# Generators of $C_{0}$-semigroups
of weighted composition operators

Eva A. Gallardo-Gutiérrez  E. A. Gallardo-Gutiérrez
Departamento de Análisis Matemático y
Matemática Aplicada
Facultad de CC. Matemáticas,
Universidad Complutense de Madrid,
Plaza de Ciencias N${}^{\underline{\Tiny o}}$ 3, 28040 Madrid, Spain
and Instituto de Ciencias Matemáticas ICMAT(CSIC-UAM-UC3M-UCM),
Madrid, Spain<EMAIL_ADDRESS>,  Aristomenis G. Siskakis  A. G.
Siskakis
Department of Mathematics
Aristotle University of Thessaloniki
54124 Thessalonıki, Greece<EMAIL_ADDRESS>and  Dmitry Yakubovich  D.
V. Yakubovich
Departamento de Matemáticas,
Universidad Autónoma de Madrid,
Cantoblanco, 28049 Madrid, Spain
and Instituto de Ciencias Matemáticas ICMAT (CSIC-UAM-UC3M-UCM),
Madrid, Spain<EMAIL_ADDRESS>
###### Abstract.

We prove that in a large class of Banach spaces of analytic functions in the
unit disc $\mathbb{D}$ an (unbounded) operator $Af=G\cdot f^{\prime}+g\cdot f$
with $G,\,g$ analytic in $\mathbb{D}$ generates a $C_{0}$-semigroup of
weighted composition operators if and only if it generates a
$C_{0}$-semigroup. Particular instances of such spaces are the classical Hardy
spaces. Our result generalizes previous results in this context and it is
related to cocycles of flows of analytic functions on Banach spaces. Likewise,
for a large class of non-separable Banach spaces $X$ of analytic functions in
$\mathbb{D}$ contained in the Bloch space, we prove that no non-trivial
holomorphic flow induces a $C_{0}$-semigroup of weighted composition operators
on $X$. This generalizes previous results in [6] and [1] regarding
$C_{0}$-semigroup of (unweighted) composition operators.

###### Key words and phrases:

Weighted composition operators, $C_{0}$-semigroups, Cocycles of Holomorphic
Flows

###### 2010 Mathematics Subject Classification:

47B33 (primary), 30C45 (secondary)

First author is partially supported by Plan Nacional I+D grant no.
PID2019-105979GB-I00, Spain

Third author is partially supported by Spanish Ministry of Science, Innovation
and Universities (grant no. PGC2018-099124-B-I00).
The first and the third author also acknowledge support from “Severo Ochoa
Programme for Centres of Excellence in R&D” (CEX2019-000904-S) of the Ministry
of Economy and Competitiveness of Spain and by the European Regional
Development Fund and from the Spanish National Research Council, through the
“Ayuda extraordinaria a Centros de Excelencia Severo Ochoa” (20205CEX001).

##  1\. Introduction

A natural question in the study of semigroups of operators is to characterize
their generators, whenever they are defined. Recently, in [2] and [14]
generators of semigroups of composition operators were characterized in the
context of a large variety of Banach spaces of analytic functions in the unit
disc $\mathbb{D}$. In particular, if an (unbounded) operator $Af=G\cdot
f^{\prime}$ on the classical Hardy space $H^{p}$, $1\leq p<\infty$ generates a
$C_{0}$-semigroup, this is a semigroup of composition operators. See also the
previous works [3], [4] or [9].

The study of semigroups of composition operators on various function spaces of
analytic functions traces back to the pioneering work of Berkson and Porta [8]
in the late seventies. As a particular instance, they gave a characterization
of the generators of semigroups of composition operators acting on the
classical Hardy spaces $H^{p}$ induced by holomorphic flows of analytic self-
maps of the unit disc $\mathbb{D}$ showing, in turns, that these semigroups
are always strongly continuous. Later on, Cowen [11] and Siskakis [26] found
applications of this theory to the study of Cesàro and other averaging
operators. Semigroups of composition operators acting on other holomorphic
function spaces as well as the inducing holomorphic flows have been
extensively studied in the last three decades (we refer to the recent
monograph [7] for more on flows).

Nevertheless, in what refers to semigroups of weighted composition operators
the panorama differs completely. Seminal works in the context of Hardy spaces
were [28] and [22] by the end of the eighties. More recently, in 2012, the
authors of [20] studied which strongly continuous semigroups of operators on
Banach spaces of analytic functions arise from holomorphic flows, where
semigroups of weighted composition operators play an important role.

Our work has two main results, which in some sense, complement each other. The
first one characterizes generators of strongly continuous semigroup (i.e.
$C_{0}$-semigroups) of weighted composition operators acting on a large class
of Banach spaces $X$ of analytic functions in the unit disc $\mathbb{D}$. More
precisely, in Section 3 we will show that whenever the polynomials are dense
in the Banach space $X$, an (unbounded) operator $A$ acting on $X$ like

$Af=G\cdot f^{\prime}+g\cdot f,$

with $G$ and $g$ analytic in $\mathbb{D}$ generates a $C_{0}$-semigroup of
weighted composition operators if and only if it generates a
$C_{0}$-semigroup. In order to prove it, we will make use of some of the ideas
in [14], along with the concept of cocycles of flows of analytic functions on
Banach spaces.

Our second result deals with classes of Banach spaces $X$ where polynomials
are no longer dense, namely, with spaces satisfying

$H^{\infty}\subseteq X\subseteq\mathcal{B},$

where $\mathcal{B}$ denotes the Bloch space and $H^{\infty}$ is the algebra of
bounded analytic functions on $\mathbb{D}$.

In particular, in Section 4 we will prove that in such a case, a nontrivial
holomorphic flow cannot give rise to a non-trivial weighted composition
$C_{0}$-semigroup. This extends previous results in [6] and [1] regarding
composition operators $C_{0}$-semigroups, that is, the unweighted case.

##  2\. Preliminaries

Recall that a one-parameter family $\Phi=\\{\varphi_{t}\\}_{t\geq 0}$ of
analytic self-maps of $\mathbb{D}$ is called a holomorphic flow (or
holomorphic semiflow by some authors) provided that it is a continuous family
that has a semigroup property with respect to composition, namely

  1. 1)

$\varphi_{0}(z)=z$, $\forall z\in{\mathbb{D}}$;

  2. 2)

$\varphi_{t+s}(z)=\varphi_{t}\circ\varphi_{s}(z)$, $\forall t,s\geq
0,\,\forall z\in{\mathbb{D}}$;

  3. 3)

For any $s\geq 0$ and any $z\in{\mathbb{D}}$, $\lim_{t\to
s}\varphi_{t}(z)=\varphi_{s}(z)$,

(see [25], for instance). In the trivial case, $\varphi_{t}(z)=z$ for all
$t\geq 0$. Otherwise, we say that $\Phi$ is nontrivial.

