^{p}_{\omega}$ such that $f(z)=z^{m}g(z)$. By the monotonicity
of $M_{p}(r,f)$, we have

$\displaystyle\|g\|_{A_{\omega}^{p}}^{p}$
$\displaystyle\leq\int_{0}^{r_{m}}M_{p}^{p}(r_{m},g)\omega(r)rdr+\int_{r_{m}}^{1}M_{p}^{p}(r,g)\omega(r)rdr$
$\displaystyle\leq
r_{m}^{-mp}\left(\frac{\int_{0}^{r_{m}}\omega(r)rdr}{\int_{r_{m}}^{1}\omega(r)rdr}+1\right)\int_{r_{m}}^{1}M_{p}^{p}(r,g)\omega(r)r^{mp+1}dr\lesssim\frac{m^{p}}{\widehat{\omega}(r_{m})}\|f\|_{A_{\omega}^{p}}^{p}.$

This together with (1.5) gives

$|f(z)|=|z^{m}||g(z)|\lesssim\frac{m|z|^{m}\|f\|_{A^{p}_{\omega}}}{((1-|z|)\widehat{\omega}(r_{m})\widehat{\omega}(z))^{1/p}}.$

When $m=1$, let $r_{m}=1/2$. Similar to the above statement, we get the
desired result. The proof is complete. ∎

######  Lemma 5.

Let $1\leq p<\infty$ and $\omega\in\widehat{\mathcal{D}}$. Then for any
$m\in\mathbb{N}$, we have

$\|\delta_{z}\|_{A_{\omega,m}^{p}}\leq\|\delta_{z}\|_{A^{p}_{\omega}}\lesssim
2^{m}\|\delta_{z}\|_{A_{\omega,m}^{p}},\quad|z|\geq\frac{1}{2}.$

###### Proof.

It is clear that
$\|\delta_{z}\|_{A^{p}_{\omega}}\geq\|\delta_{z}\|_{A_{\omega,m}^{p}}$. To
prove the right inequality, for an fixed $z\in\mathbb{D}$, consider function
$g(\zeta)=g_{z,p,\gamma,m}(\zeta)=\zeta^{m}\left(\frac{1-|z|^{2}}{1-\overline{z}\zeta}\right)^{(\gamma+1)/p}.$
Then for some large enough $\gamma>0$, $g\in A_{\omega,m}^{p}$. Thus, the
definition of $\delta_{z}$, (1.6), and Lemma 4 yield

$\|\delta_{z}\|_{A_{\omega,m}^{p}}\gtrsim\frac{|z|^{m}}{\left((1-|z|)\widehat{\omega}(z)\right)^{1/p}}\gtrsim\frac{1}{2^{m}}\|\delta_{z}\|_{A^{p}_{\omega}},\quad|z|\geq\frac{1}{2}.$

The proof is completed. ∎

The following result shows a close relation of the essential spectral radius
of composition operators on different weighted Bergman spaces. Probably a
classical proof originates from [23] in which only the case of $1<p<\infty$
was solved, while the case $p=1$ was missing. However, we will provide a
trivial method that allows us to deal with the case $1\leq p<\infty$ directly,
hence avoiding lots of laborious calculations. Indeed, the approach applied
here relies on an asymptotic estimate of the essential spectrum of composition
operator on the weighted Bergman space, which was demonstrated in [28, Theorem
19].

######  Lemma 6.

Let $1\leq p<\infty$, $\omega\in\widehat{\mathcal{D}}$, and let $\varphi$ be
an analytic self-map of $\mathbb{D}$. Then

$\left(r_{e,A^{p}_{\omega}}(C_{\varphi})\right)^{p}=\left(r_{e,A_{\omega}^{2}}(C_{\varphi})\right)^{2}.$

###### Proof.

Using [28, Theorem 19] directly with the special case $p=q$, we immediately
see that
$\|C_{\varphi}\|^{p}_{A^{p}_{\omega}}\asymp\|C_{\varphi}\|^{2}_{A^{2}_{\omega}}$.
By the definition of essential spectral radius
$r_{e,A^{p}_{\omega}}(C_{\varphi})=\lim_{n\rightarrow\infty}(\|C_{\varphi}^{n}\|_{e,A^{p}_{\omega}})^{\frac{1}{n}}$
and the fact $C_{\varphi_{n}}=C_{\varphi}^{n}$, replacing $\varphi$ by its
$n$th iteration $\varphi_{n}$, we have

$\|C_{\varphi_{n}}\|_{e,A^{p}_{\omega}}^{p}\asymp\|C_{\varphi_{n}}\|_{e,A_{\omega}^{2}}^{2},\quad
1\leq p<\infty.$

This easily leads to the identity we desired. ∎

The proof of Theorem 1 relies on the so-called iteration sequence. We say that
$\\{z_{k}\\}$ is an iteration sequence if there exists
$\varphi:\mathbb{D}\rightarrow\mathbb{D}$ such that $\varphi(z_{k})=z_{k+1}$.
The following lemma plays a critical role in our proof whose proof can be
found in [8, Lemmas 7.34 and 7.35].

######  Lemma 7.

Let $\varphi$ be an analytic self-map of $\mathbb{D}$, not an automorphism and
$\varphi(0)=0$. Given $0<r<1$, there exists $1\leq M_{r}<\infty$ such that if
$\\{z_{k}\\}_{-K}^{\infty}$ is an iteration sequence for $\varphi$ with
$|z_{n}|\geq r$ for some non-negative integer $n$ and $\\{w_{k}\\}_{k=-K}^{n}$
is arbitrary complex numbers, then there exists an $f\in H^{\infty}$ satisfied

$f(z_{k})=w_{k},\quad-K\leq k\leq n$

and

$\|f\|_{H^{\infty}}\leq M_{r}\sup\\{|w_{k}|:-K\leq k\leq n\\}.$

Moreover, there exists $0<b<1$ such that for any iteration sequence
$\\{z_{k}\\}$ we have

$\frac{|z_{k+1}|}{|z_{k}|}\leq b,$

whenever $|z_{k}|\leq\frac{1}{2}.$

The above lemma enables us to simplify some notation. For an iteration
sequence $\\{z_{k}\\}_{-K}^{\infty}$, from now on, suppose $|z_{0}|\geq 1/2$
and $n=\max\\{k:|z_{k}|\geq 1/4\\}.$ It follows from Lemma 7 that there is a
$1/2\leq b<1$ such that for any $k>n$, $\frac{|z_{k+1}|}{|z_{k}|}\leq b$ and
hence

$|z_{k}|\leq|z_{n}|b^{k-n},\quad k\geq n.$

Now, we are ready to characterize the spectrum of a certain composition
operator that will be used to prove Theorem 1.

######  Theorem 8.

Let $1\leq p<\infty$ and $\omega\in\widehat{\mathcal{D}}$. Suppose $\varphi$
is an analytic self-map on $\mathbb{D}$ with the fixed point $a\in\mathbb{D}$
but not an automorphism. Then

$\sigma_{A^{p}_{\omega}}(C_{\varphi})=\\{\lambda\in\mathbb{C}:|\lambda|<r_{e,A^{p}_{\omega}}(C_{\varphi})\\}\cup\\{(\varphi^{\prime}(a))^{n}\\}^{\infty}_{n=0}.$

###### Proof.

Without loss of generality, assume that $a=0$. Since all polynomials are
contained in weighted Bergman spaces, it follows from [15, Lemma 2] that
$\\{(\varphi^{\prime}(0))^{n}\\}_{n=0}^{\infty}$ is contained in the spectrum
of $C_{\varphi}$. If $\lambda\in\sigma_{A^{p}_{\omega}}(C_{\varphi})$ and
$|\lambda|>r_{e,A^{p}_{\omega}}(C_{\varphi})$, by [7, Proposition 2.2], we
know that $\lambda$ is an eigenvalue of $C_{\varphi}$. Moreover, Koenig’s
Theorem ([8, Theorem 2.63]) tells that the eigenvalues of $C_{\varphi}$ must
be of form $(\varphi^{\prime}(0))^{n}$. Therefore, to finish the proof, it
suffices to verify

$\displaystyle\\{\lambda\in\mathbb{C}:|\lambda|\leq
r_{e,A^{p}_{\omega}(C_{\varphi})}\\}\subseteq\sigma_{A^{p}_{\omega}}(C_{\varphi}).$
(2.2)

Trivially, $0\in\sigma_{A^{p}_{\omega}}$, which implies that (2.2) holds when
$r_{e,A^{p}_{\omega}(C_{\varphi})}=0$. Now, we assume
$r_{e,A^{p}_{\omega}(C_{\varphi})}>0$. We only need to deal with the case when
$0<|\lambda|<r_{e,A^{p}_{\omega}(C_{\varphi})}$ since spectrum is closed. If
we can prove that $(C_{\varphi}-\lambda I)^{*}$ is not invertible, then it
will be done. Let now $C_{m}$ be the restriction of $C_{\varphi}$ acting on
$A_{\omega,m}^{p}$. Note that $\varphi(0)=0$ implies that $A_{\omega,m}^{p}$
is an invariant subspace of $C_{\varphi}$ and hence
$\lambda\in\sigma_{A_{\omega,m}^{p}}(C_{m})\subseteq\sigma_{A^{p}_{\omega}}(C_{\varphi})$
by [8, Lemma 7.17]. Therefore, to show that $(C_{\varphi}-\lambda I)^{*}$ is
not invertible on $A^{p}_{\omega}$, it suffices to prove that $(C_{m}-\lambda
I)^{*}$ is not bounded blow.

