# Pioneering Studies on LTE eMBMS: Towards 5G Point-to-Multipoint
Transmissions

Hongzhi Chen1, De Mi1, Manuel Fuentes2, David Vargas 3, Eduardo Garro4, Jose
Luis Carcel4,
Belkacem Mouhouche2, Pei Xiao1 and Rahim Tafazolli1 Email:{hongzhi.chen, d.mi,
p.xiao<EMAIL_ADDRESS>{m.fuentes<EMAIL_ADDRESS>
<EMAIL_ADDRESS>{edgarcre<EMAIL_ADDRESS>1Institute for
Communication Systems, University of Surrey, United Kingdom  2Samsung
Electronics R&D UK, United Kingdom  3BBC R&D, United Kingdom  4Institute of
Telecommunications and Multimedia Applications, Universitat Politecnica de
Valencia, Spain

###### Abstract

The first 5G (5th generation wireless systems) New Radio Release-15 was
recently completed. However, the specification only considers the use of
unicast technologies and the extension to point-to-multipoint (PTM) scenarios
is not yet considered. To this end, we first present in this work a technical
overview of the state-of-the-art LTE (Long Term Evolution) PTM technology,
i.e., eMBMS (evolved Multimedia Broadcast Multicast Services), and investigate
the physical layer performance via link-level simulations. Then based on the
simulation analysis, we discuss potential improvements for the two current
eMBMS solutions, i.e., MBSFN (MBMS over Single Frequency Networks) and SC-PTM
(Single-Cell PTM). This work explicitly focus on equipping the current eMBMS
solutions with 5G candidate techniques, e.g., multiple antennas and millimeter
wave, and its potentials to meet the requirements of next generation PTM
transmissions.

###### Index Terms:

Point-to-Multipoint, eMBMS, MBSFN, SC-PTM, broadcast, multicast

††We would like to acknowledge the support of the University of Surrey 5GIC
(www.surrey.ac.uk/5gic) members for this work. This work was also supported in
part by the European Commission under the 5GPPP project 5G-Xcast
(H2020-ICT-2016-2 call, grant number 761498). The views expressed in this
contribution are those of the authors and do not necessarily represent the
project.

##  I Introduction

In the past few years, point-to-multipoint (PTM) technologies have been
specified by the 3rd Generation Partnership Project (3GPP) to simultaneously
deliver common content to a large amount of users, with a fixed amount of
radio resources. 3GPP has enhanced the PTM technologies based on Long Term
Evolution (LTE) networks, ever since the adoption of evolved Multimedia
Broadcast Multicast Services (eMBMS) in Release (Rel-) 9 [1]. eMBMS introduces
PTM support with small changes on the existing radio and core network
infrastructures and protocols of LTE. For instance, new physical, transport
and logic channels to enable MBSFN (MBMS over Single Frequency Networks) were
added and now coexist with other standardized channels in the specification.
Since its introduction, the current eMBMS Rel-14 system has been significantly
enhanced and therefore is considerably different from the original version in
Rel-9 [2]. The main novelties introduced up to now include the use of unused
MBSFN subframes for unicast reception; MooD (MBMS operation on Demand) which
permits to active and deactivate the broadcast service seamlessly and
automatically based on the users service consumption reporting; and the
introduction of SC-PTM (Single-Cell PTM) to increase the resource allocation
flexibility by multiplexing broadcast and unicast data on the same physical
channel, instead of using a dedicated multicast channel as MBSFN does. Very
recently, a first version of the 5G New Radio Rel-15 specification was
approved by 3GPP, where the use cases, the corresponding requirements and
technological paradigms are unicast centric or point-to-point (PTP). It is
therefore of great interest to extend them to broadcast/multicast-enabling PTM
scenarios. Derived from our prior work in [2], this work aims at revealing the
potential of state-of-the-art PTM technologies previously defined in LTE
Rel-14 including the latest fully standardized eMBMS, to meet the requirements
of the 5G use cases and scenarios.
To this end, we present a comprehensive technical overview of the LTE eMBMS
system, and describe its evaluation methodology in detail. A performance
analysis and link-level simulations are provided, based on the reference Key
Performance Indicators (KPI) and evaluation methodology defined by the ITU-R
(International Communications Union - Recommendation) for the IMT-2020
(International Mobile Telecommunication) evaluation process [3]. Therefore,
the results in this work can be served as a benchmark to compare the
performance of a potential 5G broadcast solution. The gap analysis of the
current eMBMS solutions against the potential requirement of next generation
PTM transmissions provides valuable insights into the practical system design
(note that the IMT-2020 recommendations are assumed to be a point-to-
point/unicast only solution, but we used them as the basis and potential
target for evaluating PTM solutions). We also discuss the potential of
combining the PTM solutions with 5G candidate techniques, e.g., Massive
Multiple-Input Multiple-Output (MIMO) and millimeter wave (mmWave), towards
the vertical use cases in next-generation PTM systems.

##  II Technology Overview

Figure 1: LTE eMBMS physical layer transmitter block diagram

The LTE eMBMS Rel-14 physical layer transmitter block diagram is displayed in
Fig. 1. Input transport block(s) (TB) containing data bits is(are) sent to the
Bit-Interleaved Coding and Modulation (BICM) processing chain. Each TB is
encoded with a combination of forward error correction (FEC), scrambler and
mapper (modulator). More specifically, a Cyclic Redundancy Check (CRC) bit
sequence is first attached to each TB. If the TB size is larger than the
maximum code block (CB) size, the input data bit sequence is then segmented
with an additional CRC sequence attached to each segmented CB. Each code block
is then coded using turbo codes (TC) with mother code rate equals to $1/3$.
Next, rate matching is performed such that the bits inside each CB are
interleaved, circular buffered and punctured/repeated to provide the specific
code rate (CR) related to the selected Modulation and Coding Scheme (MCS).
Bits after rate matching are then concatenated, scrambled and mapped to
constellation symbols. The available modulation schemes for LTE eMBMS are
QPSK, 16QAM, 64QAM, and 256QAM. After that, symbols are then allocated to all
the available resource elements (RE) in the corresponding subframe, and
finally, before transmission, CP-OFDM (Cyclic Prefix-Orthogonal Frequency
Division Multiplexing) is performed.
The eMBMS specification allows for two modes of operation at the physical
layer, i.e., MBSFN and SC-PTM, which are introduced in the following
subsections

###  II-A MBMS over Single Frequency Network

The multi-cell solution for LTE broadcasting is called MBSFN, which performs a
synchronized transmission of the same content to a group of cells over the
same frequency. This group of cells forms an area called Single Frequency
Network (SFN) area. Compared to SC-PTM, MBSFN eliminates inter-cell
interference within the whole service area. MBSFN provides PTM services using
a specific Physical Multicast Channel (PMCH) and occupying the entire
bandwidth. Regarding the subcarrier spacing, MBSFN can be configured with
values of 15, 7.5 and 1.25 kHz, corresponding to extended CP durations of
16.7, 33.3 and 200 $\mu$s respectively. Since a separate physical channel is
used, MBSFN employs different reference signals compared to LTE unicast and
SC-PTM. eMBMS is standardized to multicast data over a single antenna port,
and therefore it does not take advantage of the potential MIMO capability,
thus only a single codeword can be sent.

###  II-B Single Cell Point to Multipoint

SC-PTM is another LTE PTM solution which aims at providing broadcasting
service to groups of users in one single cell. To that end, the resource
allocation flexibility is increased by employing the Physical Downlink Shared
Channel (PDSCH), which is also used for unicast transmission. It should be
noted that sharing a physical channel implies to use the same carrier spacing
for both technologies. As a consequence, only 15 kHz subcarrier spacing with
normal CP is available for SC-PTM. Since different contents are transmitted in
neighbour cells, transmission power management is required in order to control
inter-cell interference,

##  III Evaluation Methodology and Analysis

The technical performance requirements about the primary usage scenarios and
their corresponding evaluation methodology for PTP transmissions are defined
in the IMT-2020 evaluation process [4]. In addition, the methodology for the
requirement we consider in this paper also follows these guidelines [3].
Following the same methodology, this study is focused on BICM spectral
efficiency (SE) that is considered in the IMT-2020 KPI set. This is because
the BICM SE provides the capacity supported by PTM solutions for different
modulation and coding combinations.

###  III-A BICM Spectral Efficiency Evaluation

BICM spectral efficiency denotes the spectral efficiency that can be achieved
by the PTM system while using different modulation schemes, excluding the
resource that are used for control information and synchronisation, i.e.,
overheads are not taken into account. The BICM spectral efficiency is
expressed in bits per RE (bits/RE) and can be calculated by:

$\text{SE}_{\text{\emph{BICM}}}=\log_{2}(M)\cdot\text{CR}\cdot
N_{\emph{Tx/Rx}}$ (1)

where $M$ denotes the modulation order, i.e., for QPSK, $M=4$, CR represents
the effective code rate corresponding to the MCS index and $N_{\emph{Tx/Rx}}$
is the number of independent information spatial streams with multiple
transmitter and receiver antennas. Note that the effective CR in LTE is
calculated as:

$\text{CR}=\frac{\text{TBS}}{N_{\emph{avail}}}$ (2)

where TBS represents the transport block size, which is fixed for each
modulation and coding index, can be found in [5], and $N_{\emph{avail}}$ is
the number of total available bits in one subframe, given by:

$N_{\emph{avail}}=m\cdot
N_{\text{\emph{RB}}}(N_{\emph{sym}}N^{\text{\emph{RB}}}_{sc}-N_{\emph{RS}})$
(3)

For (3), $m=\log_{2}(M)$, denotes the number of bits per constellation symbol,
$N_{\text{\emph{RB}}}$ is the number of resource blocks (RB) utilized within a
subframe, $N_{\emph{sym}}$ is the number of OFDM symbols per subcarrier
excluding those used for control channel, $N^{\text{\emph{RB}}}_{sc}$ is the
number of subcarriers per RB, and $N_{\emph{RS}}$ is the number of reference
signals per RB. Note that the maximum CR cannot exceed 0.925, which is the CR
associated to the maximum Channel State Indicator (CQI) 15. The maximum TB
size with SC-PTM and MBSFN is given for indexes $I_{\text{\emph{TBS}}}=33$ and
$32$ [5] and the associated CR is then 0.887 and 0.882 respectively. Also, the
required Carrier to Noise Ratio (CNR) to achieve a given BICM SE value depends
on the channel environment.

###  III-B BICM Spectral Efficiency Calculation

Figure 2: Resource Block frame structure for MBSFN (left) and SC-PTM (right),
with 15 kHz carrier spacing.

As previously mentioned, the BICM spectral efficiency depends on the
modulation order and effective code rate, each Modulation and Coding Scheme
(MCS) provides a different CR directly related to a particular TB size. Fig. 2
shows the different frame structure for a single RB for both configurations,
i.e. MBSFN and SC-PTM [5]. Note that the carrier spacing selected is 15 kHz
for illustration purposes, and the number of OFDM symbols per subcarrier with
MBSFN and SC-PTM is 12 and 14, due to the use of extended and normal CP
respectively. As Fig. 2 depicts, the number of subcarriers per RB with MBSFN
and SC-PTM are both 12. In SC-PTM, the number of OFDM symbols per subcarrier
used for the control channel is 3 for 5 MHz bandwidths and below and 2
otherwise. The available bits with SC-PTM when considering 50 RBs, can be
calculated using (3), obtaining $m\cdot 50(12\cdot 12-6)=6900m$ bits,
considering a channel bandwidth of 10 MHz. In MBSFN, 3 different
configurations can be used, with different carrier spacing. In this paper,
studies for MBSFN are focused on the standalone mode with carrier spacing 1.25
kHz. In this case, the available bits $N_{\emph{avail}}=m\cdot 50(1\cdot
144-24)=6000m$ bits. Since the maximum constellation size availbale is 256QAM,
the peak value of BICM spectral efficiency is 7.09 and 7.06 bits/RE with SC-
PTM and MBSFN respectively. Note that the same calculation can be easily
extended to MIMO. SC-PTM with 4 spatial streams (MIMO $4\times 4$) can reach
up to 28.36 bits/RE, whereas MBSFN is limited to 7.06 bits/RE since the use of
MIMO is not specified.