The infinitesimal generator of $\Phi=\\{\varphi_{t}\\}_{t\geq 0}$ is the
function $G$, defined by

$G(z)=\lim_{t\to
0^{+}}\frac{\varphi_{t}(z)-z}{t}=\frac{\partial\varphi_{t}(z)}{\partial
t}\arrowvert_{t=0}.$

This limit exists uniformly on compact subsets of $\mathbb{D}$ and for each
$z\in\mathbb{D}$, the mapping $w:\,t\to\varphi_{t}(z)$ satisfies the
differential equation

$\frac{d}{dt}w(t)=G(w(t))$

$\mbox{ with }w(0)=z$ (see [8]).

Associated to the holomorphic flow $\Phi=\\{\varphi_{t}\\}_{t\geq 0}$ is the
family of composition operators $\\{C_{t}\\}_{t\geq 0}$, defined on the space
of analytic functions on $\mathbb{D}$ by

$C_{t}f=f\circ\varphi_{t}.$

Clearly, $\\{C_{t}\\}_{t\geq 0}$ has the semigroup property:

  1. 1)

$C_{0}=I$ (the identity operator);

  2. 2)

$C_{t}C_{s}=C_{t+s}$ for all $t,s\geq 0$.

Moreover, recall that if an operator semigroup $\\{T_{t}\\}_{t\geq 0}$ acts on
an (abstract) Banach space $X$, then it is called _strongly continuous_ or
_$C_{0}$ -semigroup_, if it satisfies

$\lim_{t\to 0^{+}}T_{t}f=f$

for any $f\in X$. As mentioned in the introduction, Berkson and Porta [8]
proved that every holomorphic flow always induces a strongly continuous
semigroup on the classical Hardy spaces $H^{p}$ and extensions to other spaces
of analytic functions have been extensively studied in the last years.

Given a $C_{0}$-semigroup $\\{T_{t}\\}_{t\geq 0}$ on a Banach space $X$,
recall that its generator is the closed and densely defined linear operator
$A$, defined by

$Af=\lim_{t\to 0^{+}}\frac{T_{t}f-f}{t}$

with domain ${{\mathcal{D}}(A)}=\\{f\in X:\lim_{t\to
0^{+}}\frac{T_{t}f-f}{t}\enspace\text{exists}\\}$. The semigroup is determined
uniquely by its generator.

On the other hand, given a holomorphic flow $\Phi=\\{\varphi_{t}\\}_{t\geq 0}$
in $\mathbb{D}$, a multiplicative _cocycle_ for $\Phi$ is a continuous
complex-valued function $m:[0,+\infty)\times\mathbb{D}\rightarrow\mathbb{C}$
such that

  1. 1)

$m(t,\cdot)$ is analytic on $\mathbb{D}$ for all $t\geq 0$;

  2. 2)

$m(0,z)=1$, $\forall z\in{\mathbb{D}}$;

  3. 3)

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

For the sake of simplicity, we will denote $m_{t}(z)=m(t,z)$ for all $t\geq 0$
and $z\in\mathbb{D}$.

The third equation above is often called the _cocycle identity_ and it implies
that $m_{0}(z)$ is either $1$ or $0$, so the second equation is simply a non-
triviality condition. König [22] investigated weighted holomorphic flows on
the unit disc and provided a characterization of smooth cocycles in the Hardy
space. In addition, he proved that the cocycle identity implies that $m$ is
not vanishing (see [22, Lemma 2.1]).

A particular instance of a cocycle for a holomorphic flow
$\Phi=\\{\varphi_{t}\\}_{t\geq 0}$ is a _coboundary_ , that is, a continuous
complex-valued function $m$ on $[0,+\infty)\times\mathbb{D}$ such that there
exists holomorphic function $\alpha$ on $\mathbb{D}$, non-vanishing except
possibly at the common fixed point of $\varphi_{t}$, satisfying

(2.1) $m(t,z)=\frac{\alpha(\varphi_{t}(z))}{\alpha(z)}\mbox{ for all
}(t,z)\in[0,+\infty)\times\mathbb{D}.$

Cocycles and, in particular, coboundaries arise naturally in the theory of
weighted composition operators semigroups. Observe that given a holomorphic
flow $\Phi=\\{\varphi_{t}\\}_{t\geq 0}$ and a cocycle $m$ for $\Phi$, the
weighted composition operators $\\{W_{t}:t\geq 0\\}$ defined on the space of
analytic functions on $\mathbb{D}$ by

(2.2)
$W_{t}f(z)=m_{t}(z)(C_{t}f)(z)=m_{t}(z)f(\varphi_{t}(z)),\qquad(z\in\mathbb{D})$

form a semigroup. Indeed the converse also holds. Namely, (2.2) defines a
semigroup if and only if $m$ is a cocycle for $\Phi$. In addition, if $m$ is a
coboundary then (2.1) easily yields that

$W_{t}=M_{\frac{1}{\alpha}}C_{t}\,M_{\alpha}$

for every $t\geq 0$, where $M_{\alpha}$ is the operator of multiplication by
$\alpha(z)$. This says that for every $t\geq 0$, $W_{t}$ and $C_{t}$ are
similar as operators on the space of all analytic functions on $\mathbb{D}$.

Throughout the rest of the manuscript, $X$ will be a Banach space of
holomorphic functions on $\mathbb{D}$ and ${\mathcal{L}}(X)$ will denote the
space of bounded linear operators acting on $X$. The space consisting of all
analytic functions on ${\mathbb{D}}$ is denoted by
$\operatorname{Hol}({\mathbb{D}})$ while $\mathcal{O}(\overline{\mathbb{D}})$
stands for the set of all functions, holomorphic on (a neighborhood of) the
closed unit disc $\overline{\mathbb{D}}$ endowed with the usual topologies
(regarding $\mathcal{O}(\overline{\mathbb{D}})$ we refer to [5, pp. 81], for
instance). We will always assume the following natural condition:

$(\star)\hskip
170.71652pt{\mathcal{O}}(\overline{\mathbb{D}})\hookrightarrow{}X\hookrightarrow{}\operatorname{Hol}({\mathbb{D}}),$

being both embeddings continuous.

In order to complete this preliminary section, we reproduce a theorem
essentially contained in [22] for $H^{p}$ spaces (see also [28] for the case
of coboundaries). The proof works as well for Banach spaces $X$ satisfying
$(\star)$.