Let now $r=1/4$. By Lemma 7, we can choose two constants $b$ and $M$
satisfying $1/2\leq b<1$ and $1\leq M<\infty$. Fix $m$ sufficiently large such
that

$\frac{b^{m}}{|\lambda|}<\frac{1}{SM}.$

Here, $S$ is large enough and will be fixed later. For any iteration sequence
$\\{z_{k}\\}_{-K}^{\infty}$, we can define a linear functional $L_{\lambda}$
on $A_{\omega,m}^{p}$ by

$L_{\lambda}(f)=\sum_{k=-K}^{\infty}\lambda^{-k}f(z_{k}).$

Indeed, by the hypothesis that $|z_{k}|<1/4$ when $k>n$ and Lemma 4, we deduce
that

$\begin{split}\left|\sum_{k=n+1}^{\infty}\lambda^{-k}f(z_{k})\right|\lesssim&\sum_{k=n+1}^{\infty}|\lambda|^{-k}\frac{m|z_{k}|^{m}}{((1-|z_{k}|)\widehat{\omega}(r_{m})\widehat{\omega}(z_{k}))^{1/p}}\|f\|_{A^{p}_{\omega}}\\\
\lesssim&\frac{m\|f\|_{A^{p}_{\omega}}}{(\widehat{\omega}(r_{m}))^{\frac{1}{p}}}\sum_{k=n+1}^{\infty}\frac{|z_{k}|^{m}}{|\lambda|^{k}}.\end{split}$

Here, $r_{m}=(\frac{1}{m})^{\frac{1}{m}}$. However, since
$|z_{k}|\leq|z_{n}|b^{k-n}$ and $\frac{b^{m}}{|\lambda|}<1$, $L_{\lambda}$ is
a bounded linear functional on $A_{\omega,m}^{p}$. Moreover, a simple
calculation gives

$(C_{m}^{*}-\lambda)L_{\lambda}=-\lambda^{K+1}\delta_{z_{-K}}.$

Then Lemma 7 indicates that exists $h\in H^{\infty}$ with
$\|h\|_{H^{\infty}}\leq M$ such that:

  1. (i)

When $k=0$ and $k=n$, $|h(z_{k})|=1$.

  2. (ii)

When $k=0$ and $k=n$,
$\frac{z_{k}^{m}h(z_{k})}{\lambda^{k}(1-\overline{z_{0}}z_{k})^{\frac{\gamma+1}{p}}}>0$.

  3. (iii)

When $k\neq 0$ and $-K\leq k<n$, $h(z_{k})=0$.

Therefore, for a sufficiently large $\gamma$ depending $\omega$, we have

$L_{\lambda}\left(\frac{z^{m}(1-|z_{0}|^{2})^{\frac{\gamma}{p}}h(z)}{(\widehat{\omega}(z_{0}))^{1/p}(1-\overline{z_{0}}z)^{\frac{\gamma+1}{p}}}\right)=\sum_{k=-K}^{\infty}\lambda^{-k}\frac{z_{k}^{m}(1-|z_{0}|^{2})^{\frac{\gamma}{p}}h(z_{k})}{(\widehat{\omega}(z_{0}))^{1/p}(1-\overline{z_{0}}z_{k})^{\frac{\gamma+1}{p}}}=\sum_{k=-K}^{\infty}J_{k,h}.$

Note that $J_{k,h}(-K\leq k\leq n)$ are zeros unless $k=0,n$. Also,

$J_{0,h}=\frac{|z_{0}|^{m}}{(\widehat{\omega}(z_{0}))^{1/p}(1-|z_{0}|^{2})^{\frac{1}{p}}}$

and

$J_{n,h}=\frac{|z_{n}|^{m}(1-|z_{0}|^{2})^{\frac{\gamma}{p}}}{|\lambda|^{n}|(\widehat{\omega}(z_{0}))^{1/p}|1-z_{0}z_{n}|^{\frac{\gamma+1}{p}}}\geq\frac{|z_{n}|^{m}(1-|z_{0}|^{2})^{\frac{\gamma}{p}}}{2^{\frac{\gamma+1}{p}}|\lambda|^{n}(\widehat{\omega}(z_{0}))^{1/p}}.$

Since

$\sum_{k=n+1}^{\infty}\left(\frac{b^{m}}{|\lambda|}\right)^{k-n}<\frac{\frac{1}{SM}}{1-\frac{1}{SM}}\leq\frac{1}{(S-1)M},$

we obtain

$\begin{split}\left|\sum_{k=n+1}^{\infty}J_{k,h}\right|=&\left|\sum_{k=n+1}^{\infty}\lambda^{-k}\frac{z_{k}^{m}(1-|z_{0}|^{2})^{\frac{\gamma}{p}}h(z_{k})}{(\widehat{\omega}(z_{0}))^{1/p}(1-\overline{z_{0}}z_{k})^{\frac{\gamma+1}{p}}}\right|\\\
\leq&(\frac{4}{3})^{\frac{\gamma+1}{p}}\frac{M|z_{n}|^{m}(1-|z_{0}|^{2})^{\frac{\gamma}{p}}}{|\lambda|^{n}(\widehat{\omega}(z_{0}))^{1/p}}\sum_{k=n+1}^{\infty}\left(\frac{b^{m}}{|\lambda|}\right)^{k-n}\\\
\leq&(\frac{4}{3})^{\frac{\gamma+1}{p}}\cdot\frac{1}{S-1}\cdot\frac{|z_{n}|^{m}(1-|z_{0}|^{2})^{\frac{\gamma}{p}}}{|\lambda|^{n}(\widehat{\omega}(z_{0}))^{1/p}}\\\
\leq&(\frac{1}{3})^{\frac{\gamma+1}{p}}\cdot\frac{|z_{n}|^{m}(1-|z_{0}|^{2})^{\frac{\gamma}{p}}}{|\lambda|^{n}(\widehat{\omega}(z_{0}))^{1/p}}.\end{split}$

Here, we choose a large enough constant $S>0$ such that
$\frac{4^{\frac{\gamma+1}{p}}}{S-1}<1$. Therefore,

$\displaystyle\left|L_{\lambda}\left(\frac{z^{m}(1-|z_{0}|^{2})^{\frac{\gamma}{p}}h}{(\widehat{\omega}(z_{0}))^{1/p}(1-\overline{z_{0}}z)^{\frac{\gamma+1}{p}}}\right)\right|$
$\displaystyle\geq J_{0,h}+J_{n,h}-\left|\sum_{k=n+1}^{\infty}J_{k,h}\right|$
$\displaystyle\geq\frac{|z_{0}|^{m}}{(\widehat{\omega}(z_{0}))^{1/p}(1-|z_{0}|^{2})^{\frac{1}{p}}}.$

Write
$g(z)=\frac{(1-|z_{0}|^{2})^{\frac{\gamma}{p}}}{(\widehat{\omega}(z_{0}))^{1/p}(1-\overline{z_{0}}z)^{\frac{\gamma+1}{p}}}\in
A^{p}_{\omega}.$ Then we have

$\left\|\frac{z^{m}(1-|z_{0}|^{2})^{\frac{\gamma}{p}}h}{(\widehat{\omega}(z_{0}))^{1/p}(1-\overline{z_{0}}z)^{\frac{\gamma+1}{p}}}\right\|_{A_{\omega,m}^{p}}=\|z^{m}gh\|_{A_{\omega,m}^{p}}\leq
CM,$

and further,

$\|L_{\lambda}\|_{A_{\omega,m}^{p}}\geq\frac{|L_{\lambda}(z^{m}gh)|}{CM}\gtrsim\frac{|z_{0}|^{m}}{(\widehat{\omega}(z_{0}))^{1/p}(1-|z_{0}|^{2})^{\frac{1}{p}}}\asymp\|\delta_{z_{0}}\|_{A_{\omega,m}^{p}}.$