##  IV Link-Level Simulation Evaluation

Link-level results for the BICM spectral efficiency as a function of the
required CNR for SC-PTM and MBSFN are presented in this section. Different
scenarios i.e., channel types have been evaluated in order to assess the
impact of the configurations adopted. The selected Quality of Service (QoS)
metric is block error rate (BLER) lower than 0.1%. A subcarrier spacing of
$\Delta_{f}=1.25$ kHz is always used with MBSFN, in order to study the
potential advantages of this mode compared to SC-PTM ($\Delta_{f}=15$ kHz).
The simulation parameters we considered in this paper are aligned with the
IMT-2020 evaluation guidelines [3].

###  IV-A Additive White Gaussian Noise Channel

Figure 3: BICM SE vs. CNR (dB) for AWGN channel.

Fig. 3 depicts the comparison of the two PTM schemes in Additive White
Gaussian Noise (AWGN) channel as well as the channel capacity. Both
transmitter and receiver are considered equipping single antenna. Although SC-
PTM and MBSFN are both PTM solutions, we consider a simplified model where one
user inside one single cell is simulated. Note that this simplified scenario
may present the considered PTM solution in its best light. This can be
expressed from two aspects: 1) The channel been simulated is an AWGN channel;
2) The data rate of a PTM system depends on the weakest user’s data rate,
which means that a PTP case represents the best scenario in terms of the
achievable data rate of the system. From the results, we can see that the
overall BICM SE follows the same trend regardless of the PTM technology used,
and the clearly visible gap between the achievable BICM SE and channel
capacity can be observed.

Figure 4: BICM SE vs. CNR (dB) for MIMO/SIMO i.i.d Rayleigh channel.

###  IV-B Independent Identical Distributed Rayleigh Channel

Fig. 4 shows the BICM spectral efficiency for the independent and identically
distributed (i.i.d.) Rayleigh MIMO channel. In this simulation, SC-PTM
utilizes 2-by-2 MIMO, while MBSFN employs a $1\times 2$ Single-Input Multiple-
Output (SIMO) scheme. Minimum Mean Square Error (MMSE) demapper has been used
with SC-PTM in order to cope with computation complexity limitations with
higher modulation orders. For SC-PTM, representative MCS indexes from 0 to 27
with step 3 are used. From Fig. 4, spatial multiplexing of multiple codewords,
i.e., SC-PTM with MIMO, provides a higher BICM SE compare to the diversity
gain offered by multiple receive antennas with only one transmitted codeword,
e.g., MBSFN with SIMO, and the gap increases at the high CNR region. As
calculated in section. III the BICM SE for MBSFN is limited to 7 bits/RE, SC-
PTM can increase its limit to 9 bits/RE, respectively.

##  V Discussion: PTM Potential Enhancements

###  V-A Combining MBSFN with Smart Antenna Techniques

In Fig. 4, it can be seen that the performance gap between multiplexing
multiple codewords and diversity transmission/receiving of a single codeword
is significant, especially in the high CNR regime. This indicates that
enabling MIMO techniques in MBSFN scenarios have potential to increase the
system throughput/SE dramatically. Combining MBSFN with MIMO provides
diversity transmission and multiplexing. It is then possible to assume that we
have an SFN area comprised of $M$ cells, where each cell has one base station
(BS) equipped with two transmit antennas. Thus the same codewords are
multiplexed onto the two transmit antennas in each cell, and therefore each of
the $M$ BSs can broadcast the multiplexed signal to all users inside the SFN
area. Assuming there are $N$ single-antenna users in the SFN area, we can form
a 2M-by-N distributed MIMO scheme. When $M$ goes larger. i.e., in the case of
a increased SFN area, the considered system can be categorized as massive MIMO
[6]. It is worth mentioning that, the bigger the SFN area is, the harder to
perform synchronized transmission. Also the frame structure need to be
adopted, e.g., CP length, subcarrier spacing.

###  V-B Operating PTM solution in Millimetre Wave band

To meet the requirement of new generation PTM transmissions with ultra high
definition video streaming, large bandwidth is preferred to provide over Gbps
bit rates and overcome the spectrum scarcity problem [4]. In this context, the
application of mmWave communications has been shown as a potential promising
solution, where a large amount of underutilized band can be leveraged to
provide the potential gigahertz transmission bandwidth [7]. However, the
extreme narrow beam formed in the traditional mmWave antenna array design is
in general not suitable for broadcasting or multicasting. Novel designs on
high-gain omnidirectional slotted-waveguide antenna arrays have been recently
proposed [8], whose radiation pattern shows the promising coverage in an
Olympic-standard arena by using only one high-gain radiating element. This can
fundamentally change the design principle of the next generation PTM
transmissions and bring new solutions, e.g., SC-PTM in small cells with
100-fold boosted capacity based on the existing standardized PTM technologies.

##  VI Conclusion

In this paper, the state-of-the-art LTE eMBMS PTM technologies, i.e., MBSFN
and SC-PTM, have been analysed. The detailed technical overview of both MBSFN
and SC-PTM and their designs have been covered. The analysis presented, based
upon the link-level simulations has revealed that regarding BICM spectral
efficiency, SC-PTM equipped with MIMO can provide up to four times more
capacity (bits/RE) than MBSFN, since the last one does not support the use of
multiple antennas with multiple codewords. This work also discussed the
potential of combining MBSFN and SC-PTM with 5G candidate techniques such as
massive MIMO and mmWave, serving as a useful guidance to improve the current
LTE eMBMS PTM technologies.

Further investigations on comparing the LTE eMBMS with other terrestrial
broadcasting standards are also needed, in order to provide a comprehensive
gap analysis on current PTM technologies.

## References

  * [1] D. Lecompte and F. Gabin, “Evolved multimedia broadcast/multicast service (embms) in lte-advanced: overview and rel-11 enhancements,” _IEEE Communications Magazine_ , vol. 50, no. 11, pp. 68–74, Nov. 2012.
  * [2] 5G-PPP 5G-Xcast project, “LTE-Advanced Pro Broadcast Radio Access Network Benchmark,” _Deliverable D3.1_ , Nov. 2017.
  * [3] International Telecommunications Union (ITU), Draft New Report ITU-R M., “Guidelines for evaluation of radio interface technologies for IMT-2020,” Oct. 2017.
  * [4] ITU, Draft New Report ITU-R M., “Minimum requirements related to technical performance for IMT-2020 radio interface(s),” Feb. 2017.
  * [5] 3GPP TR 36.213 v14.2.0, “Evolved Universal Terrestrial Radio Access (E-UTRA); Physical layer procedures (Release 14),” Mar. 2017.
  * [6] E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” _IEEE Communications Magazine_ , vol. 52, no. 2, pp. 186–195, Feb. 2014.
  * [7] T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobile communications for 5G cellular: It will work!” _IEEE Access_ , vol. 1, pp. 335–349, May 2013.
  * [8] H. R. D. Filgueiras, I. F. da Costa, S. A. Cerqueira, R. A. Santos, and J. R. Kelly, “Mechanically reconfigurable slotted-waveguide antenna array for 5G networks,” in _2017 SBMO/IEEE MTT-S International Microwave and Optoelectronics Conference (IMOC)_ , Aug. 2017, pp. 1–5.


# Semigroups of weighted composition operators on spaces of holomorphic
functions

I. Chalendar and J.R. Partington

Isabelle Chalendar, Université Gustave Eiffel, LAMA, (UMR 8050), UPEM, UPEC,
CNRS, F-77454, Marne-la-Vallée, France.
<EMAIL_ADDRESS>

Jonathan R. Partington, School of Mathematics, University of Leeds, Leeds LS2
9JT, UK<EMAIL_ADDRESS>

Mathematics Subject Classification (2020): 30H10, 30H20, 30D05, 47B33, 47D06

##  1 Introduction

This paper is based on three hours of lectures given by the first author in
the “Focus Program on Analytic Function Spaces and their Applications” July 1
– December 31, 2021, organized by the Fields Institute for Research in
Mathematical Sciences.

The goal of this paper is to give an introduction to the properties of
discrete and continuous $C_{0}$-semigroups of (weighted) composition operators
on various spaces of analytic functions.

To that aim we detail the structure of semiflows of analytic functions on the
open unit disc $\mathbb{D}$ and their generators, which provides information
on the generators of continuous semigroups of composition operators on Banach
spaces $X$ embedding in $\mathop{\rm Hol}\nolimits(\mathbb{D})$, the Fréchet
space of holomorphic functions on $\mathbb{D}$.

An initial motivation for studying such semigroups is a better understanding
of an specific universal operator. Moreover, adding a weight to a composition
operator is motivated by the fact that such operators describe isometries on
non-Hilbertian Hardy spaces and they appear automatically when the Banach
spaces $X$ are replaced by Banach spaces of holomorphic functions on another
domain such as the right half-plane.

Thanks to the analysis of spectral properties, we deduce the asymptotic
behaviour of discrete semigroups of composition operators on various Banach
spaces such as the Hardy spaces, the weighted Bergman spaces, Bloch type
spaces or standard weighted Bergman space of infinite order. As a byproduct we
obtain characterization of the properties of isometry and similarity to
isometry, still for composition operators. We then describe the limit at
infinity for continuous semigroups of composition operators.

Compactness (immediate and eventual) and analyticity of semigroups of
composition operators are then considered on the Hardy space
$H^{2}(\mathbb{D})$, even though other classes of Banach spaces may also be
considered. References are given for more complete information.

Finally, we provide some perspectives for semigroups of composition operators
on $H^{2}(\mathbb{C}_{+})$, where $\mathbb{C}_{+}$ is the right-half-plane, as
well as an analysis of semigroups of composition operators on the Fock space.
The latter case can be treated in a complete way since the non-trivial
semiflows involved are necessarily expressed using polynomials of degree one.

###### Contents

  1. 1 Introduction
  2. 2 Background
    1. 2.1 Strongly continuous semigroups of operators: definition and characterization
    2. 2.2 Analytic semiflows on a domain and models for semiflows on $\mathbb{D}$
    3. 2.3 Models for analytic flows on $\mathbb{D}$
    4. 2.4 $C_{0}$-semigroups of composition operators
    5. 2.5 Spaces on which semigroups are not $C_{0}$
  3. 3 Motivation
    1. 3.1 Universal operators
    2. 3.2 Isometries
    3. 3.3 Change of domain
  4. 4 Asymptotic behaviour of $T^{n}$ or $T_{t}$
    1. 4.1 The discrete unweighted case
    2. 4.2 The continuous unweighted case
    3. 4.3 Weighted composition operators
    4. 4.4 Isometry and similarity to isometry
    5. 4.5 Generators
  5. 5 Compactness and analyticity
    1. 5.1 Immediate and eventual compactness
    2. 5.2 Compact analytic semigroups
  6. 6 An outlook on $\mathbb{C}_{+}$ and $\mathbb{C}$
    1. 6.1 The right halfplane $\mathbb{C}_{+}$
    2. 6.2 The complex plane $\mathbb{C}$
      1. 6.2.1 Composition operators
      2. 6.2.2 Weighted composition operators

##  2 Background

###  2.1 Strongly continuous semigroups of operators: definition and
characterization

We recall some of the standard facts about one-parameter semigroups of
operators, which may be found in many places, such as [29] and [43].

