%~Mouliné par MaN_auto v.0.34.0 (b0f8651d) 2025-07-04 15:37:55
\documentclass[CRMATH,Unicode,biblatex,published]{cedram}

\TopicFR{Analyse fonctionnelle}
\TopicEN{Functional analysis}

\addbibresource{CRMATH_Noor_20250372.bib}

\usepackage{mathtools}

\newcommand{\C} {\mathbb C}
\newcommand*{\dd}{\mathrm{d}}

\DeclareMathOperator{\vspan}{span}
\DeclareMathOperator{\Imag}{Im}
\DeclareMathOperator{\Orb}{Orb}

%%%---------------------------------------------------------------------------

%%% BAR & OVERLINE

%%% Better widebar macro
\makeatletter
\let\save@mathaccent\mathaccent
\newcommand*\if@single[3]{%
  \setbox0\hbox{${\mathaccent"0362{#1}}^H$}%
  \setbox2\hbox{${\mathaccent"0362{\kern0pt#1}}^H$}%
  \ifdim\ht0=\ht2 #3\else #2\fi
  }
%The bar will be moved to the right by a half of \macc@kerna, which is computed by amsmath:
\newcommand*\rel@kern[1]{\kern#1\dimexpr\macc@kerna}
%If there's a superscript following the bar, then no negative kern may follow the bar;
%an additional {} makes sure that the superscript is high enough in this case:
\newcommand*\widebar[1]{\@ifnextchar^{{\wide@bar{#1}{0}}}{\wide@bar{#1}{1}}}
%Use a separate algorithm for single symbols:
\newcommand*\wide@bar[2]{\if@single{#1}{\wide@bar@{#1}{#2}{1}}{\wide@bar@{#1}{#2}{2}}}
\newcommand*\wide@bar@[3]{%
  \begingroup
  \def\mathaccent##1##2{%
%Enable nesting of accents:
    \let\mathaccent\save@mathaccent
%If there's more than a single symbol, use the first character instead (see below):
    \if#32 \let\macc@nucleus\first@char \fi
%Determine the italic correction:
    \setbox\z@\hbox{$\macc@style{\macc@nucleus}_{}$}%
    \setbox\tw@\hbox{$\macc@style{\macc@nucleus}{}_{}$}%
    \dimen@\wd\tw@
    \advance\dimen@-\wd\z@
%Now \dimen@ is the italic correction of the symbol.
    \divide\dimen@ 3
    \@tempdima\wd\tw@
    \advance\@tempdima-\scriptspace
%Now \@tempdima is the width of the symbol.
    \divide\@tempdima 10
    \advance\dimen@-\@tempdima
%Now \dimen@ = (italic correction / 3) - (Breite / 10)
    \ifdim\dimen@>\z@ \dimen@0pt\fi
%The bar will be shortened in the case \dimen@<0 !
    \rel@kern{0.6}\kern-\dimen@
    \if#31
      \overline{\rel@kern{-0.6}\kern\dimen@\macc@nucleus\rel@kern{0.4}\kern\dimen@}%
      \advance\dimen@0.4\dimexpr\macc@kerna
%Place the combined final kern (-\dimen@) if it is >0 or if a superscript follows:
      \let\final@kern#2%
      \ifdim\dimen@<\z@ \let\final@kern1\fi
      \if\final@kern1 \kern-\dimen@\fi
    \else
      \overline{\rel@kern{-0.6}\kern\dimen@#1}%
    \fi
  }%
  \macc@depth\@ne
  \let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar
  \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax}%
  \macc@set@skewchar\relax
  \let\mathaccentV\macc@nested@a
%The following initialises \macc@kerna and calls \mathaccent:
  \if#31
    \macc@nested@a\relax111{#1}%
  \else
%If the argument consists of more than one symbol, and if the first token is
%a letter, use that letter for the computations:
    \def\gobble@till@marker##1\endmarker{}%
    \futurelet\first@char\gobble@till@marker#1\endmarker
    \ifcat\noexpand\first@char A\else
      \def\first@char{}%
    \fi
    \macc@nested@a\relax111{\first@char}%
  \fi
  \endgroup
}
\makeatother