We are now ready to make a judicious choice of iteration sequence. Recall that

$r_{e,A^{p}_{\omega}}(C_{\varphi})=\lim_{n\rightarrow\infty}\|C_{\varphi_{n}}\|_{e,A^{p}_{\omega}}^{1/n}=\inf_{n\in\mathbb{N}}\|C_{\varphi_{n}}\|_{e,A^{p}_{\omega}}^{1/n}.$

Since $\varphi$ is univalent, by Proposition 18 and Theorem 19 in [28], we
have

$\|C_{\varphi_{n}}\|_{e,A_{\omega}^{2}}^{2}=\limsup_{|w|\rightarrow
1^{-}}\frac{\omega^{\star}(\varphi_{n}^{-1}(w))}{\omega^{\star}(w)}=\limsup_{|w|\rightarrow
1^{-}}\frac{\omega^{\star}(w)}{\omega^{\star}(\varphi_{n}(w))}.$

For any $\lambda$ with $0<|\lambda|<r_{e,A^{p}_{\omega}}(C_{\varphi})$, choose
$\tau$ satisfying $|\lambda|<\tau<r_{e,A^{p}_{\omega}}(C_{\varphi})$. Applying
Lemma 6, we deduce

$r_{e,A^{p}_{\omega}}(C_{\varphi})=\lim_{n\rightarrow\infty}\|C_{\varphi_{n}}\|_{e,A^{p}_{\omega}}^{1/n}=\lim_{n\rightarrow\infty}\left(\|C_{\varphi_{n}}\|_{e,A_{\omega}^{2}}^{2/p}\right)^{1/n}.$

That is, there exists $N$, such that for any $n>N$,

$\|C_{\varphi_{n}}\|_{e,A_{\omega}^{2}}^{2/p}>\tau^{n}.$

Thus, for any $K\geq N$, we can choose $w$ approaching the boundary of
$\mathbb{D}$ such that

  1. (i)

$\left(\frac{\omega^{\star}(|w|)}{\omega^{\star}(|\varphi_{K}(w)|)}\right)^{1/p}\geq\tau^{K}$.

  2. (ii)

$|\varphi_{K}(w)|\geq 1/2$.

  3. (iii)

$\frac{\|\delta_{\varphi_{K}(w)}\|_{A_{\omega,m}^{p}}}{\|\delta_{w}\|_{A_{\omega,m}^{p}}}\gtrsim\frac{1}{2^{m}}\frac{\|\delta_{\varphi_{K}(w)}\|_{A^{p}_{\omega}}}{\|\delta_{w}\|_{A^{p}_{\omega}}}.$

Combining (i), (iii), (2.1) and Lemma 5, we deduce that

$\frac{\|\delta_{\varphi_{K}(w)}\|_{A_{\omega,m}^{p}}}{\|\delta_{w}\|_{A_{\omega,m}^{p}}}\gtrsim\frac{1}{2^{m}}\tau^{K}.$

With this choice of $w$, we get an iteration sequence
$\\{z_{k}\\}_{k=-K}^{\infty}$ by setting $z_{-K}=w$ so that
$z_{0}=\varphi_{K}(w)$ and $|z_{0}|\geq 1/2$. At this point, the positive
integer $K$ is still arbitrary, except for the requirement that $K\geq N$. Our
estimates say

$\begin{split}\frac{\|(C^{*}_{m}-\lambda)L_{\lambda}\|_{A_{\omega,m}^{p}}}{\|L_{\lambda}\|_{A_{\omega,m}^{p}}}\lesssim&\frac{|\lambda|^{K+1}\|\delta_{z_{-K}}\|}{\|\delta_{z_{0}}\|}\lesssim|\lambda|\left(\frac{\lambda}{\tau}\right)^{K}.\end{split}$

If $K\geq N$ large enough, then
$\left(\frac{\lambda}{\tau}\right)^{K}\rightarrow 0$, which means that
$(C^{*}_{m}-\lambda)$ is not bounded below. The proof is completed. ∎

We first assert that each semigroup $(\varphi_{t})_{t\geq 0}$ induces a
strongly continuous composition semigroup on $A^{p}_{\omega}$, provided $1\leq
p<\infty$ and $\omega\in\widehat{\mathcal{D}}$. Although the proof is
standard, we still present it for the sake of the completeness of the
statement.

######  Proposition 9.

Let $1\leq p<\infty$ and $\omega\in\widehat{\mathcal{D}}$. Suppose
$(\varphi_{t})_{t\geq 0}$ is a semigroup of analytic self-maps of $\mathbb{D}$
with the infinitesimal generator $G$. Then the corresponding composition
semigroup $(C_{t})_{t\geq 0}$ is strongly continuous on $A^{p}_{\omega}$ with
the infinitesimal generator $\Gamma$:

$\Gamma f=Gf^{\prime}$ (2.3)

on the domain

$D(\Gamma)=\\{f\in A^{p}_{\omega}:Gf^{\prime}\in A^{p}_{\omega}\\}.$

Moreover, $(C_{t})_{t\geq 0}$ is uniformly continuous on $A^{p}_{\omega}$ if
and only if $(\varphi_{t})_{t\geq 0}$ is trivial.

###### Proof.

Since $\omega\in\widehat{\mathcal{D}}$, the polynomials are dense in
$A^{p}_{\omega}$. Therefore for any $f\in A^{p}_{\omega}$, there exists a
sequence of polynomials $\\{P_{n}\\}$ such that
$\lim_{n\rightarrow\infty}\|f-P_{n}\|_{A^{p}_{\omega}}=0$. It follows from
triangle inequality that

$\displaystyle\|C_{t}f-f\|_{A^{p}_{\omega}}$
$\displaystyle\leq\|C_{t}f-C_{t}P_{n}\|_{A^{p}_{\omega}}+\|C_{t}P_{n}-P_{n}\|_{A^{p}_{\omega}}+\|f-P_{n}\|_{A^{p}_{\omega}}$
$\displaystyle\leq(\|C_{t}\|+1)\|f-P_{n}\|_{A^{p}_{\omega}}+\|C_{t}P_{n}-P_{n}\|_{A^{p}_{\omega}}.$

According to [28, Theorem 15 and (4.7)], we obtain
$\sup_{t\in[0,1]}\|C_{t}\|<\infty$. Therefore, to prove $\lim_{t\rightarrow
0^{+}}\|C_{t}f-f\|_{A^{p}_{\omega}}=0$, it suffices to prove
$\lim_{t\rightarrow 0^{+}}\|C_{t}P-P\|_{A^{p}_{\omega}}=0$ for each polynomial
$P$. Equivalently we only need to show that for each $n\geq 0$
$\lim_{t\rightarrow 0^{+}}\|(\varphi_{t})^{n}-e_{n}\|_{A^{p}_{\omega}}=0$,
where $e_{n}(z)=z^{n}$, while it can be easily obtained by Lebesgue dominated
convergence theorem. Thus $(C_{t})_{t\geq 0}$ is strongly continuous on
$A^{p}_{\omega}$.

By definition, the domain of $\Gamma$ is

$D(\Gamma)=\\{f\in A^{p}_{\omega}:\lim_{t\rightarrow
0^{+}}\frac{C_{t}(f)-f}{t}~{}~{}\mbox{exists~{}~{}in}~{}A^{p}_{\omega}\\}.$

Let now $D=\\{f\in A^{p}_{\omega}:Gf^{\prime}\in A^{p}_{\omega}\\}$. We are
going to show that if $f\in D(\Gamma)$, then $Gf^{\prime}\in A^{p}_{\omega}$.
Indeed, if $f\in D(\Gamma)$, then $\Gamma(f)\in A^{p}_{\omega}$ and

$\lim_{t\rightarrow
0^{+}}\left\|\frac{C_{t}f-f}{t}-\Gamma(f)\right\|_{A^{p}_{\omega}}=0.$

Since for a fixed $0<r<1$ the well known inequality $M_{\infty}(r,f)\lesssim
M_{p}(\frac{1+r}{2},f)(1-r)^{-\frac{1}{p}}$ yields

$M_{\infty}^{p}(r,f)\lesssim\frac{\|f\|^{p}_{A^{p}_{\omega}}}{(1-r)\widehat{\omega}(r)},$

convergence in the norm of $A^{p}_{\omega}$ implies the pointwise convergence.
Therefore, for every $z\in\mathbb{D}$,

$\displaystyle\Gamma f(z)$ $\displaystyle=\lim_{t\rightarrow
0^{+}}\frac{f(\varphi_{t}(z))-f(z)}{t}=\lim_{t\rightarrow
0^{+}}\frac{f(\varphi_{t}(z))-f(\varphi_{0}(z)}{t}$
$\displaystyle\frac{\partial(f(\varphi_{t}(z)))}{\partial
t}\bigg{|}_{t=0}=G(z)f^{\prime}(z).$

That is, $G(z)f^{\prime}(z)=\Gamma f(z)\in A^{p}_{\omega}$ and
$D(\Gamma)\subseteq D$. On the other hand, for $\lambda\in\rho(\Gamma)$, the
resolvent set of $\Gamma$, we have

$D=\\{f\in A^{p}_{\omega}:Gf^{\prime}\in A^{p}_{\omega}\\}=\\{f\in
A^{p}_{\omega}:Gf^{\prime}-\lambda f\in A^{p}_{\omega}\\}=R(\lambda,\Gamma),$

where $R(\lambda,\Gamma)=(\lambda I-\Gamma)^{-1}$ is the resolvent operator of
$\Gamma$. Since $R(\lambda,\Gamma)(A^{p}_{\omega})\subseteq D(\Gamma)$,
$D\subseteq D(\Gamma)$. Hence, $D=D(\Gamma)$.