######  Definition 2.1.

A semigroup $(T_{t})_{t\geq 0}$ of operators on a Banach space $X$ is a family
of bounded operators satisfying:
(i) $T_{0}={\rm Id}$, the identity operator, and (ii) $T_{t+s}=T_{t}T_{s}$ for
all $s,t\geq 0$.
If, in addition, it satisfies:
(iii) $T_{t}x\to x$ as $t\to 0^{+}$ for all $x\in X$, then it is called a
strongly continuous or $C_{0}$-semigroup.

A uniformly continuous semigroup $(T_{t})_{t\geq 0}$ is one satisfying
$\|T_{t}-{\rm Id}\|\to 0$ as $t\to 0^{+}$.

Associated with this is the notion of an infinitesimal generator, or simply
generator. We define an (in general unbounded) operator $A$ whose domain is

$D(A):=\\{x\in X:\lim_{t\to 0^{+}}\frac{T_{t}x-x}{t}\quad\hbox{exists}\\},$

and then

$Ax:=\lim_{t\to 0^{+}}\frac{T_{t}x-x}{t}\qquad\hbox{for}\quad x\in D(A).$

Moreover, the generator of a $C_{0}$-semigroup characterizes completely a
semigroup, that is two $C_{0}$-semigroups are equal if and only if their
generators are equal.

By the uniform boundedness principle, each $C_{0}$-semigroup is uniformly
bounded on each compact interval. As a corollary, for every $C_{0}$-semigroup
$(T_{t})_{t\geq 0}$, there exists $w\in\mathbb{R}$ and $M\geq 1$ such that

$\|T_{t}\|\leq Me^{wt}\mbox{ for all }t\geq 0.$

Contractive $C_{0}$-semigroups are the ones for which one can take $M=1$ and
$w=0$, whereas quasicontractive $C_{0}$-semigroups are the ones for which one
can take $M=1$ and $w$ is arbitrary.

The domain $D(A)$ of the generator $A$ of a $C_{0}$-semigroup is always dense
in $X$ (and moreover $(A,D(A))$ is a closed operator). It is then natural to
characterize the linear operators $(A,D(A))$ that are the generator of
$C_{0}$-semigroups. For semigroups of contractions on Hilbert spaces, Lumer
and Phillips, in 1961, provided a beautiful criterion [38] (see also [29,
Theorem 3.15]).

######  Theorem 2.2 (Lumer–Phillips).

Let $(A,D(A))$ be a linear operator with dense domain on a Hilbert space $H$.
The following assertions are equivalent:

  1. (i)

$A$ generates a $C_{0}$-semigroup of contractions;

  2. (ii)

there exists $\lambda>0$ such that $(\lambda{\rm Id}-A)D(A)=H$ and for all
$x\in D(A)$,

$\mathop{\rm Re}\nolimits\langle Ax,x\rangle\leq 0;$

  3. (iii)

for all $\lambda>0$ we have $(\lambda{\rm Id}-A)D(A)=H$ and for all $x\in
D(A)$,

$\mathop{\rm Re}\nolimits\langle Ax,x\rangle\leq 0.$

Since $(T_{t})_{t\geq 0}$ is a $C_{0}$-semigroup of quasicontractions on a
Hilbert space $H$ if and only if there exists $w\geq 0$ such that
$(e^{-wt}T_{t})_{t\geq 0}$ is a $C_{0}$-semigroup of contractions, it is then
easy to deduce a version of the Lumer–Phillips Theorem for
$C_{0}$-quasicontractions.

######  Theorem 2.3.

Let $(A,D(A))$ be a linear operator with dense domain on a Hilbert space $H$.
The following assertions are equivalent:

  1. (i)

$A$ generates a $C_{0}$-semigroup of quasicontractions;

  2. (ii)

there exists $\lambda>0$ such that $(\lambda{\rm Id}-A)D(A)=H$ and

$\sup_{x\in D(A),\|x\|\leq 1}\mathop{\rm Re}\nolimits\langle
Ax,x\rangle<\infty;$

  3. (iii)

for all $\lambda>0$ we have $(\lambda{\rm Id}-A)D(A)=H$ and

$\sup_{x\in D(A),\|x\|\leq 1}\mathop{\rm Re}\nolimits\langle
Ax,x\rangle<\infty.$

For generators of $C_{0}$-semigroups which are not necessarily
quasicontractive, another beautiful criterion involving the growth of the
resolvent is due to Hille and Yosida [29, Theorem 3.8].

######  Theorem 2.4.

Let $A$ be a linear operator defined on a linear subspace $D(A)$ of the Banach
space $X$, $w\in\mathbb{R}$ and $M>0$. Then $A$ generates a $C_{0}$-semigroup
$(T_{t})_{t\geq 0}$ that satisfies

$\|T_{t}\|\leq Me^{wt}$

if and only if

  * a)

$A$ is closed and $D(A)$ is dense in $X$;

  * (b)

every real $\lambda>w$ belongs to the resolvent set of $A$ and for such
$\lambda$ and for all positive integers $n$,

$\|(\lambda{\rm Id}-A)^{-n}\|\leq{\frac{M}{(\lambda-\omega)^{n}}}.$

We shall restrict ourselves to Banach spaces $X$ of functions that are
holomorphic on a domain $\Omega$ (usually the unit disc $\mathbb{D}$ but
sometimes the right half-plane $\mathbb{C}_{+}$ or the complex plane
$\mathbb{C}$) and satisfying the condition that point evaluations $f\mapsto
f(z)$ are continuous for all $z\in\Omega$. Assuming this, we have that norm
convergence of a sequence $(f_{n})$ to $f$ implies local uniform convergence
(uniform convergence on compact subsets of $\Omega$).

Recall that for suitable $\phi:\Omega\to\Omega$ holomorphic, the composition
operator $C_{\phi}:X\to X$ is defined by $(C_{\phi}f)(z)=f(\phi(z))$, for
$f\in X$ and $z\in\Omega$ (assuming that $C_{\phi}$ maps $X$ boundedly into
$X$, an issue which will be discussed later). Likewise, for suitable $w$
holomorphic on $\Omega$, the weighted composition operator $W_{w,\phi}$ on $X$
is defined by $(W_{w,\phi}f)(z)=w(z)f(\phi(z))$.

The main theme of this paper is to study $C_{0}$-semigroups of (weighted)
composition operators. However, we may also look at composition semigroups
from a non-operatorial point of view (for example, as in [16]).

###  2.2 Analytic semiflows on a domain and models for semiflows on
$\mathbb{D}$

######  Definition 2.5.

A continuous analytic semiflow on a domain $\Omega$ is a family
$(\phi_{t})_{t\geq 0}$ of holomorphic mappings from $\Omega$ to itself
satisfying:
(i) $\phi_{0}(z)=z$ for all $z\in\Omega$,
(ii) $\phi_{t+s}=\phi_{t}\circ\phi_{s}$ for all $s,t\geq 0$, and
(iii) for all $z\in\Omega$, the mapping $t\mapsto\phi_{t}(z)$ is continuous on
$[0,\infty)$.

######  Remark 2.6.

A family $(\phi_{t})_{t\geq 0}$ of holomorphic mappings from $\Omega$ to
itself satisfying only (i) and (ii) is called an _algebraic semiflow_. Such an
algebraic semiflow $(\phi_{t})_{t\geq 0}$ is continuous on $\mathbb{D}$ if and
only if there exists $a\in\mathbb{D}$ such that $\lim_{t\to
0}\phi^{\prime}_{t}(a)=1$ (see [16, Thm. 8.1.16]).

In this situation there exists a unique holomorphic function
$G:\Omega\to\mathbb{C}$ such that

$\frac{\partial\phi_{t}(z)}{\partial t}=G(\phi_{t}(z)),\,\,\,z\in\Omega\mbox{
and }t\in[0,\infty).$

This function is called the infinitesimal generator of the analytic semiflow
$(\phi_{t})_{t\geq 0}$ on $\Omega$ and we denote by $\mathop{\rm
Gen}\nolimits(\Omega)$ the set of all infinitesimal generator of analytic
semiflows on $\Omega$.

There are several complete characterization of $\mathop{\rm
Gen}\nolimits(\mathbb{D})$ in [16, Chap. 10], which is discussed in more
details in Section 4.2.

Analytic semiflows on $\mathbb{D}$ can be partitioned into two classes,
depending on the localization of their Denjoy–Wolff point $\alpha$, discussed
below in Section 4 (see [1], [24, Chap. 2] and [16, Chap. 8]).

If $\alpha\in\mathbb{D}$, by conjugating by the automorphism $b_{\alpha}$,
where

$b_{\alpha}(z):=\frac{\alpha-z}{1-\overline{\alpha}z},$

we may suppose without loss of generality that $\alpha=0$. In this case there
is a semiflow model

$\phi_{t}(z)=h^{-1}(e^{-ct}h(z)),$

where $c\in\mathbb{C}$ with $\mathop{\rm Re}\nolimits c\geq 0$, and
$h:\mathbb{D}\to\Omega$ is a conformal bijection between $\mathbb{D}$ and a
domain $\Omega\subset\mathbb{C}$, with $h(0)=0$ and $\Omega$ is spiral-like or
star-like (if $c$ is real), in the sense that

$e^{-ct}w\in\Omega\quad\hbox{for all}\quad w\in\Omega\quad\hbox{and}\quad
t\geq 0.$

If $\alpha\in\mathbb{T}$, then there exists a conformal map $h$ from
$\mathbb{D}$ onto a domain $\Omega$ such that $\Omega+it\subset\Omega$ for all
$t\geq 0$, and there is a semiflow model

$\phi_{t}(z)=h^{-1}(h(z)+it).$

###  2.3 Models for analytic flows on $\mathbb{D}$

This subsection relies heavily on Subsection 8.2 in [16].

######  Definition 2.7.

A family $(\phi_{t})_{t\geq 0}$ of analytic selfmaps of $\mathbb{D}$ is a
called a continuous (algebraic) _flow_ if

  * a)

$\phi_{t}$ is an automorphism of $\mathbb{D}$ for all $t\geq 0$;

  * b)

$(\phi_{t})_{t\geq 0}$ is a continuous (algebraic) semiflow.

If $\phi_{t}$ is an automorphism for all $t\geq 0$, we can introduce the
notation $\phi_{-t}:=\phi_{t}^{-1}$ for all $t\geq 0$ and then observe that

  * c)

$\phi_{s+t}=\phi_{s}\circ\phi_{t}$ for all $s,t\in\mathbb{R}$;

  * d)

for all $z\in\mathbb{D}$, the mapping $t\mapsto\phi_{t}(z)$ is continuous on
$\mathbb{R}$ if $(\phi_{t})_{t\geq 0}$ is a continuous flow.

In fact, d) is equivalent to the continuity of $t\mapsto\phi_{t}\in\mathop{\rm
Hol}\nolimits(\mathbb{D})$ on $\mathbb{R}$, where $\mathop{\rm
Hol}\nolimits(\mathbb{D})$ is endowed with the topology of the uniform
convergence on compacta of $\mathbb{D}$.

Here is a characterization of continuous flows in the set of continuous
semiflows [16, Thm. 8.2.4].