\let\oldbar\bar
\renewcommand*{\bar}[1]{{\mathchoice{\widebar{#1}}{\widebar{#1}}{\widebar{#1}}{\oldbar{#1}}}}

%%%---------------------------------------------------------------------------

%%% DELIMITERS

\usepackage{mathtools}
\DeclarePairedDelimiter{\parens}{\lparen}{\rparen}
\DeclarePairedDelimiter{\braces}{\{}{\}}
\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}
\DeclarePairedDelimiter{\angles}{\langle}{\rangle}

%%%---------------------------------------------------------------------------

\graphicspath{{./figures/}}

\newcommand*{\mk}{\mkern -1mu}
\newcommand*{\Mk}{\mkern -2mu}
\newcommand*{\mK}{\mkern 1mu}
\newcommand*{\MK}{\mkern 2mu}

\hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red}

\newcommand*{\relabel}{\renewcommand{\labelenumi}{(\theenumi)}}
\newcommand*{\romanenumi}{\renewcommand*{\theenumi}{\roman{enumi}}\relabel}
\newcommand*{\Romanenumi}{\renewcommand*{\theenumi}{\Roman{enumi}}\relabel}
\newcommand*{\alphenumi}{\renewcommand*{\theenumi}{\alph{enumi}}\relabel}
\newcommand*{\Alphenumi}{\renewcommand*{\theenumi}{\Alph{enumi}}\relabel}
\let\oldhat\hat
\renewcommand*{\hat}[1]{\mathchoice{\widehat{#1}}{\widehat{#1}}{\oldhat{#1}}{\oldhat{#1}}}
\let\oldforall\forall
\renewcommand*{\forall}{\mathrel{\oldforall}}

%%%---------------------------------------------------------------------------

\makeatletter
\def\@setafterauthor{%
  \vglue3mm%
\begingroup\hsize=12cm\advance\hsize\abstractmarginL\raggedright
\noindent
   \leftskip\abstractmarginL
  \normalfont\Small
  \@afterauthor\par
\endgroup
\vskip2pt plus 3pt minus 1pt
}
\makeatother

%%%---------------------------------------------------------------------------

\title{Cyclicity of composition operators on the Paley--Wiener spaces}
\alttitle{Cyclicité des opérateurs de composition sur les espaces de Paley--Wiener}

\author{\firstname{Pham} \lastname{Viet Hai}}
\address{Faculty of Mathematics and Informatics, Hanoi University of Science and Technology, Khoa Toan-Tin, Dai hoc Bach khoa Hanoi, 1 Dai Co Viet, Hanoi, Vietnam}
\email{hai.phamviet@hust.edu.vn}

\author{\firstname{Waleed} \lastname{Noor}}
\address{IMECC, Universidade Estadual de Campinas, Campinas-SP, Brazil}
\email{waleed@unicamp.br}

\author{\firstname{Osmar} \lastname{Reis Severiano}}
\address[2]{IMECC, Universidade Estadual de Campinas, Campinas-SP, Brazil}
\email{osmar.rrseveriano@gmail.com}

\thanks{P.~V. Hai is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number~101.02-2021.24. O.~R. Severiano is a postdoctoral fellow at the Programa de Matemática and is supported by UNICAMP (Programa de Pesquisador de Pós-Doutorado PPPD)}
\CDRGrant[Vietnam National Foundation for Science and Technology Development (NAFOSTED)]{101.02-2021.24}

\keywords{\kwd{Cyclic operator} \kwd{composition operator} \kwd{Paley--Wiener space}}
\altkeywords{\kwd{Opérateur cyclique} \kwd{opérateur de composition} \kwd{espace Paley--Wiener}}

\begin{abstract}
In this article we characterize the cyclicity of bounded composition operators $C_\phi f=f\circ \phi$ on the Paley--Wiener spaces of entire functions $B^2_\sigma$ for $\sigma>0$. We show that $C_\phi$ is cyclic precisely when $\phi(z)=z+b$ where either $b\in\mathbb{C}\setminus\mathbb{R}$ or $b\in\mathbb{R}$ with $0<\lvert b \rvert\leq \pi/\sigma$. We also describe when the reproducing kernels of $B^2_\sigma$ are cyclic vectors for $C_\phi$ and see that this is related to a question of completeness of exponential sequences in $L^2[-\sigma,\sigma]$. The interplay between cyclicity and complex symmetry plays a key role in this work.
\end{abstract}