If $(C_{t})_{t\geq 0}$ is uniformly continuous on $A^{p}_{\omega}$, then the
infinitesimal generator $\Gamma$ is bounded on $A^{p}_{\omega}$. To show
$(\varphi_{t})_{t\geq 0}$ is trivial, it is equivalent to show $G\equiv 0$. To
the end, we may consider polynomials $e_{n}(z)=z^{n}$. Then
$\Gamma(e_{n})=nGe_{n-1}$, and taking $n=1$ we see that $G\in A^{p}_{\omega}$.
Since $\Gamma$ is bounded on $A^{p}_{\omega}$, for $n\geq 1$, we have
$\|\Gamma e_{n}\|\lesssim\|e_{n}\|$, that is,

$n^{p}\int_{0}^{1}M^{p}_{p}(r,G)r^{(n-1)p+1}\omega(r)dr\lesssim\int_{0}^{1}r^{np+1}\omega(r)dr.$

It follows that $n^{p}M_{p}^{p}(\frac{1}{2},G)\lesssim 1$. Since $G\in
A^{p}_{\omega}$, we have $M_{p}(\frac{1}{2},G)=0$. Thus $G\equiv 0$. The proof
is complete. ∎

_Proof of_ Theorem 1. (i) If each $\varphi_{t}$ is an automorphism of the
disk, then it indicates that the resolvent of $\Gamma$ is compact and hence
the spectrum of $\Gamma$ contains only the point spectrum.

Now, differentiating both sides of (1.2) with respect to $t$ and letting
$t=0$, we get that $G(z)=G^{\prime}(b)\frac{h(z)}{h^{\prime}(z)}$. On one
hand, suppose $f\in\mathcal{H}(\mathbb{D})$ and $\lambda\neq 0$ such that
$\Gamma f=\lambda f$. Then (2.3) yields

$\frac{f^{\prime}}{f}=\lambda\frac{h^{\prime}}{G^{\prime}(b)h}.$

Choosing $r$ such that $|b|<r<1$ and $f$ has no zeros on $|z|=r$, we deduce

$\frac{1}{2\pi
i}\int_{|z|=r}\frac{f^{\prime}(z)}{f(z)}dz=\frac{\lambda}{G^{\prime}(b)}\frac{1}{2\pi
i}\int_{|z|=r}\frac{h^{\prime}(z)}{h(z)}\,dz.$

This together with the argument principle yields
$\frac{\lambda}{G^{\prime}(b)}=k$, where $k$ is any nonnegative integer. On
the other hand, notice that the differential equation
$f^{\prime}(z)=k\frac{h^{\prime}(z)}{h(z)}f(z)$ has the solution
$f(z)=ch^{k}$, $c\neq 0$. We finish the proof of (i). (ii) The proof is pretty
straightforward. Since we hypothesize that $(C_{t})_{t\geq 0}$ is eventually
norm continuous on $A^{p}_{\omega}$, the (SMT) holds naturally. Therefore, by
Theorem 8, we get for any $t>0$

$\\{e^{t\lambda}:\lambda\in\sigma(\Gamma)\\}=\\{\lambda\in\mathbb{C}:|\lambda|\leq
r_{e,A^{p}_{\omega}}(C_{\varphi_{t}})\\}\cup\\{(\varphi_{t}^{\prime}(b))^{k}\\}:k=0,1,2,3\ldots\\}\setminus\\{0\\}.$

The next step is taking the derivative of both sides of (1.2) with respect to
$z$ and taking $z=b$, we obtain

$\varphi_{t}^{\prime}(b)=e^{G^{\prime}(b)t},\quad t>0.$

This together with the fact that point
$0\in\sigma_{A^{p}_{\omega}}(C_{\varphi_{t}})$ for any $t>0$, we can get the
desire result.

##  3\. Eventually norm continuous composition semigroup

The proof of Theorem 2 does essentially rely on the result of characterizing
bounded difference of composition operators on weighted Bergman spaces. In
general, we have the following result for the case that
$\omega\in\mathcal{D}$. The proof of the sufficiency is basically similar to
the proof of [22, Proposition 4], while the necessity should be modified since
the original proof shown in [22, Proposition 6] used a wrong estimate [31,
p.795], see [20] for their corrections.

The Carleson square $S(I)$ based on an interval $I\subset\mathbb{T}$ is the
set $S(I)=\left\\{re^{it}\in\right.$ $\left.\mathbb{D}:e^{it}\in I,1-|I|\leq
r<1\right\\}$, where $|I|$ denotes the Lebesgue measure of
$I\subset\mathbb{T}$. We associate to each $a\in\mathbb{D}\backslash\\{0\\}$
the interval
$I_{a}=\left\\{e^{i\theta}:\left|\arg\left(ae^{-i\theta}\right)\right|\leq\frac{1-|a|}{2}\right\\}$,
and denote $S(a)=S\left(I_{a}\right)$.

To prove the necessity, we need the following result, whose proof can be found
in [18, Theorem 2.8].

######  Lemma 10.

Suppose $s>1$ and $0<r_{0}<1$. Then there exist $N=N\left(r_{0}\right)>1$ and
$C=C\left(s,r_{0}\right)>0$ such that

$\displaystyle\left|\frac{1}{(1-\bar{a}z)^{s}}-\frac{1}{(1-\bar{a}w)^{s}}\right|+\left|\frac{1}{\left(1-t_{N}\bar{a}z\right)^{s}}-\frac{1}{\left(1-t_{N}\bar{a}w\right)^{s}}\right|$
$\displaystyle\quad\geq C\delta(z,w)\left|\frac{1}{(1-\bar{a}z)^{s}}\right|,$

for all $z\in\Delta\left(a,r_{0}\right)$ with $1-|a|<1/(2N),t_{N}=1-N(1-|a|)$
and $w\in\mathbb{D}$.

######  Proposition 11.

Let $0<p<\infty$, $\omega\in\mathcal{D}$, and let $\nu$ be a positive Borel
measure on $\mathbb{D}$, Suppose $\varphi$ and $\psi$ be analytic self-maps of
$\mathbb{D}$. Let $\sigma(z)=|\delta(\varphi(z),\psi(z))|$. Then difference
operator $C_{\varphi}-C_{\psi}:A^{p}_{\omega}\to L^{p}_{\nu}$ is bounded if
and only if both weighted composition operators $\sigma C_{\varphi}$ and
$\sigma C_{\psi}$ are bounded from $A^{p}_{\omega}$ to $L^{p}_{\nu}$.
Moreover, there exist $\gamma_{0}=\gamma_{0}(\omega)$ such that for all
$\gamma>\gamma_{0}$

$\|C_{\varphi}-C_{\psi}\|^{p}\asymp\sup_{a\in\mathbb{D}}\frac{(1-|a|)^{\gamma}}{\widehat{\omega}(a)}\int_{\mathbb{D}}\left(\frac{\sigma(z)^{p}}{|1-\bar{a}\varphi(z)|^{\gamma+1}}+\frac{\sigma(z)^{p}}{|1-\bar{a}\psi(z)|^{\gamma+1}}\right)\omega(z)\,dA(z).$
(3.1)

###### Proof.

Suppose $\sigma C_{\varphi}$ and $\sigma C_{\psi}$ are bounded. For a fixed
$0<r<1$, let $E=\\{z\in\mathbb{D}:\sigma(z)>r\\}$. Then