######  Theorem 2.8.

Let $(\phi_{t})_{t\geq 0}$ be a continuous (algebraic) semiflow on
$\mathbb{D}$. Then it is a continuous (algebraic) flow if and only if there
exists $t_{0}>0$ such that $\phi_{t_{0}}$ is an automorphism.

The following theorem [16, Thm. 8.2.6] is an explicit description of all the
continuous flows on $\mathbb{D}$.

######  Theorem 2.9.

Let $(\phi_{t})_{t\geq 0}$ be a nontrivial continuous flow on $\mathbb{D}$.
Then $(\phi_{t})_{t\geq 0}$ has one of the following three mutually exclusive
forms:

  * 1)

There exists $\alpha\in\mathbb{D}$ and $w\in\mathbb{R}\setminus\\{0\\}$ such
that

$\phi_{t}(z)=\frac{(e^{-iwt}-|\alpha|^{2})z+\alpha(1-e^{-iwt})}{\overline{\alpha}(e^{-iwt}-1)z+1-|\alpha|^{2}e^{-iwt}},\,\,\,t\geq
0,\,\,\,z\in\mathbb{D}.$

Moreover it is the unique continuous flow of elliptic automorphisms for which
$\phi_{t}(\alpha)=\alpha$ for all $t$ and
$\phi^{\prime}_{t}(\alpha)=e^{-iwt}$.

  * 2)

There exist $\alpha_{1},\alpha_{2}\in\mathbb{T}$, $\alpha_{1}\neq\alpha_{2}$
and $c>0$ such that

$\phi_{t}(z)=\frac{(\alpha_{2}-\alpha_{1}e^{ct})z+\alpha_{1}\alpha_{2}(e^{ct}-1)}{(1-e^{ct})z+\alpha_{2}e^{ct}-\alpha_{1}},\,\,\,t\geq
0,\,\,\,z\in\mathbb{D}.$

Moreover it is the unique continuous flow of hyperbolic automorphisms for
which $\phi_{t}(\alpha_{i})=\alpha_{i}$ for all $t$ ($i=1,2$) and
$\phi^{\prime}_{t}(\alpha_{1})=e^{-ct}$.

  * 3)

There exist $\alpha\in\mathbb{T}$ and $w\in\mathbb{R}\setminus\\{0\\}$ such
that

$\phi_{t}(z)=\frac{(1-iwt)z+iw\alpha t}{-iw\overline{\alpha}tz+1+iwt}.$

Moreover it is the unique continuous flow of parabolic automorphisms for which
$\phi_{t}(\alpha)=\alpha$ for all $t$ and
$\phi^{\prime\prime}_{t}(\alpha)=2itw\overline{\alpha}$.

###  2.4 $C_{0}$-semigroups of composition operators

Clearly a semiflow $(\phi_{t})_{t\geq 0}$ induces a semigroup of composition
operators (on $\mathbb{D}$ these are bounded, by Littlewood’s subordination
theorem [37]), and the following condition gives a way of testing the strong
continuity.

######  Proposition 2.10.

Let $E$ be a dense subspace of a Banach space $X$ and $(T_{t})_{t\geq 0}$ a
semigroup of bounded operators on $X$ such that there exists a $\delta>0$ with
$\sup_{0\leq t\leq\delta}\|T_{t}\|<\infty$. Then $(T_{t})$ is a
$C_{0}$-semigroup on $X$ if and only if $\lim_{t\to 0}\|T_{t}f-f\|_{X}=0$ for
all $f\in E$.

In particular, if the polynomials are dense in $X$ then it is enough to check
that $\lim_{t\to 0}\|T_{t}e_{n}-e_{n}\|_{X}=0$ for all $n=0,1,2,\ldots$, where
$e_{n}(z)=z^{n}$.

###### Proof.

Clearly, the “only if” condition holds. Conversely, if $\lim_{t\to
0}\|T_{t}f-f\|_{X}=0$ for all $f\in E$, let $M:=\sup_{0\leq
t\leq\delta}\|T_{t}\|$, and let $\epsilon>0$ be given and $f\in X$. We may
find a $p\in E$ such that $\|f-p\|<\dfrac{\epsilon}{2(M+1)}$. Then

$\displaystyle\|T_{t}f-f\|$
$\displaystyle\leq\|T_{t}f-T_{t}p\|+\|T_{t}p-p\|+\|p-f\|$ $\displaystyle\leq
M\|f-p\|+\epsilon/2+\|p-f\|<\epsilon$

for sufficently small $t$. ∎

###  2.5 Spaces on which semigroups are not $C_{0}$

In Proposition 2.10 we have seen a sufficient condition for a semiflow to
induce a $C_{0}$-semigroup of composition operators, and in the Hardy,
Dirichlet and Bergman spaces we do indeed arrive at such a semigroup.

Recently, Gallardo-Gutiérrez, Siskakis and Yakubovich [32] have shown that
weighted composition operators $(W_{w_{t},\phi_{t}})_{t\geq 0}$ do not form a
nontrivial $C_{0}$-semigroup on spaces $X$ satisfying $H^{\infty}\subset
X\subset\mathcal{B}$, where $\mathcal{B}$ is the Bloch space. This includes
the case $X=\rm BMOA$. The proof is based on estimates for derivatives of
interpolating Blaschke products.

For composition operators, and with $X=H^{\infty}$ and $\mathcal{B}$, this
result was shown earlier by Blasco et al [15] with an argument involving the
Dunford–Pettis property. For spaces between $H^{\infty}$ and $\mathcal{B}$,
the result for composition operators was given by Anderson, Jovovic and Smith
[2] using geometric function theory.

##  3 Motivation

###  3.1 Universal operators

Rota [45, 46] introduced the concept of a universal operator.

######  Definition 3.1.

An operator $U\in\mathcal{L}(H)$ is universal if for all nonzero
$T\in\mathcal{L}(H)$ there is a closed subspace $\mathcal{M}\neq\\{0\\}$ of
$H$, an isomorphism $J:\mathcal{M}\to H$ and a
$\lambda\in\mathbb{C}\setminus\\{0\\}$ such that
$U\mathcal{M}\subset\mathcal{M}$ and $U_{|\mathcal{M}}=J^{-1}(\lambda T)J$.

That is, a universal operator is a “model” for all $T\in\mathcal{L}(H)$.
Universal operators are of interest in the study of the invariant subspace
problem, whether every operator on a separable infinite-dimensional Hilbert
space has a nontrivial closed invariant subspace. This has a positive solution
if and only every minimal invariant subspace of a given universal operator is
finite-dimensional. We refer to [21] for more details and examples.

Caradus [17] gives a convenient sufficient condition for a Hilbert space
operator to be universal.

######  Theorem 3.2.

Let $U\in\mathcal{L}(H)$ be such that:
(i) $\dim\ker U=\infty$; and
(ii) $U$ is surjective.
Then $U$ is universal.

Consider now the hyperbolic automorphisms

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

with $r\in(-1,1)$. It is helpful to consider them using the parametrization

$\psi_{t}(z):=\phi_{r}(z)\qquad\hbox{with}\quad r=\frac{1-e^{-t}}{1+e^{-t}}$

for $t\in\mathbb{R}$.

It was shown by Nordgren, Rosenthal and Wintrobe [42] that for $r\neq 0$ the
operator $C_{\phi_{r}}-{\rm Id}$ is universal on $H^{2}$. Of course
$C_{\phi_{r}}$ and $C_{\phi_{r}}-{\rm Id}$ have the same invariant subspaces.

A simpler proof using the fact that $C_{\phi_{r}}$ can be embedded in a
$C_{0}$-group was given by Cowen and Gallardo-Gutiérrez [25].

Some work on the invariant subspaces of such operators is due to Matache [39],
Mortini [41] and Gallardo-Gutiérrez and Gorkin [31].

###  3.2 Isometries

Another application of weighted composition operators goes back to Banach
[11], who showed that every surjective isometry $F$ of the space $C(K)$ of
continuous complex functions on a compact metric space $K$ has the form

$F(f)=w(f\circ\phi),$

where $w\in C(K)$ satisfies $|w|=1$ and $\phi$ is a homeomorphism of $K$.

The Hardy space $H^{2}(\mathbb{D})$ is a Hilbert space, and thus has many
(linear) isometries; however for other $p$ with $1<p<\infty$ there are
relatively few, and they are expressible as weighted composition operators. In
[27] deLeeuw, Rudin, and Wermer gave a description of the isometric
surjections of $H^{1}(\mathbb{D})$, which arise from conformal mappings of
$\mathbb{D}$ onto $\mathbb{D}$.

Moreover, Forelli [30] gave the following theorem, which does not assume
surjectivity.

######  Theorem 3.3.

Suppose that $p\neq 2$ and that $T:H^{p}(\mathbb{D})\to H^{p}(\mathbb{D})$ is
a linear isometry. Then there are a non-constant inner function $\phi$ and a
function $F\in H^{p}(\mathbb{D})$ such that

$Tf=F(f\circ\phi)$ (1)

for $f\in H^{p}(\mathbb{D})$.

###  3.3 Change of domain

It is well known that composition operators on the Hardy space
$H^{2}(\mathbb{C}_{+})$ of the right half-plane are unitarily equivalent to
weighted composition operators on $H^{2}(\mathbb{D})$. For example the
following explicit formula is given in [20].

######  Proposition 3.4.

Let $M$ denote the self-inverse bijection from $\mathbb{D}$ onto
$\mathbb{C}_{+}$ given by $M(z)=\dfrac{1-z}{1+z}$, and let
$\Psi:\mathbb{C}_{+}\to\mathbb{C}_{+}$ be holomorphic. Then the composition
operator $C_{\Psi}$ on $H^{2}(\mathbb{C}_{+})$ is unitarily equivalent to the
operator $L_{\Phi}:H^{2}(\mathbb{D})\to H^{2}(\mathbb{D})$ defined by

$L_{\Phi}f(z)=\frac{1+\Phi(z)}{1+z}f(\Phi(z)),$

where $\Phi=M\circ\Psi\circ M$.

So for example the $C_{0}$-group $(T_{t})_{t\in\mathbb{R}}$ on
$H^{2}(\mathbb{C}_{+})$ given by $T_{t}g(z)=g(e^{t}z)$ $(z\in\mathbb{C}_{+})$
for $g\in H^{2}(\mathbb{C}_{+})$ is unitarily equivalent to the weighted
composition group $(S_{t})_{t\in\mathbb{R}}$ given by

$S_{t}f(z)=\frac{2}{1+z+e^{t}(1-z)}f\left(\frac{1+z-e^{t}(1-z)}{1+z+e^{t}(1-z)}\right).$

Formulae for general domains are given, for example, in [35, Prop. 2.1]. If
$W$ is a weighted composition operator between two Hardy–Smirnoff spaces
$E^{2}(\Omega_{1})$ and $E^{2}(\Omega_{2})$, with $\Omega_{1}$ and
$\Omega_{2}$ conformally equivalent to the disc $\mathbb{D}$, then $W$ is
unitarily equivalent to a weighted composition operator on
$H^{2}(\mathbb{D})$. Similar formulae are given for Bergman spaces.

##  4 Asymptotic behaviour of $T^{n}$ or $T_{t}$

###  4.1 The discrete unweighted case

For a fixed composition operator $C_{\phi}$ there are several possible modes
of convergence for the sequence $(C^{n}_{\phi})_{n\geq 1}$, some of which we
now list in progressively weaker order.