######  Theorem 2.1.

Let $\\{W_{t}\\}_{t\geq 0}$ be a weighted composition $C_{0}$-semigroup acting
on $X$ defined by (2.2), and let $G$ be the infinitesimal generator of
$\\{\varphi_{t}\\}$. Then the following holds.

  1. (i)

The equation

(2.3) $g(z)=\frac{\partial m_{t}(z)}{\partial t}\arrowvert_{t=0},\qquad\mbox{
for }z\in\mathbb{D}$

defines an analytic function in ${\mathbb{D}}$, and $m_{t}$ is a cocycle for
$\Phi$. Moreover, $m_{t}$ is expressed by

(2.4) $m_{t}(z)=\exp\left(\int_{0}^{t}g(\varphi_{s}(z))ds\right),\qquad t\geq
0,\,z\in\mathbb{D}.$

  2. (ii)

The generator $A$ of the semigroup $\\{W_{t}\\}_{t\geq 0}$ and its domain are
given by

(2.5) $Af=G\cdot f^{\prime}+g\cdot f,\quad{{\mathcal{D}}(A)}=\left\\{f\in
X:\;G\cdot f^{\prime}+g\cdot f\in X\right\\}.$

Roughly speaking, our main result in Section 3 can be seen as the converse of
this theorem.

Finally, we remark that whenever the cocycle $m$ for $\Phi$ is _bounded_ ,
that is, for every $t\geq 0$ the analytic function $m_{t}$ is bounded on
$\mathbb{D}$, the weighted composition operator $W_{t}$ is a bounded operator
on the Hardy spaces $H^{p}$ for every $1\leq p\leq\infty$. In the opposite
direction, if $g$ is analytic with $\operatorname{Re}g(z)<M<\infty$ on
$\mathbb{D}$, then the above formulas induce a strongly continuous semigroup
$\\{W_{t}\\}_{t\geq 0}$.

##  3\. $C_{0}$-semigroups of weighted composition operators: Generators

In this section we will prove that in a large class of Banach spaces $X$ of
analytic functions in the unit disc $\mathbb{D}$, including Hardy spaces
$H^{p}$ $(1\leq p<\infty)$, an (unbounded) operator $A$ defined by (2.5)
generates a $C_{0}$-semigroup of weighted composition operators if and only if
it generates a $C_{0}$-semigroup of operators.

######  Theorem 3.1.

Suppose $X$ is a Banach space of analytic functions on ${\mathbb{D}}$,
satisfying $(\star)$.

Let $\\{S_{t}\\}_{t\geq 0}$ be a $C_{0}$-semigroup on $X$, whose generator $A$
is defined by (2.5), where $G,\,g$ are analytic functions in ${\mathbb{D}}$.
Then there exists a holomorphic flow $\Phi=\\{\varphi_{t}\\}_{t\geq 0}$, whose
infinitesimal generator is $G$ and a cocycle $m$ for $\Phi$, satisfying (2.3)
and (2.4), such that $S_{t}$ is the weighted composition operator

$S_{t}f(z)=m_{t}(z)f(\varphi_{t}(z))$

for every $t\geq 0,z\in\mathbb{D}$ and $f\in X$.

In order to prove the result, as it was mentioned in the Introduction, we will
make use of some ideas in [14]. Note that the natural assumption $(\star)$ on
$X$ was also made in [14].

###### Proof.

Assume the generator $A$ of the $C_{0}$-semigroup $\\{S_{t}\\}_{t\geq 0}$ is
given by $Af=G\cdot f^{\prime}+g\cdot f$ where
$G,\,g\in\operatorname{Hol}({\mathbb{D}})$. We begin by posing the Cauchy
Problem as in [14]

$None$ $\begin{cases}\displaystyle\frac{\partial\varphi_{t}(z)}{\partial
t}=G(\varphi_{t}(z))\\\ \varphi_{0}(z)=z\qquad\qquad\qquad\big{(}z\in
D(0,r)=\\{z\in{\mathbb{D}}:\;|z|<r\\}\big{)},\end{cases}$

where $r\in(0,1)$ is a fixed radius. Hence, upon applying the theory of
ordinary differential equations in complex domain, there exists $t_{0}>0$ and
an analytic solution $\\{\varphi_{t}(z)\\}$ of (CP), defined for $z\in D(0,r)$
and all complex $t$, $|t|<t_{0}$. In particular, for these values of $z$ and
$t$, $|\varphi_{t}(z)|<1$.

Moreover, this solution is unique in the class of smooth functions and has the
semigroup property:

(3.1) $\varphi_{t+s}(z)=\varphi_{t}\circ\varphi_{s}(z)$

whenever $t,s,t+s\in(-t_{0},t_{0})$ and $z,\varphi_{s}(z)\in D(0,r)$.

Choose $r^{\prime}\in(0,r)$ such that $\varphi_{t}(z)\in D(0,r)$ for
$t\in(-t_{0},t_{0})$ and $z\in D(0,r^{\prime})$ (here we may need to replace
$t_{0}$ with a smaller positive number). Then

(3.2) $\varphi_{-t}\circ\varphi_{t}(z)=z\qquad\text{if $t\in(-t_{0},t_{0})$
and $z\in D(0,r^{\prime})$.}$

First, we prove the following:

Claim 1: For any $f\in{{\mathcal{D}}(A)}$

(3.3) $\qquad
S_{t}f(z)=\exp\left(\int_{0}^{t}g(\varphi_{s}(z))ds\right)\,(f\circ\varphi_{t})(z),\qquad
z\in D(0,r),\,0\leq t<t_{0}.$

###### Proof of Claim 1.

By (CP) we have

$\frac{\partial\varphi_{-t}(z)}{\partial t}=-G(\varphi_{-t}(z)),$

for any $z\in D(0,r)$ and any $t\in(0,t_{0})$. Let us fix
$f\in{{\mathcal{D}}(A)}$ and denote by

$f_{t}(z)=S_{t}f(z),\quad z\in{\mathbb{D}},\;t\geq 0.$

Calculating the derivative of $f_{t}\circ\varphi_{-t}(z)$ with respect to $t$
we obtain