$\begin{split}\|C_{\varphi}(f)-C_{\psi}(f)\|^{p}_{L^{p}_{\nu}}&=\left(\int_{E}+\int_{\mathbb{D}\setminus
E}\right)|f(\varphi(z))-f(\psi(z))|^{p}\nu(z)\,dA(z)\\\
&\leq\frac{1}{r}\int_{E}\sigma(z)|f(\varphi(z))-f(\psi(z))|^{p}\nu(z)\,dA(z)+I\\\
&\leq\frac{1}{r}(\|\sigma C_{\varphi}f\|^{p}_{L^{p}_{\nu}}+\|\sigma
C_{\psi}f\|^{p}_{L^{p}_{\nu}})+I,\quad f\in A^{p}_{\omega},\end{split}$ (3.2)

where $I=\int_{\mathbb{D}\setminus
E}|f(\varphi(z))-f(\psi(z))|^{p}\nu(z)\,dA(z)$. To estimate $I$, for a
measurable function $h$ and an analytic self-map $\varphi$, we define the
pull-back measure $\varphi_{*}(h)(M)=\int_{\varphi^{-1}(M)}h\,dA$ for each
measurable set $M\subseteq\mathbb{D}$. Now, using the [21, Lemma 1] and
Fubini’s theorem, we deduce that for any $r<R<1$

$\begin{split}I&=\int_{\mathbb{D}\setminus
E}|f(\varphi(z))-f(\psi(z))|^{p}\nu(z)\,dA(z)\\\
&\lesssim\int_{\mathbb{D}\setminus
E}\frac{\sigma(z)^{p}}{(1-|\varphi(z)|)^{2}}\int_{\Delta(\varphi(z),R)}|f(\zeta)|^{p}\,dA(\zeta)\nu(z)\,dA(z)\\\
&\leq\int_{\mathbb{D}}\frac{|f(\zeta)|^{p}}{(1-|\zeta|)^{2}}\int_{\varphi^{-1}(\Delta(\zeta,R))}\sigma(z)^{p}\nu(z)\,dA(z)\,dA(\zeta)\\\
&=\int_{\mathbb{D}}|f(\zeta)|^{p}\frac{\varphi_{*}(\sigma^{p}\nu)(\Delta(\zeta,R))}{(1-|\zeta|)^{2}}\,dA(\zeta).\end{split}$
(3.3)

By the definition of the pull-back measure, we see that $\|\sigma
C_{\varphi}f\|_{L^{p}_{\nu}}=\|f\|_{L^{p}_{\varphi_{*}(\sigma^{p}\nu)}}$.
Therefore, for any $a\in\mathbb{D}$, it follows from the hypothesis
$\omega\in\mathcal{D}\subseteq\widehat{\mathcal{D}}$ that

$\int_{S(a)}\frac{\varphi_{*}(\sigma^{p}\nu)(\Delta(\zeta,R))}{(1-|\zeta|)^{2}}\,dA(\zeta)\lesssim\varphi_{*}(\sigma^{p}\nu)(S(a))\lesssim\omega(S(a)),\quad|a|>R.$
(3.4)

Combining (3.3), (3.4) and the Carleson measure characterization for
$A^{p}_{\omega}$ [25, Theorem 3.3], we get

$I\lesssim\sup_{a\in\mathbb{D}}\frac{\varphi_{*}(\sigma^{p}\nu)(S(a))}{\omega(S(a))}{\|f\|^{p}_{A^{p}_{\omega}}}.$

Since $\sigma C_{\varphi}$ and $\sigma C_{\psi}$ are bounded, both
$\varphi_{*}(\sigma^{p}\nu)$ and $\psi_{*}(\sigma^{p}\nu)$ are $p$-Carleson
measure for $A^{p}_{\omega}$. (3.2), the above estimate and [25, Theorem 3.3]
yield

$\|C_{\varphi}-C_{\psi}\|^{p}\lesssim\sup_{a\in\mathbb{D}}\frac{\varphi_{*}(\sigma^{p}\nu)(S(a))}{\omega(S(a))}+\sup_{a\in\mathbb{D}}\frac{\psi_{*}(\sigma^{p}\nu)(S(a))}{\omega(S(a))}.$
(3.5)

Then the upper bound can be acquired from (3.6), which will be proved a bit
later.

Conversely, suppose $C_{\varphi}-C_{\psi}$ is bounded. For any
$a\in\mathbb{D}$, consider the function

$f_{a}(z)=\left(\frac{1-|a|}{1-\overline{a}z}\right)^{\frac{\gamma+1}{p}}\omega(S(a))^{-\frac{1}{p}},\quad
z\in\mathbb{D}.$

It follows from [25, Lemma 2.1] that there exists
$\gamma_{0}=\gamma_{0}(\omega)$ such that for all $\gamma>\gamma_{0}$,
$\|f_{a}\|_{A^{p}_{\omega}}\asymp 1$. A standard calculation based on [25,
Theorem 3.3] shows that if a positive measure $\mu$ is a p-Carleson measure
for $A^{p}_{\omega}$, then for those $\gamma$ the identity operator
$Id:A^{p}_{\omega}\to L^{p}_{\mu}$ satisfies

$\|I_{d}\|^{p}_{A^{p}_{\omega}\to
L^{p}_{\mu}}\asymp\sup_{a\in\mathbb{D}}\|f_{a}\|^{p}_{L^{p}_{\mu}}.$ (3.6)

Now, fix a $\gamma>\gamma_{0}$ and an $r\in(0,1)$. Then choose such $N=N(r)$
and $t_{N}$ appearing in Lemma 10, we have $\|f_{t_{N}a}\|\asymp 1$. When
$|a|>1-\frac{1}{2N}$, the hypothesis and Lemma 10 yield

$\begin{split}&\quad\|C_{\varphi}-C_{\psi}\|^{p}\gtrsim\|(C_{\varphi}-C_{\psi})f_{a}\|_{L^{p}_{\nu}}^{p}+\|(C_{\varphi}-C_{\psi})f_{t_{N}a}\|_{L^{p}_{\nu}}^{p}\\\
&=\int_{\mathbb{D}}\left(\left|\left(\frac{1-|a|}{1-\overline{a}\varphi(z)}\right)^{\frac{\gamma+1}{p}}-\left(\frac{1-|a|}{1-\overline{a}\psi(z)}\right)^{\frac{\gamma+1}{p}}\right|^{p}\right.\\\
&\quad\left.+\left|\left(\frac{1-|a|}{1-t_{N}\overline{a}\varphi(z)}\right)^{\frac{\gamma+1}{p}}-\left(\frac{1-|a|}{1-t_{N}\overline{a}\psi(z)}\right)^{\frac{\gamma+1}{p}}\right|^{p}\right)\frac{\nu(z)}{\omega(S(a))}\,dA(z)\\\
&\gtrsim\int_{\varphi^{-1}(\Delta(a,r))}\left(\left|\left(\frac{1-|a|}{1-\overline{a}\varphi(z)}\right)^{\frac{\gamma+1}{p}}-\left(\frac{1-|a|}{1-\overline{a}\psi(z)}\right)^{\frac{\gamma+1}{p}}\right|\right.\\\
&\quad\left.+\left|\left(\frac{1-|a|}{1-t_{N}\overline{a}\varphi(z)}\right)^{\frac{\gamma+1}{p}}-\left(\frac{1-|a|}{1-t_{N}\overline{a}\psi(z)}\right)^{\frac{\gamma+1}{p}}\right|\right)^{p}\frac{\nu(z)}{\omega(S(a))}\,dA(z)\\\
&\gtrsim\frac{(1-|a|)^{\gamma+1}}{\omega(S(a))}\int_{\varphi^{-1}(\Delta(a,r))}\frac{\sigma(z)^{p}}{|1-\overline{a}\varphi(z)|^{\gamma+1}}\nu(z)\,dA(z)\asymp\frac{\varphi_{*}(\sigma^{p}\nu)(\Delta(a,r))}{\omega(S(a))}.\end{split}$
(3.7)

The same step shows that for $\psi$, we have

$\|C_{\varphi}-C_{\psi}\|^{p}\gtrsim\frac{\psi_{*}(\sigma^{p}\nu)(\Delta(a,r))}{\omega(S(a))}.$
(3.8)

When $|a|\leq 1-\frac{1}{2N}$, then first $\omega(S(a))$ is uniformly bounded
and

$\sigma(z)=\frac{|\varphi(z)-\psi(z)|}{|1-\overline{\varphi(z)}\psi(z)|}\lesssim|\varphi(z)-\psi(z)|,\quad\varphi(z)\in\Delta(a,r)~{}\text{or}~{}\psi(z)\in\Delta(a,r).$