  * •

Norm convergence. There exists an operator $P\in\mathcal{L}(X)$ such that
$\|C_{\phi}^{n}-P\|\to 0$.

  * •

Strong convergence. There exists an operator $P\in\mathcal{L}(X)$ such that
$\|C_{\phi}^{n}x-Px\|\to 0$ for all $x\in X$.

  * •

Weak convergence. There exists an operator $P\in\mathcal{L}(X)$ such that
$C_{\phi}^{n}x\to Px$ weakly for all $x\in X$.

In each case $P$ is the projection onto $\mathop{\rm
fix}\nolimits(C_{\phi}):=\\{x\in X:C_{\phi}x=x\\}$ along the decomposition

$X=\mathop{\rm fix}\nolimits(C_{\phi})\oplus\overline{\mathop{\rm
Im}\nolimits({\rm Id}-C_{\phi})}.$

The following theorem from [7] helps with the analysis.

######  Theorem 4.1.

Let $T\in\mathcal{L}(X)$ with $\sup_{n}\|T^{n}\|<\infty$. Then the following
are equivalent.
(i) $P:=\lim T^{n}$ exists and $P$ is a finite-rank operator.
(ii) (a) The essential spectral radius $r_{e}(T)$ satisfies $r_{e}(T)<1$;
(b) $\sigma_{p}(T)\cap\mathbb{T}\subset\\{1\\}$; and
(c) if $1\in\sigma(T)$ then $1$ is a pole of the resolvent $(z{\rm
Id}-T)^{-1}$ of order 1.

In this case $P$ is the residue at 1.

We sketch the proof.

###### Proof.

(i) $\implies$ (ii):
Let $X_{1}=PX$, $X_{2}=({\rm Id}-P)X$ and $T_{i}=T_{X_{i}}$ for $i=1,2$.

Then $\|T_{2}^{n}\|_{\mathcal{L}(X_{2})}\to 0$ as $n\to\infty$, and hence
$r(T_{2})<1$.

Since $\sigma(Y)=\sigma(T_{1})\cup\sigma(T_{2})=\\{1\\}\cup\sigma(T_{2})$ and

$(\lambda{\rm Id}-T)^{-1}=\begin{cases}\frac{1}{\lambda-1}&\hbox{on }X_{1},\\\
(\lambda{\rm Id}-T_{2})^{-1}&\hbox{on }X_{2},\end{cases}$

we see that (ii) follows.

(ii) $\implies$ (i):
Let $P$ be the residue at $1$, and let $X_{1}=PX$, $X_{2}=({\rm Id}-P)X$ and
$T_{i}=T_{X_{i}}$ for $i=1,2$.

Then $\sigma(T_{1})=\\{1\\}$ and $\sigma(T_{2})=\sigma(T)\setminus\\{1\\}$ by
(a) and (b).

Thus $r(T_{2})<1$ and so $\|T_{2}^{n}\|_{\mathcal{L}(X_{2})}\to 0$.

It follows from (c) that $T_{1}$ is diagonalisable and thus $T_{1}={\rm Id}$.
∎

Recall that $X\hookrightarrow\mathop{\rm Hol}\nolimits(\mathbb{D})$ means that
$X$ embeds continuously in $\mathop{\rm Hol}\nolimits(\mathbb{D})$, which
means that, for all $\lambda\in\mathbb{D}$, $\delta_{\lambda}:f\mapsto
f(\lambda)$ from $X$ to $\mathbb{C}$ is bounded.

Arendt and Batty [3] have given criteria for strong convergence. More
recently, from [5] we mention the following theorem.

######  Theorem 4.2.

Let $\phi:\mathbb{D}\to\mathbb{D}$ be holomorphic,
$C_{\phi}\in\mathcal{L}(X)$, where $X$ is a Banach space such that
$X\hookrightarrow\mathop{\rm Hol}\nolimits(\mathbb{D})$. Then
$(C_{\phi}^{n})_{n}$ converges uniformly if and only if $r_{e}(C_{\phi})<1$.

So it remains to study the essential spectral radius of $C_{\phi}$. To this
end we write $\phi^{(n)}=\underbrace{\phi\circ\ldots\phi}_{\hbox{ $n$
factors}}$.

#### Denjoy–Wolff theory

A mapping $\phi:\mathbb{D}\to\mathbb{D}$ is an elliptic automorphism if it has
the form

$\phi=\psi_{a}\circ R_{\theta}\circ\psi_{a},$

where $a\in\mathbb{D}$ and
$\psi_{a}(z)=\dfrac{a-z}{1-\overline{a}z}=\psi_{a}^{-1}(z)$, and
$R_{\theta}(z)=e^{i\theta}z$ with $\theta\in\mathbb{R}$.

Then the classical Denjoy–Wolff theorem states that provided $\phi$ is not an
elliptic automorphism, the sequence $(\phi^{(n)})_{n}$ converges uniformly to
some $\alpha\in\overline{\mathbb{D}}$ on each compact subset of $\mathbb{D}$.

Such an $\alpha$ is called the Denjoy–Wolff point of $\phi$ and is sometimes
denoted by $\mathop{\rm DW}\nolimits(\phi)$.

Case 1. $|\alpha|=1$.

######  Theorem 4.3.

Suppose that $\mathbb{C}[z]\subset X$ and suppose that each $L\in X^{\prime}$
with $L(e_{n})=L(e_{1})^{n}$ has the form $L(f)=\delta_{z_{0}}(f):=f(z_{0})$
for some $z_{0}\in\mathbb{D}$. Then $\sup_{n}\|C_{\phi}^{n}\|=\infty$.
Therefore even weak convergence of $(C_{\phi}^{n})$ cannot occur.

###### Proof.

If $\sup_{n}\|C_{\phi}^{n}\|=M<\infty$ then

$|f(\alpha)|=\lim_{n\to\infty}|f(\phi^{(n)}(0))|\leq\|\delta_{0}\|M\|f\|,$

so $\delta_{\alpha}=\delta_{z_{0}}$ for some $z_{0}\in\mathbb{D}$, which is
absurd. ∎

Case 2. $|\alpha|<1$.

The natural candidate for $P$ is given by $Pf=f(\alpha)=\delta_{\alpha}(f){\bf
1}$, a rank-one operator.

######  Theorem 4.4.

For composition operators on $H^{p}(\mathbb{D})$, with $1\leq p<\infty$, the
sequence $(C_{\phi}^{n})$ converges uniformly and strongly if and only if
$|\alpha|<1$ and $\phi$ is not inner. It converges weakly if and only if
$|\alpha|<1$.

We proved that $r_{e}(C_{\phi})<1$ on $H^{2}$ if and only if $\phi$ is not
inner and $|\alpha|<1$. When $\phi$ is inner and $|\alpha|<1$ then $C_{\phi}$
is similar to an isometry (this is a necessary and sufficient condition [12]).
We deduce the result for $H^{p}(\mathbb{D})$ from a theorem of Shapiro [48],
namely $r_{e,H^{2}}(C_{\phi}))^{2}=r_{e,H^{p}}(C_{\phi}))^{p}$. For $p=1$
there is the inequality “$>$”. Hence $r_{e,H^{2}}(C_{\phi}))<1$ implies that
$r_{e,H^{p}}(C_{\phi}))<1$.

We may summarise the results obtained on various Banach spaces in a table [5,
6].

Space |  Uniform |  Strong |  Weak
---|---|---|---
$H^{p}(\mathbb{D})$ |  $\phi$ not inner, $|\alpha|<1$ |  $\phi$ not inner, $|\alpha|<1$ |  $|\alpha|<1$
$A^{p}_{\beta}(\mathbb{D})$, $\beta>-1$, $1\leq p<\infty$ |  $|\alpha|<1$ |  $|\alpha|<1$ |  $|\alpha|<1$
$\mathcal{B}_{0},\mathcal{B}^{\gamma}$ |  $|\alpha|<1$ |  $|\alpha|<1$ |  $|\alpha|<1$
$H_{\nu_{p}}^{\infty}$, $0<p<\infty$ |  $|\alpha|<1$ |  $|\alpha|<1$ |  $|\alpha|<1$

We write $dA$ for the normalized Lebesgue area measure on $\mathbb{D}$, i.e.
$dA(re^{i\theta})=\frac{1}{\pi}rdrd\theta$. The _standard weighted Bergman
space_ , $A_{\beta}^{p}(\mathbb{D})$, $\beta\geq-1$, $p\geq 1$ is the space of
all holomorphic functions $f:\mathbb{D}\to\mathbb{C}$ such that

$\int_{\mathbb{D}}|f(z)|^{p}(1-|z|^{2})^{\beta}dA(z)<\infty.$

Every $A_{\beta}^{p}$ is a Banach space when $1\leq p<\infty$ with norm the
$p^{th}$ root of above integral, denoted by $\|f\|_{A_{\beta}^{p}}$.
The _unweighted Bergman space_ , $A^{p}$ is obtained when $\beta=0$.
The _standard Hardy space_ $H^{p}(\mathbb{D})$ are obtained when $\beta=-1$.

Here $\mathcal{B}_{0}$ and $\mathcal{B}_{\gamma}$ are variations on the Bloch
space, namely,

$\mathcal{B}_{0}=\\{f\in\mathop{\rm Hol}\nolimits(\mathbb{D}):\lim_{|z|\to
1}(1-|z|^{2})|f^{\prime}(z)|=0\\}$

and

$\mathcal{B}^{\gamma}=\\{f\in\mathop{\rm
Hol}\nolimits(\mathbb{D}):\sup_{z\in\mathbb{D}}(1-|z|^{2})^{\gamma}|f^{\prime}(z)|<\infty\\}.$

For $p>0$, the _standard weighted Bergman space of infinite order_ ,
$H_{\nu_{p}}^{\infty}(\mathbb{D})$ (or $H_{\nu_{p}}^{\infty}$), is the Banach
space of all holomorphic functions $f:\mathbb{D}\to\mathbb{C}$ such that

$\|f\|_{H_{\nu_{p}}^{\infty}}:=\sup_{z\in\mathbb{D}}\nu_{p}(z)|f(z)|<\infty,$

with the norm as defined above, where $\nu_{p}(z)=(1-|z|^{2})^{p}$.

###  4.2 The continuous unweighted case

We now address the question whether we can deduce the asymptotic behaviour of
a $C_{0}$ semigroup $(C_{\phi_{t}})_{t\geq 0}$ from properties of its
generator.

As we shall explain in more detail below, for $X\hookrightarrow\mathop{\rm
Hol}\nolimits(\mathbb{D})$ with $(C_{\phi_{t}})_{t\geq 0}$ a $C_{0}$-semigroup
induced by an analytic semiflow $(\phi_{t})_{t\geq 0}$ with

$G(z)=\lim_{t\to 0}\dfrac{\phi_{t}(z)-z}{t},$

the generator $A$ is given by $Af=Gf^{\prime}$ ($f\in D(A)$) with dense domain
$D(A)=\\{f\in X:Gf^{\prime}\in X\\}$.

Various well-known properties of analytic semiflows are the following, which
can be found in the recent book [16].

  1. 1.

For all $t\geq 0$, $\phi_{t}$ is injective.

  2. 2.

If there is a $t_{0}>0$ such that $\phi_{t_{0}}$ is an (elliptic)
automorphism, then $\phi_{t}$ is an elliptic automorphism for all $t>0$.

  3. 3.

For all semiflows that are not elliptic automorphisms, there is a unique
$\alpha\in\overline{\mathbb{D}}$ such that $\lim_{t\to
0}\|\phi_{t}-\alpha\|_{\infty,K}=0$ for all compact subsets
$K\subset\mathbb{D}$. Such an $\alpha$ is called the Denjoy–Wolff point of
$(\phi_{t})_{t\geq 0}$ (see also Section 4.1).