$\displaystyle\displaystyle\frac{\partial(f_{t}\circ\varphi_{-t})(z)}{\partial
t}$ $\displaystyle=$ $\displaystyle\displaystyle\frac{\partial f_{t}}{\partial
t}\big{(}\varphi_{-t}(z)\big{)}+\frac{\partial f_{t}}{\partial
z}\big{(}\varphi_{-t}(z)\big{)}\,\frac{\partial\varphi_{-t}(z)}{\partial t}$
$\displaystyle=$
$\displaystyle\big{(}Af_{t}\big{)}\big{(}\varphi_{-t}(z)\big{)}-\frac{\partial
f_{t}}{\partial z}(\varphi_{-t}(z))G(\varphi_{-t}(z))$ $\displaystyle=$
$\displaystyle\displaystyle G(\varphi_{-t}(z))\frac{\partial f_{t}}{\partial
z}(\varphi_{-t}(z))+g(\varphi_{-t}(z))f_{t}(\varphi_{-t}(z))-\frac{\partial
f_{t}}{\partial z}(\varphi_{-t}(z))G(\varphi_{-t}(z)),$

where the second line follows since $A$ is the generator of the semigroup, and
hence $\displaystyle\frac{\partial f_{t}}{\partial t}=Af_{t}$. Accordingly,

(3.4) $\frac{\partial(f_{t}\circ\varphi_{-t})(z)}{\partial
t}=g(\varphi_{-t}(z))f_{t}(\varphi_{-t}(z)),\quad z\in D(0,r),\;0\leq
t<t_{0}.$

For $z\in D(0,r)$, let us write $Y(t)=(f_{t}\circ\varphi_{-t})(z)$ and
$\psi(t)=g(\varphi_{-t}(z))$. Then (3.4) is the differential equation

$Y^{\prime}(t)=\psi(t)\,Y(t),$

so

$Y(t)\,\exp\left(-\int_{0}^{t}\psi(u)\,du\right)=C_{z},$

where $C_{z}$ is a constant (depending on $z$). Thus, for each $z\in D(0,r)$

$(f_{t}\circ\varphi_{-t})(z)=C_{z}\,\exp\left(\int_{0}^{t}g(\varphi_{-u}(z))\,du\right).$

For $t=0$ we have $C_{z}=f_{0}(\varphi_{0}(z))=f(z)$ and therefore

(3.5)
$(f_{t}\circ\varphi_{-t})(z)=\exp\left(\int_{0}^{t}g(\varphi_{-u}(z))\,du\right)\,f(z)$

for $z\in D(0,r)$. Replacing $z$ by $\varphi_{t}(z)$ in (3.5) we deduce by
means of analyticity that

$\displaystyle f_{t}(z)$ $\displaystyle=$
$\displaystyle\exp\left(\int_{0}^{t}g(\varphi_{t-u}(z))\,du\right)\,f(\varphi_{t}(z))$
$\displaystyle=$
$\displaystyle\exp\left(\int_{0}^{t}g(\varphi_{s}(z))\,ds\right)\,f(\varphi_{t}(z)),$

for all $z\in D(0,r)$. This proves Claim 1. ∎

Now, let $f\in X$. There exists a sequence
$\\{f_{n}\\}\subset{{\mathcal{D}}(A)}$ such that $f_{n}\to f$ in $X$ as
$n\to\infty$. Hence, $f_{n}\to f$ uniformly on compact subsets of $\mathbb{D}$
as $n\to\infty$, and therefore

$\lim_{n\to\infty}S_{t}f_{n}(z)=S_{t}f(z)$

for each $z\in\mathbb{D}$ and $t\geq 0$. Accordingly, by Claim 1, for any
$f\in X$

(3.6)
$S_{t}f(z)=\exp\left(\int_{0}^{t}g(\varphi_{s}(z))ds\right)\,(f\circ\varphi_{t})(z),\qquad
z\in D(0,r),\,0\leq t<t_{0}.$

Now, upon applying (3.6) to the constant function $\mathbbm{1}(z)=1$, we
deduce

(3.7)
$S_{t}(\mathbbm{1})(z)=\exp\left(\int_{0}^{t}g(\varphi_{s}(z))ds\right)\,$

for $z\in D(0,r),\,0\leq t<t_{0}.$ Define

(3.8) $m_{t}(z):=S_{t}(\mathbbm{1})(z)\quad z\in{\mathbb{D}},\,t\geq 0.$

Clearly, this definition agrees with (3.7) if $0\leq t<t_{0}$ and $|z|<r$.
Therefore $m_{t}(z)\neq 0$ for $0\leq t<t_{0}$ and $|z|<r$. Moreover,
$m_{t}\in X$ and in particular, $m_{t}$ is analytic on ${\mathbb{D}}$, for any
$t\geq 0$.

Now apply (3.6) to the identity function $\operatorname{id}$,
$\operatorname{id}(z)\equiv z$. We get

$S_{t}(\operatorname{id})(z)=\exp\left(\int_{0}^{t}g(\varphi_{s}(z))ds\right)\,\varphi_{t}(z)\in
X$

for $z\in D(0,r)$ and $0\leq t<t_{0}$. Therefore

(3.9) $\varphi_{t}(z)=\frac{1}{m_{t}(z)}\,S_{t}(\operatorname{id})(z).$

for every $z$, $|z|<r$ and $0\leq t<t_{0}$. Hence for any such $t$,
$\varphi_{t}$ continues to a meromorphic function in $\mathbb{D}$.

Analogously, applying (3.6) to the functions $z^{n}$ for $n\geq 2$, it follows
for every $a\in\mathbb{D}$

(3.10) $(\varphi_{t}(a))^{n}=\frac{1}{m_{t}(z)}\,S_{t}(z^{n})(a),$

for $0\leq t<t_{0}$. If for some $0\leq t<t_{0}$, $\varphi_{t}$ had a pole in
a point $a_{0}\in\mathbb{D}$, then $\varphi_{t}^{n}$ would have at least $n$th
order pole at $a_{0}$ for any $n\geq 1$, which contradicts (3.10). Therefore
$\varphi_{t}$ is analytic in $\mathbb{D}$ for any $t$, $0\leq t<t_{0}$.

Now, a similar argument to that in [14, Claim 2] along with (3.10) yields that
there exists a positive $t_{1}\leq t_{0}$ such that $|\varphi_{t}(z)|<1$ for
all $z\in{\mathbb{D}}$ and all $0\leq t<t_{1}$.

Finally, observe that from (3.1) we get

$\varphi_{s}\circ\varphi_{t}(z)=\varphi_{s+t}(z)$

for $s,t\geq 0$, $s+t<t_{1}$ and all $z\in{\mathbb{D}}$. Since the family
$\\{\varphi_{t}\\}$ satisfies (CP), it follows that it can be continued to a
holomorphic flow, defined on $[0,+\infty)\times{\mathbb{D}}$ (see [25,
Proposition 3.3.1]). Now we can assert that the right hand side of (3.6)
defines a semigroup on the space $\operatorname{Hol}({\mathbb{D}})$. Since
this semigroup coincides with the semigroup $\\{S_{t}\\}$ for $0\leq t<t_{0}$,
if follows that (3.6) holds for all $t>0$ and all $z\in{\mathbb{D}}$. In
particular, by applying (3.6) to the function $\mathbbm{1}$, we get (2.4).