Therefore, we deduce

$\begin{split}\frac{\varphi_{*}(\sigma^{p}\nu)(\Delta(a,r))}{\omega(S(a))}&=\frac{1}{\omega(S(a))}\int_{\varphi^{-1}(\Delta(a,r))}\left|\frac{\varphi(z)-\psi(z)}{1-\overline{\varphi(z)}\psi(z)}\right|^{p}\nu(z)\,dA(z)\\\
&\lesssim\int_{\mathbb{D}}|\varphi(z)-\psi(z)|^{p}\nu(z)\,dA(z)\lesssim\|C_{\varphi}-C_{\psi}\|^{p}.\end{split}$
(3.9)

Similarly,

$\frac{\psi_{*}(\sigma^{p}\nu)(\Delta(a,r))}{\omega(S(a))}\lesssim\|C_{\varphi}-C_{\psi}\|^{p}.$
(3.10)

Combining (3.7), (3.8), (3.9), and (3.10), we see that for any
$a\in\mathbb{D}$,

$\|C_{\varphi}-C_{\psi}\|^{p}\gtrsim\frac{\varphi_{*}(\sigma^{p}\nu)(\Delta(a,r))}{\omega(S(a))}+\frac{\psi_{*}(\sigma^{p}\nu)(\Delta(a,r))}{\omega(S(a))}.$
(3.11)

Now, since $\omega\in\mathcal{D}$, we see that there exist $r=r(\omega)$ such
that for any $z\in\mathbb{D}$, $\omega(S(z))\asymp\omega(\Delta(z,r))$. This
together with (1.9) yields a positive $\mu$ is a Carleson measure for
$A^{p}_{\omega}$ if and only if

$\sup_{z\in\mathbb{D}}\frac{\mu(\Delta(z,r))}{\omega(\Delta(z,r))}<\infty$

and moreover

$\|Id\|^{p}_{A^{p}_{\omega}\to
L^{p}_{\mu}}\asymp\sup_{z\in\mathbb{D}}\frac{\mu(\Delta(z,r))}{\omega(\Delta(z,r))},$

which combines with (3.11) and (3.6) complete the proof. ∎

_Proof of_ Theorem 2. By the definition of the eventual continuity and a
direct application of Proposition 11, we are in a position to get the result.
$\Box$

We underline here that the proof of the sufficiency only requires the
hypothesis that $\omega\in\widehat{\mathcal{D}}$ instead of
$\omega\in\mathcal{D}$, which means that any semigroup $(\varphi_{t})_{t\geq
0}$ satisfying (1.8) induces the eventually continuous composition semigroup
on $A^{p}_{\omega}$, provided $\omega\in\widehat{\mathcal{D}}$. This fact
guarantees the existence of such $(\varphi_{t})_{t\geq 0}$ and the amount is
huge.

######  Corollary 12.

Let $1\leq p<\infty$, $\omega\in\widehat{\mathcal{D}}$. Suppose
$(\varphi_{t})_{t\geq 0}$ is a semigroup of analytic self-maps of $\mathbb{D}$
and $(C_{t})_{t\geq 0}$ is the corresponding composition semigroup on
$A^{p}_{\omega}$. If there exists a $t_{0}$ such that
$\|\varphi_{t_{0}}\|_{H^{\infty}}<1$, then $(C_{t})_{t\geq 0}$ is eventually
norm continuous on $A^{p}_{\omega}$.

###### Proof.

By the hypothesis and the property of semigroup $(\varphi_{t})_{t\geq 0}$, we
see that $\|\varphi_{t}\|_{H^{\infty}}<1$ for all $t>t_{t_{0}}$. Therefore, we
deduce

$\lim_{s\to t}\delta(\varphi_{s}(z),\varphi_{t}(z))=\lim_{s\to
t}\left|\frac{\varphi_{s}(z)-\varphi_{t}(z)}{1-\overline{\varphi_{s}(z)}\varphi_{t}(z)}\right|=0,\quad
z\in\mathbb{D},~{}~{}~{}s,t>t_{0}.$

Then the application of Theorem 2 (sufficiency holds for the case
$\omega\in\widehat{\mathcal{D}}$) and the dominant convergence theorem verify
the result, as required. ∎

Another regularity property of a strongly continuous semigroup we are
considering now is the so-called eventual compactness. Recall that the
composition semigroup $(C_{t})_{t\geq 0}$ induced by $(\varphi_{t})_{t\geq 0}$
is eventually compact on a Banach space $X$ if there exists a $t_{0}>0$ such
that for all $t>t_{0}$, $C_{\varphi_{t}}$ is compact on $X$. It follows from
[13, Lemma 5] that an eventually compact semigroup is always eventually norm
continuous. We have the following description for the eventual compactness of
composition semigroup on $A^{p}_{\omega}$, provided
$\omega\in\widehat{\mathcal{D}}$.

######  Theorem 13.

Let $1\leq p<\infty$, $\omega\in\widehat{\mathcal{D}}$. Suppose
$(\varphi_{t})_{t\geq 0}$ is a semigroup of analytic self-maps of $\mathbb{D}$
and $(C_{t})_{t\geq 0}$ is the corresponding composition semigroup on
$A^{p}_{\omega}$. Then $(C_{t})_{t\geq 0}$ is eventually compact on
$A^{p}_{\omega}$ if only if there exists a $t_{0}>0$ such that for all
$t>t_{0}$

$\limsup_{|z|\rightarrow
1^{-}}\frac{\omega^{\star}(z)}{\omega^{\star}(\varphi_{t}(z))}=0.$

###### Proof.

Since each $\varphi_{t}$ is univalent on $\mathbb{D}$, a direct application of
the combination of [28, Corollary 21] and [28, Lemma 23] completes the proof.
∎

One has seen that the eventual compactness of an operator semigroup implies
its eventual norm continuity, while the converse does not hold anymore.
Indeed, it turns out that the eventual compactness of an operator semigroup is
equivalent to its eventual norm continuity and additionally, the compactness
of its generator, see [13, Lemma 5.11] for the proof. In general, if the
generator of a strongly continuous operator semigroup has a compact resolvent,
then this semigroup has some very nice properties. For instance, the spectrum
of the generator only consists point spectrum. In particular, for the
composition semigroup on weighted Bergman spaces with doubling weights, we can
also characterize the compactness of the resolvent of its generator, which
will be presented in the next section through studying a certain integral
operator.

##  4\. A novel integral operator on $A^{p}_{\omega}$

For a strongly continuous composition semigroup $(C_{t})_{t\geq 0}$ on a
Banach space $X$ with the generator $\Gamma$, and for any
$\lambda\in\rho(\Gamma)$, the resolvent $R(\lambda,\Gamma)$ of $\Gamma$ can be
generally represented as

$R(\lambda,\Gamma)(f)=\int_{0}^{\infty}e^{-\lambda t}C_{t}(f)\,dt,\quad f\in
X.$

However, if the Denjoy-Wolff point of the induced semigroup
$(\varphi_{t})_{t\geq 0}$ point is 0, then $R(\lambda,\Gamma)$ has a concrete
representation:

######  Lemma 14.

Let $1\leq p<\infty$, $\omega\in\widehat{\mathcal{D}}$, and let
$(\varphi_{t})_{t\geq 0}$ be a non-trivial semigroup of self-maps on
$\mathbb{D}$ with Denjoy-Wolff point 0, infinitesimal generator $G$ and
Koenigs function $h$. Suppose $(C_{t})_{t\geq 0}$ is the corresponding
composition semigroup on $A^{p}_{\omega}$, with the generator $\Gamma$. Then
for all $\lambda\in\rho(\Gamma)$ the resolvent operator of $\Gamma$ has the
following representation:

$R(\lambda,\Gamma)f(z)=-\frac{1}{G^{\prime}(0)}\frac{1}{(h(z))^{\frac{\lambda}{-G^{\prime}(0)}}}\int_{0}^{z}f(\zeta)(h(\zeta))^{\frac{\lambda}{-G^{\prime}(0)}-1}h^{\prime}(\zeta)d\zeta.$

In particular, $-G^{\prime}(0)$ belongs to $\rho(\Gamma)$ and hence

$R(-G^{\prime}(0),\Gamma)f(z)=-\frac{1}{G^{\prime}(0)h(z)}\int_{0}^{z}f(\zeta)h^{\prime}(\zeta)d\zeta.$
(4.1)

###### Proof.

Let

$R:=-\frac{1}{G^{\prime}(0)}\frac{1}{(h(z))^{\frac{\lambda}{-G^{\prime}(0)}}}\int_{0}^{z}f(\zeta)(h(\zeta))^{\frac{\lambda}{-G^{\prime}(0)}-1}h^{\prime}(\zeta)d\zeta.$