\begin{altabstract}
Dans cet article, nous caractérisons la cyclicité des opérateurs de composition bornée $C_\phi f=f\circ \phi$ sur les espaces de Paley--Wiener des fonctions entières $B^2_\sigma$ pour $\sigma>0$. Nous montrons que $C_\phi$ est cyclique précisément lorsque $\phi(z)=z+b$ où $b\in\mathbb{C}\setminus\mathbb{R}$ ou $b\in\mathbb{R}$ avec $0<\lvert b \rvert\leq \pi/\sigma$. Nous décrivons également lorsque les noyaux reproducteurs de $B^2_\sigma$ sont des vecteurs cycliques pour $C_\phi$ et voyons que cela est lié à une question de complétude des suites exponentielles dans $L^2[-\sigma,\sigma]$. L'interaction entre cyclicité et symétrie complexe joue un rôle clé dans ce travail.
\end{altabstract}

\dateposted{2025-07-21}
\begin{document}

\maketitle

\section{Introduction}

A bounded linear operator $T$ on a separable Hilbert space $\mathcal{H}$ is \emph{cyclic} with \emph{cyclic vector} $f\in\mathcal{H}$ if the linear span of the \emph{orbit} $\Orb(f,T)=\braces[\big]{T^nf:n=0,1,2,\dotsc}$ is dense in $\mathcal{H}$. Similarly $T$ is \emph{supercyclic} if all scalar multiples of elements in $\Orb(f,T)$ are dense in $\mathcal{H}$, and \emph{hypercyclic} if the orbit itself is dense. Cyclicity is a central theme in linear dynamics and has been studied widely due to its connection with the Invariant Subspace Problem~(ISP). See the recent monographs~\cite{Bayart-Matheron} and~\cite{Linear_Chaos} to learn more about linear dynamics.

On the other hand a bounded linear operator $T$ on $\mathcal{H}$ is \emph{complex symmetric} if there exists an orthonormal basis for $\mathcal{H}$ with respect to which $T$ has a self-transpose matrix representation. An equivalent definition also exists. A \emph{conjugation} is a conjugate-linear operator $J\colon\mathcal{H}\to\mathcal{H}$ that satisfies the conditions:
\begin{enumerate}\alphenumi
\item $J$ is \emph{isometric}: $\angles{Jf,Jg}=\angles{g,f} \ \forall f,g\in\mathcal{H}$;
\item $J$ is \emph{involutive}: $J^2=I$.
\end{enumerate}
We say that $T$ is $J$-\emph{symmetric} if $JT=T^*J$, and complex symmetric if there exists a conjugation~$J$ with respect to which $T$ is $J$-symmetric. Complex symmetric operators are generalizations of complex symmetric matrices and of normal operators, and their study was initiated by Garcia, Putinar and Wogen~\cite{Garcia_1, Garcia_2, Garcia_3, Garcia_4}. If $JT=T^*J$ for some operator $T$ and conjugation $J$, then $T$ is cyclic (supercyclic) if and only if $T^*$ is cyclic (supercyclic). The conjugation $J$ serves as a bijection between the cyclic vectors of $T$ and $T^*$.

Let $\mathcal{S}$ be a space of functions defined on a set $\Omega$. A \emph{composition operator} $C_{\phi}$ on $\mathcal{S}$ with \emph{symbol} $\phi\colon\Omega\to\Omega$ is defined as
\[
C_{\phi}f=f\circ \phi, \quad f\in \mathcal{S}.
\]
The cyclicity of composition operators has been studied on several holomorphic function spaces (see~\cite{Bayart, Gallardo, Noor}) and has become an active research topic within linear dynamics. In~\cite{chacon2007composition}, Chácon, Chácon and Giménez initiated the study of composition operators on the classical Paley--Wiener space $B^2_{\pi}$. They prove that $C_\phi$ is bounded on $B^2_{\pi}$ precisely when
\begin{equation}\label{b1}
\phi(z)=az+b, \quad \text{where} \ a\in \mathbb{R} \ \text{with} \ 0<\abs{a}\leq 1 \ \text{and} \ b\in \mathbb{C}.
\end{equation}
More recently Ikeda, Ishikawa and Yoshihiro~\cite{Ikeda} show that this is true even in the general context of reproduction kernel Hilbert spaces of entire functions on $\mathbb{C}^n$.