The following classical theorem of Berkson and Porta [13] describes a
semigroup in terms of its generator.

######  Theorem 4.5.

let $(\phi_{t})_{t\geq 0}$ be an analytic semiflow on $\mathbb{D}$. Then the
generator

$G(z):=\lim_{t\to 0}\dfrac{\phi_{t}(z)-z}{t}$

exists for all $z\in\mathbb{D}$. Also $G\in\mathop{\rm
Hol}\nolimits(\mathbb{D})$ and

$G(z)=(\alpha-z)(1-\overline{\alpha}z)F(z)$ (2)

where $F\in\mathop{\rm Hol}\nolimits(\mathbb{D})$ and $\mathop{\rm
Re}\nolimits F\geq 0$ (this implies that
$F\in\bigcap_{0<p<1}H^{p}(\mathbb{D})$ and hence has radial limits almost
everywhere on $\mathbb{T}$). Reciprocally, any $G$ of the form (2) is the
generator of a semiflow.

Another characterization was given in [9, Thm. 3.9]. If one knows a priori
that $G\in H^{1}(\mathbb{D})$, the necessary and sufficient condition is that
$\mathop{\rm Re}\nolimits(\overline{z}G^{*}(z))\leq 0$ a.e. on $\mathbb{T}$,
where $G^{*}$ denotes the radial limit of $G$. The proof relies on the
following observation which can be seen as a maximum principle for harmonic
functions. If $h:\mathbb{D}\to\mathbb{C}$ is in $H^{1}(\mathbb{D})$, then
$h(z)=\frac{1}{2\pi}\int_{0}^{2\pi}\mathop{\rm
Re}\nolimits\left(\frac{e^{it}+z}{e^{it}-z}\right)f^{*}(e^{it})dt$, where
$h^{*}$ stand for the radial limit of $h$, and thefore if $\mathop{\rm
Re}\nolimits(h^{*}(e^{it}))\leq 0$ a.e. on $\mathbb{T}$, then $\mathop{\rm
Re}\nolimits(h(z))\leq 0$ for all $z\in\mathbb{D}$.

We have already mentioned that $F$ and thus $G$ are always in
$H^{p}(\mathbb{D})$ for $0<p<1$, which implies the existence of $G^{*}$ in
$L^{p}(\mathbb{T})$. Nevertheless it is not possible to improve [9, Thm. 3.9]
since $G(z):=-z\left(\frac{z-1}{z+1}\right)$ is in $H^{p}(\mathbb{D})$ for all
$p<1$, satisfies $\mathop{\rm Re}\nolimits(\overline{z}G^{*}(z))\leq 0$ a.e.
on $\mathbb{T}$ and is not the generator of a semiflow using (2).

Combining [16, Thm. 10.2.6] which is called “Abate’s formula” and [16, Thm.
10.2.10], we obtain the following characterization of generators.

######  Theorem 4.6.

Let $G:\mathbb{D}\to\mathbb{C}$ be holomorphic function. The following
assertions are equivalent:

  * (i)

$G$ is the generator of an analytic semiflow on $\mathbb{D}$;

  * (ii)

$\mathop{\rm Re}\nolimits(2\overline{z}G(z)+(1-|z|^{2})G^{\prime}(z))\leq 0$
for all $z\in\mathbb{D}$;

  * (iii)

$\limsup_{z\in\mathbb{D},z\to\xi}\mathop{\rm
Re}\nolimits(\overline{z}G(z))\leq 0$ for all $\xi\in\mathbb{T}$.

Now, from results on $H^{p}(\mathbb{D})$ and similar spaces in Section 4.1 we
obtain the following result.

######  Theorem 4.7.

Let $X=H^{p}(\mathbb{D})$ or other classical Banach spaces (e.g. Bergman,
Dirichlet). Let $(C_{\phi_{t}})_{t\geq 0}$ be a $C_{0}$-semigroup of
composition operators on $X$ with generator $A:f\mapsto Gf^{\prime}$. Then the
semigroup converges uniformly if and only if it converges strongly, and this
is if and only if $G$ has a zero in $\mathbb{D}$, and there exists
$\xi\in\mathbb{T}$ such that $\limsup_{z\in\mathbb{D},z\to\xi}\mathop{\rm
Re}\nolimits(\overline{z}G(z))<0$.

Indeed, $G$ has a zero in $\mathbb{D}$ if and only if either $\phi_{t}$ is an
elliptic automorphism for each $t$ or else the Denjoy–Wolff point $\alpha$
lies in $\mathbb{D}$. The condition
$\limsup_{z\in\mathbb{D},z\to\xi}\mathop{\rm Re}\nolimits(\overline{z}G(z))<0$
implies that $(\phi_{t})_{t\geq 0}$ is not a semiflow of automorphisms. For if
$(\phi_{t})_{t\geq 0}$ were a semigroup of automorphisms, then both $G$ and
$-G$ would generate semigroups and so we would have
$\lim_{z\in\mathbb{D},z\to\xi}\mathop{\rm Re}\nolimits(\overline{z}G(z))=0$
everywhere on $\mathbb{T}$. Note that since $\phi_{t}$ is injective, it is not
inner precisely when it is not an automorphism.

###  4.3 Weighted composition operators

Iterates of weighted composition operators on holomorphic function spaces $X$
that embed continuously in $\mathop{\rm Hol}\nolimits(\mathbb{D})$ are studied
in [22]. Some of the most relevant theorems here are the following, where
again $W_{w,\phi}$ is the weighted composition operator $f\mapsto
w(f\circ\phi)$. The methods of proof are similar to those of [5]. We assume
first that $\phi$ is not an elliptic automorphism and that $W_{w,\phi}$ is at
least power bounded.

######  Theorem 4.8.

With $X$ and $W_{w,\phi}$ as above, suppose also that $\alpha:=\mathop{\rm
DW}\nolimits(\phi)\in\mathbb{D}$. Then the sequence $(W_{w,\phi}^{n})$
converges weakly as $n\to\infty$ if and only if (i) $|w(\alpha)|<1$, or (ii)
$|w(\alpha)|=1$ and $\sup_{n}\|W_{w,\phi}^{n}\|<\infty$.

######  Theorem 4.9.

Under the hypotheses of Theorem 4.8, suppose that either $|w(\alpha)|<1$, or
(ii) $|w(\alpha)|=1$ and $\sup_{n}\|W_{w,\phi}^{n}\|<\infty$. Then
$(W_{w,\phi}^{n})$ converges uniformly if and only if $r_{e}(W_{w,\phi})<1$.

For elliptic automorphisms the story is slightly different.

######  Theorem 4.10.

Let $\phi$ be an elliptic automorphism of infinite order with fixed point
$\alpha\in\mathbb{D}$ and $w\in A(\mathbb{D})$ bounded away from 0 on
$\mathbb{D}$. Suppose that $\sup_{n}\|W_{w,\phi}^{n}\|<\infty$. Then
$(W_{w,\phi}^{n})$ converges uniformly if and only if $|w(\alpha)|<1$.

###  4.4 Isometry and similarity to isometry

As a byproduct of the results in Section 4.1 we have characterizations of the
properties isometry and similarity to an isometry for composition operators
[5, 6].

Spaces |  $C_{\varphi}$ isometric |  $C_{\varphi}$ similar to an isometry
---|---|---
$H^{p}(\mathbb{D}),1\leq p<\infty$ |  $\varphi$ inner and $\varphi(0)=0$ |  $\varphi$ inner and $\exists b\in\mathbb{D}$ with $\varphi(b)=b$
$A^{p}_{\beta}(\mathbb{D}),\beta>-1,1\leq p<\infty$ |  $\varphi$ rotation |  $\varphi$ elliptic automorphism
$\mathcal{B}$ |  $\varphi(0)=0$ and $\tau_{\varphi}^{\infty}=1$ |  $\exists b\in\mathbb{D}$ with $\varphi(b)=b$ and $\tau_{\varphi}^{\infty}=1$
$\mathcal{B}_{0}$ or $\mathcal{B}^{\alpha}$, $\alpha\neq 1$ |  $\varphi$ rotation |  $\varphi$ elliptic automorphism
$H^{\infty}_{\nu_{p}},0<p<\infty$ |  $\varphi$ rotation |  $\varphi$ elliptic automorphism

Here

$\tau_{\phi}^{\infty}:=\sup_{z\in\mathbb{D}}\frac{1-|z|^{2}}{1-|\phi(z)|^{2}}|\phi^{\prime}(z)|.$

###  4.5 Generators

Suppose now that $(T_{t})_{t\geq 0}$ is a $C_{0}$-semigroup of weighted
composition operators $T_{t}=w_{t}C_{\phi_{t}}$ acting on a suitable space of
functions on a domain $\Omega$. We have already discussed results describing
generators in the unweighted case in Theorems 4.5 and 4.6, namely
$Af=Gf^{\prime}$, where

$G(z):=\lim_{t\to 0}\dfrac{\phi_{t}(z)-z}{t}.$ (3)

The converse, as shown in work by Arendt and Chalendar [4] and Gallardo and
Yakubovich [33], associates with a holomorphic function $G$ a semiflow
satisfying $u^{\prime}(t)=G(u(t))$, with $u(0)=z$.

######  Theorem 4.11.

Let $X$ be a function space with continuous point evaluations on a domain
$\Omega$. If either
(i) If $(z_{n})\subset\Omega$ such that $z_{n}\to z\in\Omega\cup\\{\infty\\}$
and $\lim_{n\to\infty}f(z_{n})$ exists in $\mathbb{C}$ for all $f\in X$ then
$z\in\Omega$, or
(ii) $\Omega=\mathbb{D}$ and $\mathop{\rm
Hol}\nolimits(\overline{\mathbb{D}})\subset X\subset\mathop{\rm
Hol}\nolimits(\mathbb{D})$ with continuous embeddings,
then the semigroup associated with $G$ is a (quasicontractive) semigroup of
composition operators.

Note that $\mathop{\rm Hol}\nolimits(\overline{\mathbb{D}})$ embeds
continuously in $X$ means that for each $\varepsilon>0$, there exists a
positive constant $c(\varepsilon)$ such that

$\|f\|_{X}\leq c(\epsilon)\sup_{|z|\leq 1+\epsilon}|f(z)|,\,\,f\in
A(D(0,1+\varepsilon)),$

where $A(D(0,1+\varepsilon))$ is the algebra of all holomorphic functions in
$D(0,1+\varepsilon)$, the open disc centered at $0$ and of radius
$1+\varepsilon$, and continuous on the closure of $D(0,1+\varepsilon)$.

For weighted composition operators on $\mathbb{D}$, a semigroup of the form
$T_{t}=w_{t}C_{\phi_{t}}$ has infinitesimal generator $Af=Gf^{\prime}+gf$,
where now

$g(z)=\frac{\partial w_{t}(z)}{\partial t}_{t=0}$

and $G$ is once again given by (3). Assuming Condition (ii) in Theorem 4.11,
it is shown in [32, Thm. 3.1] that every semigroup with generator of the form
$Af=Gf^{\prime}+gf$ is a semigroup of weighted composition operators. See also
[14] for a generalization of [4].

##  5 Compactness and analyticity

This section is mainly extracted from [10]. We restrict our study to the
semigroups of composition operators on the Hardy space $H^{2}(\mathbb{D})$,
even though some of the results can be extended to non-Hilbertian Hardy spaces
or weighted Hardy spaces such as the Dirichlet space.