It is elementary to compute that $(\lambda I-\Gamma)R=R(\lambda I-\Gamma)=I$,
which shows that $R$ is the resolvent operator of $\Gamma$. Since the Denjoy-
Wolff point of $(\varphi_{t})_{t\geq 0}$ is 0, it is easy to see that
$\text{Re}\,(-G^{\prime}(0))\geq 0$. If $\text{Re}\,(-G^{\prime}(0))>0$, since
$C_{\varphi}$ is uniformly bounded by [28, Theorem 15 and (4.7)], we have

$\omega_{0}:=\lim_{t\rightarrow\infty}\frac{\log\|C_{t}\|}{t}=0.$

So $-G^{\prime}(0)\in\rho(\Gamma)$ by [24, Theorem 1.10, p42]. If
$-G^{\prime}(0)$ is a pure imaginary number, write $G(z)=-i\alpha z$, where
$\alpha\in\mathbb{R}\setminus\\{0\\}$ and $\Gamma f=-i\alpha zf^{\prime}(z)$.
In this case, $(i\alpha-\Gamma)(f)=g$ has the unique analytic solution

$f(z)=\frac{1}{i\alpha z}\int_{0}^{z}g(\zeta)d\zeta.$

It is not difficult to see that the operator

$g\to\frac{1}{i\alpha z}\int_{0}^{z}g(\zeta)d\zeta$

is bounded on $A^{p}_{\omega}$. Hence $-G^{\prime}(0)$ belongs to
$\rho(\Gamma)$. The proof is complete. ∎

Let us return to $R(-G^{\prime}(0),\Gamma)$ defined in (4.1) for a moment. In
this definition, the function $h$ is a certain spiral-like function with
$h(0)=0$, as the Koenigs function of the semigroup $h$, and hence $h$ is
certainly univalent on $\mathbb{D}$. Since the resolvent is always bounded,
such $R(-G^{\prime}(0),\Gamma)$ is bounded on $A^{p}_{\omega}$. Moreover, its
compactness plays an essential role in characterizing the compactness of the
general $R(\lambda,\Gamma)$. Nevertheless, such an operator can be well-
defined for any univalent function $h$ with $h(0)=0$. Therefore, it seems that
it is even more natural to characterize bounded and compact $R_{h}$ on
$A^{p}_{\omega}$ with $\omega\in\widehat{\mathcal{D}}$ for the full range
$0<p<\infty$, where $R_{h}$ is defined as

$R_{h}(f)(z)=\frac{1}{h(z)}\int_{0}^{z}f(\zeta)h^{\prime}(\zeta)\,d\zeta,\quad
f\in\mathcal{H}(\mathbb{D}),$

provided $h$ is univalent function on $\mathbb{D}$ with $h(0)=0$.

Before presenting the result for the above question, we first recall that for
$\alpha\geq 1$ and $\omega\in\widehat{\mathcal{D}}$,
$\mathcal{C}^{\alpha}(\omega^{\star})$ consists of all
$f\in\mathcal{H}(\mathbb{D})$ such that

$\|f\|^{2}_{\mathcal{C}^{\alpha}(\omega^{\star})}=|g(0)|^{2}+\sup_{I\in\mathbb{T}}\frac{\int_{S(I)}|f^{\prime}(z)|^{2}\omega^{\star}(z)dA(z)}{(\omega(S(I)))^{\alpha}}<\infty;$

$\mathcal{C}_{0}^{\alpha}(\omega^{\star})$ consists of all
$f\in\mathcal{H}(\mathbb{D})$ such that

$\lim_{|I|\rightarrow
0}\frac{\int_{S(I)}|f^{\prime}(z)|^{2}\omega^{\star}(z)dA(z)}{(\omega(S(I)))^{\alpha}}=0.$

If $\alpha=1$, then by [26] the relation among these spaces, $\mathcal{B}$
($\mathcal{B}_{0}$) and $BMOA$ ($\mathord{\rm VMOA}$) is the following:

$\mathord{\rm
BMOA}\subset\mathcal{C}^{1}(\omega^{\star})\subset\mathcal{B},\quad\mathord{\rm
VMOA}\subset\mathcal{C}_{0}^{1}(\omega^{\star})\subset\mathcal{B}_{0}.$ (4.2)

Our conclusion can be stated as follows, which contains the results of Theorem
3.

######  Theorem 15.

Let $0<p<\infty$ and $\omega\in\widehat{\mathcal{D}}$. Suppose $h$ is a
univalent function on $\mathbb{D}$ satisfying $h(0)=0$. Then $R_{h}$ is
automatically bounded on $A^{p}_{\omega}$. Moreover, the following statements
are equivalent:

  * (i)

$R_{h}$ is compact on $A^{p}_{\omega}$;

  * (ii)

$\log\frac{h(z)}{z}\in\mathcal{C}^{1}_{0}(\omega^{\star})$;

  * (iii)

$\log\frac{h(z)}{z}\in\mathcal{B}_{0}$;

  * (iv)

$\log\frac{h(z)}{z}\in\mathord{\rm VMOA}$.

###### Proof.

Since $h$ is univalent with $h(0)=0$ by the hypothesis, the equivalence among
(ii), (iii) and (iv) is obvious by the relation (4.2) and a result in [30].
Therefore, it suffices to verify that $R_{h}$ is always bounded and (i) and
(ii) are equivalent. To this end, we are going to use a method applied in the
proof of [36, Theorem 6.1], which points out that $R_{h}$ can be decomposed as
follows.

$M_{z}P_{h}=R_{h}M_{z},\quad Q_{h}=P_{h}+Q_{h}P_{h},$ (4.3)

where

$M_{z}f(z)=zf(z),$ $P_{h}f(z)=\frac{1}{zh(z)}\int_{0}^{z}f(\zeta)\zeta
h^{\prime}(\zeta)d\zeta,$

and

$Q_{h}f(z)=\frac{1}{z}\int_{0}^{z}f(\zeta)\frac{\zeta
h^{\prime}(\zeta)}{h(\zeta)}d\zeta.$

Moreover, it is elementary to see that

$Q_{h}f(z)=J(f)(z)+L_{h}M_{z}(f)(z),$ (4.4)

where $J(f)(z)=\frac{1}{z}\int_{0}^{z}f(\zeta)d\zeta$ and

$L_{h}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\zeta)\left(\log\frac{h(\zeta)}{\zeta}\right)^{\prime}d\zeta.$
(4.5)

Now, since $h$ is univalent with $h(0)=0$, a basic calculation based on the
well-known inequality for a univalent function [11, Theorem 2.7] shows that
$\left(\log\frac{h(z)}{z}\right)^{\prime}\lesssim\frac{1}{1-|z|}+1$ and hence
$\log\frac{h(z)}{z}\in\mathcal{B}$ and then certainly belongs to
$\mathcal{C}^{1}(\omega^{\star})$. Then applying the [25, Theorem 6.8 (i)], we
immediately see that $L_{h}$ is bounded on $A^{p}_{\omega}$. Moreover,
trivially, $J$ is also bounded on $A^{p}_{\omega}$. Thus, $Q_{h}$ is bounded
on $A^{p}_{\omega}$ by (4.4), and so are $P_{h}$ and $R_{h}$ by the identities
(4.3).

Similarly, to prove that (i) and (ii) are equivalent, we only need to notice
that $L_{h}$ in (4.5) is compact if and only if
$\log\frac{h(z)}{z}\in\mathcal{C}^{1}_{0}(\omega^{\star})$. Then since $J$ is
compact on $A^{p}_{\omega}$, $L_{h}$ is compact if and only if $Q_{h}$ is
compact by (4.4). Likewise, by (4.3), $Q_{h}$ is compact if and only if
$P_{h}$ is compact if and only if $R_{h}$ is compact. Combining those facts,
we are in a position to finish the proof. ∎

As a corollary of the above theorem, we can get some characterizations for
compact resolvent of composition semigroup action on $A^{p}_{\omega}$.