The rest of the proof runs as in [14], which yields the statement of the Main
Theorem. ∎

######  Remark 3.2.

Note that, by hypotheses, the weighted composition operators act boundedly on
$X$. Nevertheless, we point out that even in the Hardy spaces $H^{p}$ it is
unknown how to characterize bounded weighted composition operators in terms of
the inducing symbols (see [15] for related discussions).

######  Remark 3.3.

It is to be noticed that our results have consequences for spaces of analytic
functions on domains other than the disc. More precisely let
$\Omega\subsetneq{\mathbb{C}}$ be a simply connected domain and suppose $X$ is
a Banach space consisting of functions analytic on $\Omega$, which satisfies
the analogous condition $(\star)$ for $\Omega$. Suppose $\\{S_{t}\\}_{t\geq
0}$ is a $C_{0}$-semigroup acting on $X$, with infinitesimal generator of the
form

$Af=G\cdot f^{\prime}+g\cdot f,\quad\mathcal{D}(A)=\\{f\in X:G\cdot
f^{\prime}+g\cdot f\in X\\}$

where $G$ and $g$ are analytic on $\Omega$. Let $h:{\mathbb{D}}\to\Omega$ is a
fixed conformal map onto $\Omega$. Consider the Banach space
$X({\mathbb{D}})=\\{F=f\circ h:f\in X\\}$ with the transferred norm
$\|F\|_{X({\mathbb{D}})}=\|f\|_{X}$ and assume that $X({\mathbb{D}})$
satisfies $(\star)$. The composition operator $C_{h}(f)=f\circ h$ is an onto
isometry between $X$ and $X({\mathbb{D}})$, with $C_{h}^{-1}=C_{h^{-1}}$. It
is then clear that the family of operators $\\{T_{t}\\}_{t\geq 0}$,

$T_{t}=C_{h}\circ S_{t}\circ C_{h^{-1}}$

is a $C_{0}$-semigroup on $X({\mathbb{D}})$. Its infinitesimal generator
$\Gamma$, for $F\in\mathcal{D}(\Gamma)$, is

$\displaystyle\Gamma F$ $\displaystyle=\lim_{t\to
0^{+}}\frac{T_{t}F-F}{t}=\lim_{t\to 0^{+}}\frac{C_{h}\circ S_{t}\circ
C_{h^{-1}}(F)-F}{t}$ $\displaystyle=\lim_{t\to 0^{+}}\frac{S_{t}f\circ
h-f\circ h}{t}=(Af)\circ h,\quad\text{where}\,f=F\circ h^{-1}\in X,$
$\displaystyle=G\circ h\cdot f^{\prime}\circ h+g\circ h\cdot f\circ h$
$\displaystyle=\frac{1}{h^{\prime}}G\circ h\cdot F^{\prime}+g\circ h\cdot F$
$\displaystyle=G_{1}\cdot F^{\prime}+g_{1}\cdot F,$

where $G_{1}(z)=\frac{1}{h^{\prime}(z)}G(h(z))$ and $g_{1}(z)=g(h(z))$ are
analytic in ${\mathbb{D}}$. Thus by Theorem 3.1 there is a holomorphic flow
$\\{\varphi_{t}\\}$ and a cocycle $m$ for $\\{\varphi_{t}\\}$ such that

$T_{t}F(z)=m_{t}(z)F(\varphi_{t}(z)),\quad t\geq 0,\,z\in{\mathbb{D}},\,F\in
X({\mathbb{D}}).$

So we deduce that

$\displaystyle S_{t}(f)(z)$ $\displaystyle=(C_{h^{-1}}\circ T_{t}\circ
C_{h})(f)(z)$ $\displaystyle=m_{t}(h^{-1}(z))(f\circ h\circ\varphi_{t}\circ
h^{-1})(z)$ $\displaystyle=\mu_{t}(z)f(\psi_{t}(z))$

for $f\in X$ and $z\in\Omega$ where $\psi_{t}=h\circ\varphi_{t}\circ h^{-1}$
is a holomorphic flow in $\Omega$ and $\mu_{t}(z)=m_{t}(h^{-1}(z))$ is a
cocycle in $\Omega$ for $\\{\psi_{t}\\}$.

More generally, let $h$ is as before and let $\tau$ be a zero-free holomorphic
function on ${\mathbb{D}}$. Then the same observation applies if $W_{h,\tau}$
is the weighted composition operator $W_{h,\tau}(f)=\tau\cdot(f\circ h)$, and
the space $X({\mathbb{D}})$, defined as $X({\mathbb{D}})=\\{W_{h,\tau}f:\;f\in
X\\}$, satisfies $(\star)$. This remark can be applied, for instance, to Hardy
and to Smirnov spaces of bounded and unbounded domains (see [12] for the
corresponding definitions).

##  4\. Absence of non-trivial $C_{0}$-semigroups of weighted composition
operators in non-separable Banach function spaces

As mentioned in the introduction, in this section we deal with Banach spaces
$X$ of analytic functions on $\mathbb{D}$ such that

(4.1) $H^{\infty}\subseteq X\subseteq\mathcal{B}.$

Recall that the Bloch space $\mathcal{B}$ is a Banach space endowed with the
norm

(4.2)
$\|f\|_{\mathcal{B}}=|f(0)|+\sup_{z\in\mathbb{D}}|f^{\prime}(z)|(1-|z|^{2}),\qquad(f\in\mathcal{B}).$

It will be showed below that such space $X$ is always non-separable (see
Remark 4.4).

In [6], the authors showed that no non-trivial holomorphic flow
$\Phi=\\{\varphi_{t}\\}_{t\geq 0}$ induces a $C_{0}$-semigroup of composition
operators on $H^{\infty}$ or $\mathcal{B}$ by considering an argument
involving the Dunford-Pettis property. More recently, Anderson, Jovovic and
Smith [1] proved that the same holds for the space BMOA as well as for any
Banach space $X$ satisfying (4.1) by means of a geometric function theory
argument. Based on it, we will prove the following

######  Theorem 4.1.

Let $X$ be a Banach space of analytic functions on $\mathbb{D}$ such that
$H^{\infty}\subseteq X\subseteq\mathcal{B}$. Assume that
$\Phi=\\{\varphi_{t}\\}_{t\geq 0}$ is a nontrivial holomorphic flow. Let
$\\{W_{t}\\}_{t\geq 0}$ be any weighted composition semigroup induced by
$\Phi$, namely, suppose there exists a cocycle $m$ for $\Phi$ such that

(4.3) $W_{t}f(z)=m_{t}(z)f(\varphi_{t}(z)),\qquad(z\in\mathbb{D}),$

for any $f\in X$. Then $\\{W_{t}\\}_{t\geq 0}$ is no longer a strongly
continuous semigroup in $X$.