The cyclicity and complex symmetry of $C_\phi$ on the Hardy--Hilbert space $H^2(\mathbb{C}_+)$ of the half-plane were characterized by Noor and Severiano~\cite{Noor} for affine symbols $\phi$. In this article we study the cyclicity and complex symmetry of composition operators on the \emph{Paley--Wiener spaces} $B^2_{\sigma}$ for all $\sigma>0$. From results in~\cite{Ikeda} it follows that the only bounded composition operators $C_\phi$ on $B^2_{\sigma}$ for $\sigma>0$ are those induced by symbols of the form~\eqref{b1}. The Paley--Wiener spaces $B^2_\sigma$ are isometrically embedded into $H^2(\mathbb{C}_+)$ as so-called \emph{model subspaces} $K_\Theta$ (see~\cite[p.~305]{Nikolski}) defined as
\[
K_\Theta\coloneqq H^2(\mathbb{C}_+)\ominus\Theta H^2(\mathbb{C}_+) \quad \text{where} \ \Theta(z)=e^{i\sigma z}.
\]
Therefore one can view this work as the study of $C_\phi$ on some model subspaces of $H^2(\mathbb{C}_+)$. In contrast with $H^2(\mathbb{C}_+)$, our results show the existence of non-normal complex symmetric $C_\phi$ (see Theorem~\ref{normal}), and that the cyclicity of $C_\phi$ and its adjoint \smash{$C_\phi^*$} in $B^2_\sigma$ depends on $\sigma>0$ (see Theorem~\ref{cyclic}). In particular, we show that no $C_\phi$ is supercyclic on any $B^2_\sigma$. Finally we characterize the reproducing kernels $(k_w)_{w\in\mathbb{C}}$ in $B^2_\sigma$ that are cyclic vectors for $C_\phi$ and show that this is closely related to the completeness of exponential sequences $(e^{i\lambda_n t})_{n\in\mathbb{N}}$ in $L^2[-\sigma,\sigma]$ (see Theorem~\ref{cyclic vector}). The latter is a central question in non-harmonic Fourier analysis (see~\cite{young}). Some of the main results are summarized in the following table.

\begin{table}[h]
\centering
\caption{Main results for $C_\phi$ on $B^2_\sigma$ where $\phi(z)=az+b$.}\label{tab:example}
\begin{tabular}{|l|l|}
\hline $C_\phi$ cyclic & $a=1$ with $b\in\mathbb{C}\setminus\mathbb{R}$ or $0<\abs{b}\leq\pi/\sigma$, $b\in\mathbb{R}$ \\
\hline $C^*_\phi$ cyclic & $0<\abs{a}<1$, $b\in\mathbb{C}$ or $C_\phi$ cyclic \\
\hline $C_\phi$ supercyclic & never \\
\hline $C_\phi$ complex symmetric & $a=1$, $b\in\mathbb{C}$ or $a=-1$, $b\in\mathbb{C}$ \\
\hline $C_\phi$ normal & $a=1$, $b\in\mathbb{C}$ or $a=-1$, $b\in\mathbb{R}$ \\

\hline
\end{tabular}
\end{table}

Before we begin, it is necessary to mention an error in~\cite{chacon2007composition} during the computation of an adjoint formula for $C_\phi$ on $B^2_\pi$ which then leads to an incomplete description of normal $C_\phi$. This is discussed and corrected in Section~\ref{section_2}.