######  Corollary 16.

Let $1\leq p<\infty$ and $\omega\in\widehat{\mathcal{D}}$. Suppose
$(\varphi_{t})_{t\geq 0}$ is a semigroup of analytic self-maps of $\mathbb{D}$
with Denjoy-Wolff point $0$, infinitesimal generator $G$, and associated
Koenigs function $h$. Denote by $\Gamma$ the infinitesimal generator if the
corresponding composition semigroup $(C_{t})_{t\geq 0}$ on $A^{p}_{\omega}$
and denote by $R(\lambda,\Gamma)$ the resolvent operator for
$\lambda\in\rho(\Gamma)$. Then the following statements are equivalent:

  * (i)

$R(\lambda,\Gamma)$ is compact on $A^{p}_{\omega}$;

  * (ii)

$\log\frac{h(z)}{z}\in\mathcal{C}^{1}_{0}(\omega^{\star})$;

  * (iii)

$\log\frac{h(z)}{z}\in\mathcal{B}_{0}$;

  * (iv)

$\log\frac{h(z)}{z}\in\mathord{\rm VMOA}$;

  * (v)

For any $\zeta\in\partial\mathbb{D}$,

$\lim_{z\to\zeta}\left|\frac{G(z)}{z-\zeta}\right|=\infty.$

###### Proof.

The well-known resolvent equation

$R(\lambda,\Gamma)-R(\mu,\Gamma)=(\mu-\lambda)R(\lambda,\Gamma)R(\mu,\Gamma)\quad\lambda,\mu\in\rho(\Gamma)$

shows that $R(\lambda,\Gamma)$ is compact on $A^{p}_{\omega}$ for all
$\lambda\in\rho(\Gamma)$ if and only if it is compact for a certain
$\lambda_{0}\in\rho(\Gamma)$. So, by Lemma 14, to prove the compactness of
$R(\lambda,\Gamma)$, it suffices to identify the compactness of
$R(-G^{\prime}(0),\Gamma)$, or equivalently, the compactness of $R_{h}$
defined as before. Therefore, a direct application of Theorem 15 gives the
equivalence of (i), (ii), (iii) and (iv). In addition, the equivalence of (v)
and (iii) follows from [36, Theorem 6.1]. The proof is complete. ∎

Another attempt for $R_{h}$ is to characterize its spectrum on
$A^{p}_{\omega}$ for any univalent function $h$ with $h(0)=0$. This question
is generally challenging for those $h$, but it seems promising for some
special $h$ due to the spectral mapping theorem for the resolvent if $h$ is a
Koenigs function of some semigroups.

First, recall that a univalent function
$h:\mathbb{D}\rightarrow\Omega:=h(\mathbb{D})$ is said to be $\mu$-spiral-like
if there is a number $\mu\in\mathbb{C}$ with $\text{Re}\mu>0$ such that for
each $\omega\in\Omega$ and $t\geq 0$ the point $e^{-t\mu}\omega$ also belongs
to $\Omega$. See more information about spiral-like functions in [32]. Then
the class $\varphi_{t}(z):=h^{-1}(e^{-\mu t}h(z))$ becomes a one-parameter
semigroup with the Koneigs function $h$. Moreover, the Denjoy-Wolff point of
such $(\varphi_{t})_{t\geq 0}$ is 0 and infinitesimal generator
$G=-\mu\frac{h(z)}{h^{\prime}(z)}$ with $G^{\prime}(0)=-\mu$. Therefore, by
Lemma 14, $\mu$ is in the resolvent set of $\Gamma$, the generator of the
corresponding $(C_{t})_{t\geq 0}$, and the resolvent $R(\mu,\Gamma)$ can be
written as

$R(\mu,\Gamma)=\frac{1}{\mu
h(z)}\int_{0}^{z}f(\zeta)h^{\prime}(\zeta)\,d\zeta=\frac{1}{\mu}R_{h},\quad
f\in A^{p}_{\omega}.$ (4.6)

Next, the following spectral mapping theorem for the resolvent (see [13,
p.161]) asserts that

$\sigma_{A^{p}_{\omega}}(R(\mu,\Gamma))\setminus\\{0\\}=(\mu-\sigma_{A^{p}_{\omega}}(\Gamma))^{-1}:=\left\\{\frac{1}{\mu-\lambda}:\lambda\in\sigma_{A^{p}_{\omega}}(\Gamma)\right\\}$
(4.7)

for each $\lambda\in\rho(\Gamma)$. Combining those facts, we may get the
following result:

######  Proposition 17.

Let $1\leq p<\infty$, $\Re{\mu}>0$ and $\omega\in\widehat{\mathcal{D}}$.
Suppose $h$ is a $\mu$-spiral-like function on $\mathbb{D}$ satisfying
$h(0)=0$. Then the following statements hold:

  * (i)

If $\log\frac{h(z)}{z}\in\mathcal{B}_{0}$, then

$\sigma_{A^{p}_{\omega}}(R_{h})=\left\\{\frac{1}{k+1},~{}h^{k}\in
A^{p}_{\omega},~{}k=0,1,2,\cdots\right\\}\cup\\{0\\};$

  * (ii)

For any $t>0$, $\sigma_{A^{2}_{\omega}}(R_{h})$ satisfies the following
identity:

$\left\\{e^{t\left(\mu-\frac{\mu}{\sigma_{A^{2}_{\omega}}(R_{h})}\right)}:\lambda\in\sigma(\Gamma)\right\\}\cup\\{0\\}=\\{\lambda\in\mathbb{C}:|\lambda|\leq
r_{e,A^{2}_{\omega}}(C_{\varphi_{t}})\\}\cup\\{e^{-k\mu
t}:k=0,1,2,3\ldots\\},$

where $\varphi_{t}(z):=h^{-1}(e^{-\mu t}h(z))$ for all $z\in\mathbb{D}$ and
$\Gamma$ is the generator of the corresponding composition semigroup.

###### Proof.

(i) By the above explanation, we see that $\varphi_{t}(z)=h^{-1}(e^{-\mu
t}h(z))$ is a one-parameter semigroup of the self-maps with the Denjoy-Wolff
point 0. Let $\Gamma$ be the generator of the corresponding composition
semigroup on $A^{p}_{\omega}$. Since $\log\frac{h(z)}{z}\in\mathcal{B}_{0}$,
by Corollary 16, we see that $R(\mu,\Gamma)$ is compact on $A^{p}_{\omega}$
and hence its spectrum only consists of point spectrum. Therefore, these facts
together with Theorem 1 and (4.7) yield

$\sigma_{A^{p}_{\omega}}(R(\mu,\Gamma))\setminus\\{0\\}=\left\\{\frac{1}{\mu-
kG^{\prime}(0)}:~{}h^{k}\in
A^{p}_{\omega}\right\\}=\left\\{\frac{1}{(k+1)\mu}:~{}h^{k}\in
A^{p}_{\omega}\right\\}.$ (4.8)

In addition, (4.6) yields $R_{h}=\mu R(\mu,\Gamma)$ and hence
$\sigma_{A^{p}_{\omega}(R_{h})}=\mu\sigma_{A^{p}_{\omega}}(R(\mu,\Gamma))$ and
we have $0\in\sigma_{A^{p}_{\omega}(R_{h})}$ since $R_{h}(f)(0)=0$ for any
$f\in\mathcal{H}(\mathbb{D})$, which combines with (4.8) concludes the result.

(ii) Since we focus on the Hilbert space $A^{2}_{\omega}$, the spectral
mapping theorem holds automatically. Then the combined application of Theorem
1, (4.6) and (4.7) completes the proof. ∎

We finish the section and the paper by presenting several examples, while
other constructions certainly can be provided.

######  Example 1.

Let $h(z)=z$. Then $h$ is a $c$-spiral-like function for any $c$ with $\Re
c>0$. By Proposition 17, we get

$\sigma_{A^{p}_{\omega}}(R_{z})=\left\\{\frac{1}{k+1},k=0,1,2,\cdots\right\\}\cup\\{0\\}.$

######  Example 2.

Let $h(z)=\log\frac{1}{1-z}$. Then $h$ is a $1$-spiral-like function. It
induces the one-parameter semigroup $(\varphi_{t})_{t\geq 0}$ with
$\varphi_{t}(z)=1-(1-z)e^{-t}$. By Proposition 17, we get

$\sigma_{A^{p}_{\omega}}(R_{z})=\left\\{\frac{1}{k+1},\left(\log\frac{1}{1-z}\right)^{k}\in
A^{p}_{\omega},k=0,1,2,\cdots\right\\}\cup\\{0\\}.$

######  Example 3.

Let $h(z)=\log\frac{1+z}{1-z}$. Then $h$ is a $1$-spiral-like function. It
induces the one-parameter semigroup $(\varphi_{t})_{t\geq 0}$ with
$\varphi_{t}(z)=\frac{(1+z)^{e^{-t}}-(1-z)^{e^{-t}}}{(1+z)^{e^{-t}}+(1-z)^{e^{-t}}}$.
We have

$\sigma_{A^{p}_{\omega}}(R_{z})=\left\\{\frac{1}{k+1},\left(\log\frac{1+z}{1-z}\right)^{k}\in
A^{p}_{\omega},k=0,1,2,\cdots\right\\}\cup\\{0\\}.$

Acknowledgements The authors would like to express their gratitude to
Professor A. G. Siskakis for his valuable suggestion on Theorem 1, which
provides a nice road to further study of Theorem 1.

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