We remark that for any reasonable Banach space $X$ of analytic functions in
${\mathbb{D}}$, the Banach space $[\varphi_{t},X]\subset X$ is defined; it is
the maximal subspace of $X$, on which the unweighted composition semigroup
$\\{C_{t}\\}$ is strongly continuous. The recent work [10] is devoted to a
description of flows $\\{\varphi_{t}\\}$ such that the spaces
$[\varphi_{t},\mathcal{B}]$ and $[\varphi_{t},BMOA]$ are the minimal ones,
that is, $[\varphi_{t},\mathcal{B}]=\mathcal{B}_{0}$ and
$[\varphi_{t},BMOA]=VMOA$.

In the proof of Theorem 4.1, we will make use the following lemma, which can
be extracted from [1], see also [13].

######  Lemma 4.2.

For any holomorphic flow $\\{\varphi_{t}\\}_{t\geq 0}$, one of the two
statements holds:

  1. (1)

There is a point $\gamma_{0}\in\partial{\mathbb{D}}$ such that

$\exists\lim_{r\to
1^{-}}\varphi_{t}(r\gamma_{0}):=\varphi_{t}(\gamma_{0}),\quad t>0,$

and $\varphi_{t}(\gamma_{0})\in{\mathbb{D}}$ for any $t>0$;

  2. (2)

$\varphi_{t}$ is an automorphism of ${\mathbb{D}}$ onto itself for any $t\geq
0$.

This is implicit in the proof of Theorem 3.1 in [1], which is based on the
König model of $\\{\varphi_{t}\\}_{t\geq 0}$. Namely, there is a Riemann map
$h$ of ${\mathbb{D}}$ onto a simply connected domain $\Omega$ with $h(0)=0$
and a constant $c$, $\operatorname{Re}c\geq 0$, such that either

(4.4) $\varphi_{t}(z)=h^{-1}\big{(}e^{-ct}h(z)\big{)},\quad
z\in{\mathbb{D}},\,t\geq 0$

or

(4.5) $\varphi_{t}(z)=h^{-1}\big{(}h(z)+ct\big{)},\quad
z\in{\mathbb{D}},\,t\geq 0.$

In the first case $\Omega$ has to be spirallike, $we^{-ct}\in\Omega$ for each
$w\in\Omega$ and $t\geq 0$. In the second one, $\Omega$ is close to convex,
$w+ct\in\Omega$ for each $w\in\Omega$ and $t\geq 0$. We refer to [24] or [25]
for more on the subject.

A key ingredient in the proof will be a result which allows us to estimate the
derivative of an infinite interpolating Blaschke products in pseudohyperbolic
discs. Of particular help will be Lemma 3.5 from the paper [17] by Girela,
Peláez and Vukotic. In order to formulate it, we recall that given
$\theta\in\mathbb{R}$ and a sequence of (not necessarily distinct) points
$\\{z_{k}\\}$ in $\mathbb{D}$ satisfying the _Blaschke condition_

$\sum_{k=1}^{\infty}(1-|z_{k}|)<\infty,$

the infinite product

$B(z)=e^{i\theta}\,\prod_{k=1}^{\infty}\frac{|z_{k}|}{z_{k}}\frac{z_{k}-z}{1-\overline{z_{k}}z}$

converges uniformly on compact subsets of $\mathbb{D}$.

For each $k\geq 1$, we will denote by $b_{z_{k}}$ the factor
$\frac{|z_{k}|}{z_{k}}\frac{z_{k}-z}{1-\overline{z_{k}}z}$. Whenever
$z_{k}=0$, the above expression $|z_{k}|/z_{k}$ is taken to be equal to $-1$,
so that $b_{z_{k}}(z)=z$ in this case. The holomorphic function $B$ is called
the _Blaschke product with zero sequence $\\{z_{k}\\}$_. It satisfies

  * i)

$B$ vanishes precisely at the points $\\{z_{k}\\}$, with the corresponding
multiplicities (that is, $\\{z_{k}\\}$ is the zero-sequence of $B$), and

  * ii)

$|B(z)|<1$ for every $z\in\mathbb{D}$.

  * iii)

$\lim_{r\to 1^{-}}|B(re^{i\theta})|=1$ almost everywhere on the boundary
$\partial\mathbb{D}$.

Recall that a sequence $\\{z_{n}\\}$ is said to be interpolating if for any
bounded sequence of complex numbers $\\{w_{n}\\}$ there exists a function
$f\in H^{\infty}$ such that $f(z_{n})=w_{n}$ for all $n$. A Blaschke product
is interpolating if its zero sequence is interpolating (in this case, all
$z_{n}$ are distinct). It is known that $B$ is interpolating if and only if
there is a constant $\delta>0$ such that $|B_{n}(z_{n})|>\delta>0$ for all
$n$, where $B_{n}=B/b_{z_{n}}$. Whenever $1-|z_{n+1}|\leq\alpha(1-|z_{n}|)$
for all $n$, where $\alpha<1$ is a constant, the sequence $\\{z_{n}\\}$ is
interpolating.

For these and other properties of Blaschke products, we refer to Garnett’s
book [16].

In the next statement, $\Delta(a,r)$ will denote the pseudohyperbolic disc

$\Delta(a,r)=\big{\\{}z\in{\mathbb{D}}:\quad\rho(z,a)<r\big{\\}},$

where $\rho(z,w)$ stands for the pseudohyperbolic distance between two points
$w,\,z$ in $\mathbb{D}$, namely,
$\rho(z,w)=\left|\frac{w-z}{1-\overline{w}z}\right|.$

######  Lemma 4.3.

Suppose $B$ is an infinite Blaschke product in ${\mathbb{D}}$. Let
$\\{a_{n}\\}$ be a part of its zeros. Suppose that it is an interpolating
sequence and that

$|(B/b_{a_{n}})(a_{n})|\geq\delta>0.$

Then there are positive constants $\alpha$ and $\beta$, which depend only on
$\delta$, such that the pseudo-discs $\Delta(a_{n},\alpha)$, $n\geq 1$ are
pairwise disjoint and

$|B^{\prime}(z)|\geq\frac{\beta}{1-|a_{n}|},\quad z\in\Delta(a_{n},\alpha).$

Though the statement of Lemma 4.3 is formally more general than the one of
Lemma 3.5 in [17], the proof is obtained by means of the same argument.