###  5.1 Immediate and eventual compactness

We recall that a semigroup $(T(t))_{t\geq 0}$ is said to be immediately
compact if the operators $T(t)$ are compact for all $t>0$. A semigroup
$(T(t))_{t\geq 0}$ is said to be eventually compact if there exists $t_{0}>0$
such that $T(t)$ is compact for all $t\geq t_{0}$. Similar definitions hold
for immediately/eventually trace-class.

The following theorem [43, Chap. 2, Thm 3.3] links immediate compactness with
continuity in norm.

######  Theorem 5.1.

Let $(T(t))_{t\geq 0}$ be a $C_{0}$-semigroup and let $A$ be its infinitesimal
generator. Then $(T(t))_{t\geq 0}$ is immediately compact if and only if
(i) the resolvent $R(\lambda,A)$ is compact for all (or for one)
$\lambda\in\mathbb{C}\setminus\sigma(A)$, and
(ii) $\lim_{s\to t}\|T(s)-T(t)\|=0$ for all $t>0$.

We begin with an elementary observation.

######  Proposition 5.2.

Suppose that for some $t_{0}>0$ one has $|\phi_{t_{0}}(\xi)|=1$ on a set of
positive measure; then $C_{\phi_{t_{0}}}$ is not compact on
$H^{2}(\mathbb{D})$ and so the semigroup $(C_{\phi_{t}})_{t\geq 0}$ is not
immediately compact.

###### Proof.

For the Hardy space, this follows since the weakly null sequence
$(e_{n})_{n\geq 0}$ with $e_{n}(z)=z^{n}$ is mapped into $(\phi_{t_{0}}^{n})$,
which does not converge to $0$ in norm. ∎

We shall now give a sufficient condition for immediate compactness of a
semigroup of composition operators, in terms of the associated function $G$.
First, we recall a classical necessary and sufficient condition for
compactness of a composition operator $C_{\phi}$ in the case when $\phi$ is
univalent [24, pp. 132, 139].

######  Theorem 5.3.

For $\phi:\mathbb{D}\to\mathbb{D}$ analytic and univalent, the composition
operator $C_{\phi}$ is compact on $H^{2}(\mathbb{D})$ if and only if

$\lim_{z\to\xi}\frac{1-|z|^{2}}{1-|\phi(z)|^{2}}=0$

for all $\xi\in\mathbb{T}$.

The following proposition collects together standard results on trace-class
composition operators [24, p. 149].

######  Proposition 5.4.

For $\phi:\mathbb{D}\to\mathbb{D}$ analytic with $\|\phi\|_{\infty}<1$, the
composition operator $C_{\phi}$ is trace-class on $H^{2}(\mathbb{D})$.

Here is an example showing that immediate and eventual compactness are
different even for semigroups of composition operators on $H^{2}(\mathbb{D})$.

######  Example 5.5.

Let $h$ be the Riemann map from $\mathbb{D}$ onto the starlike region

$\Omega:=\mathbb{D}\cup\\{z\in\mathbb{C}:0<\mathop{\rm Re}\nolimits(z)<2\mbox{
and }0<\mathop{\rm Im}\nolimits(z)<1\\},$

with $h(0)=0$. Since $\partial\Omega$ is a Jordan curve, the Carathéodory
theorem [44, Thm 2.6, p. 24] implies that $h$ extends continuously to
$\mathbb{T}$.

Let $\phi_{t}(z)=h^{-1}(e^{-t}h(z))$. Note that for $0<t<\log 2$,
$\phi_{t}(\mathbb{T})$ intersects $\mathbb{T}$ on a set of positive measure,
and thus, $C_{\phi_{t}}$ is not compact by Proposition 5.2. Moreover, for
$t>\log 2$, $\|\phi_{t}\|_{\infty}<1$, and therefore $C_{\phi_{t}}$ is compact
(actually trace-class). Figure 1 represents the image of $\varphi_{t}$ for
different values of $t$.

Figure 1.

###  5.2 Compact analytic semigroups

A $C_{0}$-semigroup $T$ will be called analytic (or holomorphic) if there
exists a sector
$\Sigma_{\theta}=\\{re^{i\alpha},r\in\mathbb{R}_{+},|\alpha|<\theta\\}$ with
$\theta\in(0,\frac{\pi}{2}]$ and an analytic mapping
$\widetilde{T}:\Sigma_{\theta}\to\mathcal{L}(X)$ such that $\widetilde{T}$ is
an extension of $T$ and

$\sup_{\xi\in\Sigma_{\theta}\cap\mathbb{D}}\|\widetilde{T}(\xi)\|<\infty.$

In both cases, the generator of $T$ (or $\widetilde{T}$) will be the linear
operator $A$ defined by

$D(A)=\left\\{x\in X,\lim_{\mathbb{R}\ni t\to 0}\frac{T(t)x-x}{t}\text{
exists}\right\\}$

and, for all $x\in D(A)$,

$Ax=\lim_{\mathbb{R}\ni t\to 0}\frac{T(t)x-x}{t}.$

In the particular case of analytic semigroups, the compactness is equivalent
to the compactness of the resolvent, by Theorem 5.1, since the analyticity
implies the uniform continuity [29, p. 109].

######  Remark 5.6.

For an analytic semigroup $(T(t))_{t\geq 0}$, being eventually compact is
equivalent to being immediately compact. Indeed, consider $Q$, the quotient
map from the bounded linear operators on a Hilbert space ${\mathcal{L}}(H)$
onto the Calkin algebra (the quotient of ${\mathcal{L}}(H)$ by the compact
operators). Then $(QT(t))_{t\geq 0}$ is an analytic semigroup which vanishes
for $t>0$ large enough, and therefore vanishes identically (this observation
is attributed to W. Arendt).

We may include that remark in the following result.

######  Theorem 5.7.

Let $G:\mathbb{D}\to\mathbb{C}$ be a holomorphic function such that the
operator $A$ defined by $Af(z)=G(z)f^{\prime}(z)$ with dense domain
$D(A)\subset H^{2}(\mathbb{D})$ generates an analytic semigroup $(T(t))_{t\geq
0}$ of composition operators. Then the following assertions are equivalent:

  1. 1.

$(T(t))_{t\geq 0}$ is immediately compact;

  2. 2.

$(T(t))_{t\geq 0}$ is eventually compact;

  3. 3.

$\forall\xi\in\mathbb{T}$,
$\lim_{z\in\mathbb{D},z\to\xi}\left|\frac{G(z)}{z-\xi}\right|=\infty$.

######  Theorem 5.8.

Let $G:\mathbb{D}\to\mathbb{C}$ be a holomorphic function such that the
operator $A$ defined by $Af(z)=G(z)f^{\prime}(z)$ has dense domain
$D(A)\subset H^{2}(\mathbb{D})$. The operator $A$ generates an analytic
semigroup of composition operators on $H^{2}(\mathbb{D})$ if and only if there
exists $\theta\in(0,\frac{\pi}{2})$ such that for all
$\alpha\in\\{-\theta,0,\theta\\}$

$\limsup_{z\in\mathbb{D},z\to\xi}\mathop{\rm
Re}\nolimits(e^{i\alpha}\overline{z}G(z))\leq 0\mbox{ for all
}\xi\in\mathbb{T}.$

Using the semiflow model, we have the following result.

######  Theorem 5.9.

Let $(C_{\phi_{t}})_{t\geq 0}$ be an immediately compact analytic semigroup on
$H^{2}(\mathbb{D})$. Then the following conditions are equivalent:
1\. There exists a $t_{0}>0$ such that $\|\phi_{t_{0}}\|_{\infty}<1$;
2\. For all $t>0$ one has $\|\phi_{t}\|_{\infty}<1$.
Therefore, if there exists a $t_{0}>0$ such that
$\|\phi_{t_{0}}\|_{\infty}<1$, then $(C_{\phi_{t}})_{t\geq 0}$ is immediately
trace-class.

It is of interest to consider the relation between immediate compactness and
analyticity for a $C_{0}$-semigroup of composition operators: this is because
compactness of a semigroup $(T(t))_{t\geq 0}$ is implied by compactness of the
resolvent together with norm-continuity at all points $t>0$, as in Theorem
5.1.

######  Example 5.10.

Consider

$G(z)=\frac{2z}{z-1},$

Now the image of the unit disc under $\overline{z}G(z)$ is contained in the
left half-plane so the operator $A:f\mapsto Gf^{\prime}$ generates a non-
analytic $C_{0}$-semigroup of composition operators $(C_{\phi_{t}})_{t\geq 0}$
on $H^{2}(\mathbb{D})$. On the other hand, it can be shown that $C_{\phi_{t}}$
is compact – even trace-class – for each $t>0$. For we have the equation

$\phi_{t}(z)e^{-\phi_{t}(z)}=e^{-2t}ze^{-z}.$

Now the function $z\mapsto ze^{-z}$ is injective on $\overline{\mathbb{D}}$;
this follows from the argument principle, for the image of $\mathbb{T}$ is
easily seen to be a simple Jordan curve. It follows that
$\|\phi_{t}\|_{\infty}<1$ for all $t>0$, and so $C_{\phi_{t}}$ is trace-class.

######  Example 5.11.

The semigroup corresponding to $G(z)=(1-z)^{2}$ is analytic but not
immediately compact. For

$\phi_{t}(z)=\frac{(1-t)z+t}{-tz+1+t}$

the Denjoy–Wolff point is $1$, so the semigroup cannot be immediately compact.

The analyticity follows on calculating $\overline{z}G(z)$ for $z=e^{i\theta}$.
We obtain $-4\sin^{2}(\theta/2)$, which gives the result by Theorem 5.8.

##  6 An outlook on $\mathbb{C}_{+}$ and $\mathbb{C}$

###  6.1 The right halfplane $\mathbb{C}_{+}$

Unlike in the case of the disc, not all composition operators on
$H^{2}(\mathbb{C}+)$ are bounded, and there are no compact composition
operators.

The following theorem was given by Elliott and Jury [28] (see also [40]).
Recall that the angular derivative $\phi^{\prime}(\infty)$ of a self-map of
$\mathbb{C}_{+}$ is defined by

$\phi^{\prime}(\infty)=\lim_{z\to\infty}\frac{z}{\phi(z)}$

######  Theorem 6.1.

Let $\phi:\mathbb{C}_{+}\to\mathbb{C}_{+}$ be holomorphic. The composition
operator $C_{\phi}$ is bounded on $H^{2}(\mathbb{C}_{+})$ if and only if
$\phi$ has finite angular derivative $0<\lambda<\infty$ at infinity, in which
case $\|C_{\phi}\|=\sqrt{\lambda}$. We also have for the essential norm that
$\|C_{\phi}\|_{e}=\|C_{\phi}\|$ so that there are no compact composition
operators on $H^{2}(\mathbb{C}_{+})$.

Berkson and Porta [13] gave the following criterion for an analytic funcction
$G$ to generate a one-parameter semiflow of analytic mappings from
$\mathbb{C}_{+}$ into itself, namely, solutions to the initial value problem

$\frac{\partial\phi_{t}(z)}{\partial t}=G(\phi_{t}(z)),\qquad\phi_{0}(z)=z,$

namely the condition

$x\frac{\partial(\mathop{\rm Re}\nolimits G)}{\partial x}\leq\mathop{\rm
Re}\nolimits G\qquad\hbox{on}\quad\mathbb{C}_{+},$

where as usual $x=\mathop{\rm Re}\nolimits z$. In this case the semigroup with
generator $A:f\mapsto Gf^{\prime}$ consists of composition operators.