We are now in position to prove Theorem 4.1.

###### Proof of Theorem 4.1.

By the Closed Graph Theorem, the inclusion $X\subseteq\mathcal{B}$ is
continuous. Hence it suffices to prove that there exist a function $f\in
H^{\infty}\subseteq X$ and a constant $\delta>0$ such that

(4.6) $\limsup_{t\to 0^{+}}\|W_{t}f-f\|_{\mathcal{B}}\geq\delta.$

Assume first that in Lemma 4.2, case (1) holds. Without loss of generality, we
can assume that $\gamma_{0}=1$. We are going to choose an increasing sequence
$r_{n}\nearrow 1$ and a decreasing one $t_{n}\searrow 0$ such that
$\\{r_{n}\\}\cup\\{\varphi_{t_{n}}(r_{n})\\}$ is an interpolating sequence in
${\mathbb{D}}$ (in particular, the sequence $\\{r_{n}\\}$ is disjoint from
$\\{\varphi_{t_{n}}(r_{n})\\}$). Moreover, we require that

$|\varphi_{t_{1}}(r_{1})|<r_{1}<\dots<\big{|}\varphi_{t_{n-1}}(r_{n-1})\big{|}<r_{n-1}<\big{|}\varphi_{t_{n}}(r_{n})\big{|}<r_{n}<\dots$

and

(4.7)
$1-r_{n}<\frac{1}{2}\big{(}1-|\varphi_{t_{n}}(r_{n})|\big{)}<\frac{1}{4}(1-r_{n-1})$

for all $n\geq 1$. (For $n=1$, we only require the first inequality.) This
will be done by an inductive construction.

Take $t_{1}>0$ arbitrarily. Since $|\varphi_{t_{1}}(1)|=\lim_{r\to
1^{-}}|\varphi_{t_{1}}(r)|<1$, one can choose $r_{1}<1$ so that

$1-r_{1}<\frac{1}{2}\big{(}1-|\varphi_{t_{1}}(r_{1})|\big{)},$

and in particular, $|\varphi_{t_{1}}(r_{1})|<r_{1}$. This is the first step of
the construction.

Suppose $t_{n-1}$ and $r_{n-1}$ have been constructed. Choose
$t_{n}\in(0,\frac{1}{2}t_{n-1})$ so that

$0<1-|\varphi_{t_{n}}(1)|<\frac{1}{2}(1-r_{n-1}).$

Then for any $r<1$ sufficiently close to $1$,

$1-|\varphi_{t_{n}}(r)|<\frac{1}{2}(1-r_{n-1}).$

Hence $r_{n}<1$ can be chosen in such way that

$1-r_{n}<\frac{1}{2}\big{(}1-|\varphi_{t_{n}}(r_{n})|\big{)}<\frac{1}{4}(1-r_{n-1}).$

This defines sequences $r_{n}\nearrow 1$ and $t_{n}\searrow 0$ with all
desired properties.

We put

$f=\widetilde{B}\widehat{B}^{2}$

where the Blaschke products $\widetilde{B}$, $\widehat{B}$ correspond to
sequences of zeros $\\{r_{n}\\}$, $\\{\varphi_{t_{n}}(r_{n})\\}$,
respectively. By (4.7), $\widetilde{B}\widehat{B}$ is an interpolating
Blaschke product.

Having in mind (4.2), we note that

$\displaystyle\|W_{t}f-f\|_{\mathcal{B}}$ $\displaystyle\geq$
$\displaystyle\sup_{z\in\mathbb{D}}\big{|}(m_{t}(f\circ\varphi_{t}))^{\prime}(z)-f^{\prime}(z)\big{|}(1-|z|^{2})$
$\displaystyle\geq$ $\displaystyle\sup_{0\leq r<1}\big{|}m_{t}(r)\cdot
f^{\prime}\big{(}\varphi_{t}(r)\big{)}\varphi_{t}^{\prime}(r)+m_{t}^{\prime}(r)f\big{(}\varphi_{t}(r)\big{)}-f^{\prime}(r)\big{|}(1-r).$

Since
$f\big{(}\varphi_{t_{n}}(r_{n})\big{)}=f^{\prime}\big{(}\varphi_{t_{n}}(r_{n})\big{)}=0$,
by substituting $r=r_{n}$, $t=t_{n}$ in the above inequality, we get

$\|W_{t_{n}}f-f\|_{\mathcal{B}}\geq|f^{\prime}(r_{n})|(1-r_{n})\geq\delta>0,$

for all $n$ (we applied Lemma 4.3 to the Blaschke product $f$ and to
$a_{n}=r_{n}$). This gives (4.6).

Now assume Case (2) in the structure Lemma 4.2. Then $\varphi_{t}$ for $t>0$
is a disc automorphism, being either elliptic, parabolic or hyperbolic, and
this classification is the same for all $t$ (see [7], for instance).

Hence, the flow $\\{\varphi_{t}\\}_{t\geq 0}$ can be continued to a
holomorphic flow $\\{\varphi_{t}:\,t\in\mathbb{R}\\}$. In the parabolic case,
we will assume (without loss of generality) that $1$ is not the Denjoy-Wolff
point of $\\{\varphi_{t}:t\geq 0\\}$. In the hyperbolic case, we will assume
that $1$ is not the Denjoy-Wolff point neither of $\\{\varphi_{t}:t\geq 0\\}$
nor of $\\{\varphi_{-t}:t\geq 0\\}$.

Then, in all three cases, for a small $t_{0}>0$, the curves
$\eta_{r}:[0,t_{0}]\to{\mathbb{D}}$,

$\eta_{r}(t):=\varphi_{t}(r),$

tend as $r\to 1^{-}$ in the $C^{1}$ metric to the curve $\eta_{1}$. This
latter curve is a one-to-one smooth parametrization of a small subarc of
$\partial{\mathbb{D}}$, one of whose endpoints is $1$.

Put $r_{n}=1-2^{-n}$ for $n\geq N_{0}$, where $N_{0}$ is large enough. If the
curve $\eta_{1}$ is contained in $\\{\operatorname{Im}z\geq 0\\}$, then there
exist $t_{n}\searrow 0$ such that $\arg(\varphi_{t_{n}}(r_{n})-1)=3\pi/4$,
$n\geq N_{0}$. If $\eta_{1}$ is contained in $\\{\operatorname{Im}z\leq 0\\}$,
then one can choose $t_{n}\searrow 0$ so that
$\arg(\varphi_{t_{n}}(r_{n})-1)=-3\pi/4$, $n\geq N_{0}$. In both cases,

$\frac{1-\operatorname{Re}\varphi_{t_{n}}(r_{n})}{2^{-n}}\to 1\quad\text{ as
}n\to\infty.$

It follows that the sequence
$\\{r_{n}\\}\cup\big{\\{}\varphi_{t_{n}}(r_{n})\big{\\}}$ is contained in a
Stoltz angle with vertex at $1$ and is separated in the pseudohyperbolic
metric $\rho(\cdot,\cdot)$. Thus, this sequence is interpolating (see [23],
Lecture VII) and the rest of the proof runs as in the Case (1). ∎

######  Remark 4.4.

Any space $X$ satisfying (4.1) is not separable. Indeed, let $B$ be an
interpolating Blaschke product with (infinitely many) zeros on the radius
$[0,1)$, for instance, at the points $1-2^{-n}$. Put
$B_{t}(z)=B(\exp(-it)*z)$. Then each Blaschke product $B_{t}$ belongs to $X$.
Using Lemma 4.3, we see that the distances $\|B_{t}-B_{s}\|_{\mathcal{B}}$,
where $t,s\in[0,2\pi)$ and $t\neq s$, are uniformly bounded from below by some
constant $\varepsilon>0$ which implies our assertion.

## References

  * [1] A. Anderson, M. Jovovic, W. Smith, Composition semigroups on BMOA and $H^{\infty}$, J. Math. Anal. Appl. 449 (2017), no. 1, 843–852.
  * [2] W. Arendt, I. Chalendar, Generators of semigroups on Banach spaces inducing holomorphic semiflows, Israel J. Math. 229 (2019), no. 1, 165–179.
  * [3] C. Avicou, I. Chalendar, J. R. Partington, A class of quasicontractive semigroups acting on Hardy and Dirichlet space, J. Evol. Equ. 15 (2015), 647–665.
  * [4] C. Avicou, I. Chalendar, J. R. Partington, Analyticity and compactness of semigroups of composition operators, J. Math. Anal. Appl. 437 (2016), 545–560.
  * [5] C. Berenstein and R. Gay, Complex Analysis and Topics in Harmonic Analysis, Springer-Verlag, New York (1995).
  * [6] O. Blasco, M. Contreras, S. Díaz-Madrigal, J. Martínez, M. Papadimitrakis, A. Siskakis, Semigroups of composition operators and integral operators in spaces of analytic functions, Ann. Acad. Sci. Fenn. Math. 38 (2013) 67–89.
  * [7] F. Bracci, M. D. Contreras and S. Díaz-Madrigal, Continuous Semigroups of Holomorphic Self-maps of the Unit Disc, Springer Monographs in Mathematics (2020).
  * [8] E. Berkson and H. Porta, Semigroups of analytic functions and composition operators. Michigan Math. J. 25 (1978), no. 1, 101–115.
  * [9] I. Chalendar, J. R. Partington, A class of quasicontractive semigroups acting on Hardy and weighted Hardy spaces. Semigroup Forum 95 (2017), no. 2, 281–292.
  * [10] N. Chalmoukis, V. Daskalogiannis, Holomorphic semigroups and Sarason’s characterization of vanishing mean oscillation (2021), arXiv:2106.01294.
  * [11] C. Cowen, Subnormality of the Cesàro operator and a semigroup of composition operators. Indiana Univ. Math. J. 33 (1984), 305–318
  * [12] P.L. Duren, _Theory of $H^{p}$ spaces._ Pure and Applied Mathematics, Vol. 38. Academic Press, New York-London, 1970.
  * [13] P. Galanopoulos, N. Merchán, A. G. Siskakis, Semigroups of composition operators in analytic Morrey spaces. Integral Equations Operator Theory 92 (2020), no. 2, Paper No. 12, 15 pp.
  * [14] E. A. Gallardo-Gutiérrez, D. V. Yakubovich, On generators of $C_{0}$-semigroups of composition operators. Israel J. Math. 229 (2019), no. 1, 487-500.
  * [15] E. A. Gallardo-Gutiérrez, R. Kumar, J. R. Partington, Boundedness, compactness and Schatten-class membership of weighted composition operators. Integral Equations Operator Theory 67 (2010), no. 4, 467-479.
  * [16] J. B. Garnett, _Bounded Analytic Functions_ , Academic Press, 1981.
  * [17] D. Girela, J. Á. Peláez, D. Vukotić, Integrability of the derivative of a Blaschke product. Proc. Edinb. Math. Soc. (2) 50 (2007), no. 3, 673–687.
  * [18] E. Hille, Ordinary differential equations in the complex domain. Wiley-Interscience (John Wiley $\&$ Sons), 1976.
  * [19] F. Jafari, T. Tonev, E. Toneva, K. Yale, Holomorphic flows, cocycles, and coboundaries. Michigan Math. J. 44 (1997), no. 2, 239–253.
  * [20] F. Jafari, Z. Slodkowski, T. Tonev, Semigroups of operators on Hardy spaces and cocycles of holomorphic flows. Complex Anal. Oper. Theory 6 (2012), no. 1, 113–119.
  * [21] F. Jafari, T. Tonev, E. Toneva, Automatic differentiability and characterization of cocycles of holomorphic flows. Proc. Amer. Math. Soc. 133 (2005), no. 11, 3389–3394.
  * [22] W. König, Semicocycles and weighted composition semigroups on $H^{p}$. Michigan Math. J. 37 (1990), no. 3, 469–476.
  * [23] N. K. Nikol’skiǐ, Treatise on the shift operator. Spectral function theory. With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller. Translated from the Russian by Jaak Peetre. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 273. Springer-Verlag, Berlin, 1986.
  * [24] Ch. Pommerenke, Univalent functions: with a chapter on quadratic differentials by Gerd Jensen, Vandenhoeck and Ruprecht in Göttingen, 1975.
  * [25] D. Shoikhet, Semigroups in Geometrical Function Theory, Springer, 2001.
  * [26] A. G. Siskakis, Composition semigroups and the Cesàro operator on $H^{p}$. J. Lond. Math. Soc. 36, 153– 164 (1987).
  * [27] A. G. Siskakis, Semigroups of composition operators on spaces of analytic functions, a review. Studies on composition operators (Laramie, WY, 1996), 229–252, Contemp. Math., 213, Amer. Math. Soc., Providence, RI, 1998.
  * [28] A. G. Siskakis, Weighted composition semigroups on Hardy spaces. Proceedings of the symposium on operator theory (Athens, 1985). Linear Algebra Appl. 84 (1986), 359–371.