Arvanitidis [8] showed that a necessary and sufficient condition for these
composition operators to be bounded is that the non-tangential limit
$\delta:=\angle\lim_{z\to\infty}G(z)/z$ exists, in which case
$\|C_{\phi_{t}}\|=e^{-\delta t/2}$ and the semigroup is quasicontractive.

In [10] it was shown that for $G$ holomorphic on $\mathbb{C}_{+}$ a necessary
condition for the operator $A:f\mapsto Gf^{\prime}$ to generate a
quasicontractive semigroup is

$\inf_{z\in\mathbb{C}_{+}}\frac{\mathop{\rm Re}\nolimits G(z)}{\mathop{\rm
Re}\nolimits z}>-\infty,$

and for a contractive semigroup this infimum is non-negative.

###  6.2 The complex plane $\mathbb{C}$

For $\mathbb{C}$ we present material on the Fock space from [23]. The Fock
space is arguably one of the most important spaces of entire functions, and it
is possible to give a complete answer to several questions involving
(weighted) composition operators.

For $1\leq\nu<\infty$ the Fock space $\mathcal{F}^{\nu}$ is defined to be the
space of entire functions $f$ on $\mathbb{C}$ such that the norm

$\|f\|_{\nu}:=\left(\frac{\nu}{2\pi}\int_{\mathbb{C}}|f(z)|^{\nu}e^{-\nu
z^{2}/2}\,dm(z)\right)^{1/\nu}$

is finite. The space $\mathcal{F}^{2}$ is a Hilbert space with orthonormal
basis $(\tilde{e}_{n})_{n=0}^{\infty}$ defined by

$\tilde{e}_{n}(z)=\frac{z^{n}}{\sqrt{n!}}.$

####  6.2.1 Composition operators

Boundedness of composition operators on $\mathcal{F}^{2}$ was characterized in
[19] as follows.

######  Theorem 6.2.

For an entire function $\phi$ the composition operator $C_{\phi}$ is bounded
on $\mathcal{F}^{2}$ if and only if either $\phi(z)=az+b$ with $|a|<1$ and
$b\in\mathbb{C}$ or $\phi(z)=az$ with $|a|=1$. In the case where $|a|<1$ the
operator $C_{\phi}$ is compact.

Thus there are relatively few bounded composition operators on
$\mathcal{F}^{2}$ and most natural questions can be answered easily. When we
come to weighted composition operators there will be much more to say.

The discussion here is based mostly on [23], although some results may also be
found in [47], which concentrates on spectra and mean ergodicity.

Considering now iterates $(C_{\phi}^{n})_{n}$ we have

$C_{\phi}^{n}f(z)=\begin{cases}f\left(a^{n}z+\frac{1-a^{n}}{1-a}b\right)&\hbox{if
}a\neq 1,\\\ f(z)&\hbox{if }a=1.\end{cases}$

We no wish to use Theorem 4.1 again, so we need to know whether $C_{\phi}$ is
power-bounded. The following formula is due to [19] in the case $\nu=2$, and
[26] in general.

$\|C_{\phi}\|=\exp\left(\frac{1}{4}\frac{|b|^{2}}{1-|a|^{2}}\right)\qquad\hbox{if}\quad\phi(z)=az+b\quad\hbox{with}\quad|a|<1.$

######  Theorem 6.3.

The asymptotics of iteration of bounded composition operators on
$\mathcal{F}^{\nu}$ are as follows.

  1. 1.

If $\phi(z)=az$ with $|a|=1$ and $a\neq 1$, then $(C^{n}_{\phi})$ consists of
unitary operators and does not converge even weakly.

  2. 2.

If $\phi(z)=az+b$ with $|a|<1$ then $(C_{\phi}^{n})$ converges in norm to the
operator $T:f\mapsto f\left(\dfrac{b}{1-a}\right)$.

The proof is short, and we include it here.

###### Proof.

1\. In the first case we have that $C_{\phi}^{n}f(z)=f(a^{n}z)$ and for
$f(z)=z$ there is clearly no convergence.

2\. In the second case

$\phi^{(n)}(z)=a^{n}z+\frac{1-a^{n}}{1-a}b=:a_{n}+b_{n}z,\hbox{ say}.$

and

$\frac{|b_{n}|^{2}}{1-|a_{n}|^{2}}\to\frac{|b|^{2}}{|1-a|^{2}}$

so $C_{\phi}$ is power bounded. Also $r_{e}(C_{\phi})=0$ since $C_{\phi}$ is
compact.

For the point spectrum, suppose that $\lambda$ is an eigenvalue and $f$ an
eigenvector. Then

$f(az+b)=\lambda f(z)$

and so $f(a_{n}+b_{n}z)=\lambda^{n}f(z)$, which implies that

$\lambda^{n}f(z)\to f(b/(1-a)).$ (4)

This means that $f$ is identically zero if $|\lambda|=1$ and $\lambda\neq 1$.

Finally if $1\in\sigma(C_{\phi})$ then $1$ is an eigenvalue since $C_{\phi}$
is compact. By (4) $f$ is constant, and $\dim\ker(C_{\phi}-{\rm Id})=1$. Hence
$1$ is a pole of the resolvent of order at most 1. Indeed, assume that the
pole order of $1$ is larger than $1$. Looking at the Jordan normal form of
$T_{0}:={C_{\varphi}}_{|X_{0}}$, where $X_{0}=P\mathcal{F}^{\nu}$ and $P$ the
residue, we see that there exists $f\in\mathcal{F}^{\nu}$ such that $({\rm
Id}-C_{\varphi})f=1_{\mathbb{C}}$. Evaluating at $z=\frac{b}{1-a}$ we obtain a
contradiction. Thus the pole order is $1$. It follows that $P$ is the
projection onto $\ker(C_{\varphi}-{\rm Id})=\mathbb{C}1_{\mathbb{C}}$ along
$\\{f\in\mathcal{F}^{\nu}:f(b/(1-a))=0\\}$. Thus
$Pf=f(b/(1-a))1_{\mathbb{C}}$. This completes the proof that $C^{n}_{\phi}\to
P$ is norm thanks to Theorem 4.1.

∎

Similarly, we can characterise $C_{0}$-semigroups of bounded composition
operators.

######  Theorem 6.4.

A $C_{0}$-semigroup $(C_{\phi_{t}})_{t\geq 0}$ of bounded composition of
$\mathcal{F}^{\nu}$ satisfies one of the following conditions for $t>0$.

  1. 1.

$\phi_{t}(z)=e^{\lambda t}z+C(e^{\lambda t}-1)$ for some
$\lambda\in\mathbb{C}_{-}=\\{z\in\mathbb{C}:\mathop{\rm Re}\nolimits z<0\\}$
and $C\in\mathbb{C}$.

  2. 2.

$\phi_{t}(z)=e^{\lambda t}z$ for some $\lambda\in i\mathbb{R}$.

The generator is given by

$Af(z)=\lambda(z+C)f^{\prime}(z),$

where $C=0$ in case (2).

Moreover $(C_{\phi_{t}})_{t\geq 0}$ converges in norm a $t\to\infty$ if and
only if $\lambda\in\mathbb{C}_{-}$, in which case the limit $T$ satisfies
$Tf=f(-C)$.

Using a theorem from [4] we get:

######  Theorem 6.5.

Every $C_{0}$-semigroup on $\mathcal{F}^{2}$ with generator of the form
$Af=Gf^{\prime}$ for some $G\in\mathop{\rm Hol}\nolimits(\mathbb{C})$ is a
semigroup of composition operators with generator satisfying

$\begin{array}[]{rll}G(z)&=az+b,&\hbox{with }\mathop{\rm Re}\nolimits
a<0,\quad\hbox{or}\\\ G(z)&=az,&\hbox{with }a\in i\mathbb{R}.\end{array}$

Moreover, the condition

$\limsup_{|z|\to\infty}\mathop{\rm Re}\nolimits\overline{z}G(z)\leq 0$ (5)

is necessary and sufficient for such $G$.

####  6.2.2 Weighted composition operators

For weighted composition operators $W_{w,\phi}$ we have the result of Le [36],
extended by Hai and Khoi [34] as follows:

######  Proposition 6.6.

The weighted composition operator $W_{w,\phi}$ with $w$ not identically zero
is bounded on $\mathcal{F}^{\nu}$ if and only if both the following conditions
hold:

  * •

$w\in\mathcal{F}^{\nu}$;

  * •

$M(w,\phi):=\sup_{z\in\mathbb{C}}|w(z)|^{2}e^{(|\phi(z)|^{2}-|z|^{2})}<\infty$.

Moreover, in this case $\phi(z)=az+b$ with $|a|\leq 1$. If $|a|=1$ then we
also have

$w(z)=w(0)e^{-\overline{b}az}$ (6)

for $z\in\mathbb{C}$.

See also [18] for a recent discussion of these results. Note that from [34] we
have the useful inequality

$\sqrt{M(w,\phi)}\leq\|W_{w,\phi}\|\leq\sqrt{M(w,\phi)}/|a|.$

Thus for power-boundedness of $W_{w,\phi}$, it is possible to give a complete
result for $|a|=1$.

######  Theorem 6.7.

Suppose that $\phi(z)=az+b$ with $|a|=1$, then $W_{w,\phi}$ is power-bounded
on $\mathcal{F}^{\nu}$ if and only if $|w(0)|\leq e^{-|b|^{2}/2}$.

The case $|a|<1$ is apparently unsolved, although this result for non-
vanishing $w$ (as in the case of semigroups) can be found in [18].

######  Theorem 6.8.

For $\phi(z)=az+b$ with $|a|<1$ and $w$ nonvanishing the operator $W_{w,\phi}$
is bounded on $\mathcal{F}^{\nu}$ if and only if

$w(z)=e^{p+qz+rz^{2}}$

and either:
(i) $|r|<\beta/2$, in which case $W_{w,\phi}$ is compact on
$\mathcal{F}^{\nu}$, or
(ii) $|r|=\beta/2$ and, with $t=q+\overline{b}a$, one has either $t=0$ or else
$t\neq 0$ and $r=-\frac{\beta}{2}\frac{t^{2}}{|t|^{2}}$. In case (ii)
$W_{w,\phi}$ is not compact on $\mathcal{F}^{\nu}$.

Finally, we can use this to characterise $C_{0}$-semigroups on the Fock space
[23].

######  Theorem 6.9.

A $C_{0}$-semigroup $T_{t}f(z)=w_{t}(z)f(\phi_{t}(z))$ ($t\geq 0$) of weighted
composition operators on the Fock space $\mathcal{F}^{\nu}$ has one of the
following two expressions:

  1. 1.

$\phi_{t}(z)=\exp(\lambda t)z+C(\exp(\lambda t)-1)$ for some
$\lambda\in\mathbb{C}_{-}$ and $C\in\mathbb{C}$; in which case
$w_{t}=e^{p_{t}+q_{t}z+r_{t}z^{2}}$, where explicit formulae for $p_{t}$,
$q_{t}$ and $r_{t}$ can be given.

  2. 2.

$\phi_{t}(z)=\exp(\lambda t)z+C(\exp(\lambda t)-1)$ for some $\lambda\in
i\mathbb{R}$ and $C\in\mathbb{C}$, in which case
$w_{t}(z)=w_{t}(0)\exp(\overline{C}(\exp(\lambda t)-1)z)$ and moreover
$w_{t}(0)=e^{\mu t}e^{|C|^{2}(e^{\lambda t}-1)}$ for some $\mu\in\mathbb{C}$.

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