\section{Preliminaries}\label{section_2}

An entire function $f$ is of \emph{exponential type} if the inequality $\abs[\big]{f(z)}\leq Ae^{B\abs{z}}$ holds for all $z\in \mathbb{C}$ and for some constants $A,B>0$. The exponential type $\sigma$ of $f$ is defined as the infimum of all $B>0$ for which this inequality holds and can be determined by
\[
\sigma=\limsup_{r\to\infty}\frac{\log M_f(r)}{r} \quad \text{where} \ M_f(r)=\max_{\abs{z}=r}\abs[\big]{f(z)}.
\]

\subsection{Paley--Wiener spaces \texorpdfstring{$B^2_{\sigma}$}{lg}}

For $\sigma>0$, the \emph{Paley--Wiener space} $B_\sigma^2$ consists of all entire functions of exponential type $\leq\sigma$ whose restrictions to $\mathbb{R}$ belong to $L^2(\mathbb{R})$. The space $B_\sigma^2$ is a reproducing kernel Hilbert space when endowed with the norm
\[
\norm{f}^2 = \int^{\infty}_{-\infty}\abs[\big]{f(t)}^2\,\dd t, \quad f\in B^2_{\sigma}.
\]
For each $w\in \mathbb{C}$, let $k_w$ denote the reproducing kernel for $B_{\sigma}^2$ at $w$ defined by
\[
k_w(z)=\frac{\sin \sigma(z-\bar{w})}{\pi(z-\bar{w})}, \quad z\in \mathbb{C}
\]
and which satisfies the basic relation $\angles{f,k_w} =f(w)$ for all $f\in B_{\sigma}^2$. The Paley--Wiener Theorem states that any $f\in B^2_\sigma$ can be represented as the \emph{inverse} Fourier transform
\[
f(z)=(\mathfrak{F}^{-1}F)(z)\coloneqq \frac{1}{\sqrt{2\pi}}\int_{-\sigma}^\sigma F(t)e^{izt}\,\dd t
\]
of some $F\in L^2[-\sigma,\sigma]$ where the Fourier transform $\mathfrak{F}\colon B^2_\sigma\to L^2[-\sigma,\sigma]$ is an isometric isomorphism. We denote $\widehat{f}\coloneqq \mathfrak{F}f$ for $f\in B^2_\sigma$ and in particular $(\widehat{k_w})(t)=e^{-i\bar{w}t}$ for $w\in\mathbb{C}$. The monographs~\cite{Branges} and~\cite{young} may be consulted for more information about these spaces.

\subsection{Composition operators} 

The composition operator $C_{\phi}$ is bounded on $B^2_{\sigma}$ if and only if its inducing symbol $\phi$ has the form
\begin{equation}\label{symbol}
\phi(z)=az+b, \quad \text{where} \ a\in \mathbb{R} \ \text{with} \ 0<\abs{a}\leq 1 \ \text{and} \ b\in \mathbb{C}
\end{equation}
with the usual action of $C_{\phi}^*$ on reproducing kernels determined by $C_{\phi}^*k_w=k_{\phi(w)}$ for $w\in \mathbb{C}$. For each $n\in \mathbb{N}$, the $n$-iterate of the self-map $\phi\colon\mathbb{C}\to \mathbb{C}$ is denoted by $\phi^{[n]}$ where
\begin{equation}\label{iterate}
\phi^{[n]}(z)=
\begin{cases}
z+n b, 										& \text{if} \ a=1, \\
a^{n} z+\frac{\parens{1-a^{n}}}{1-a}b, 	& \text{if} \ a \neq 1,
\end{cases}
\end{equation}
and $C_{\phi}^n=\C_{\phi^{[n]}}$. We see from~\eqref{iterate} that if $0<\abs{a}<1$ and $\alpha\coloneqq \frac{b}{1-a}$, then $\alpha$ is an attractive fixed point of $\phi$, that is, $\phi^{[n]}(z)\to \alpha$ as $n\to \infty$ for all $z\in \mathbb{C}$.

\subsection{Adjoints of composition operators}

In~\cite{chacon2007composition} the authors show that $C_\phi$ and $C_\phi^*$ on $B^2_\pi$ are unitarily equivalent, via the Fourier transform $\mathfrak{F}$, to a pair of weighted composition operators $\widehat{C}_{\phi}$ and $\widehat{C}_{\phi}^*$ on $L^2[-\pi,\pi]$ respectively. For simplicity they assume $a>0$ which is sufficient for their results. However, as we shall see in the next section, the cases $a>0$ and $a<0$ lead to very distinct outcomes for the complex symmetry and cyclicity of $C_\phi$. Therefore for the sake of completeness we provide a proof of their result for all $0<\abs{a}\leq 1$.

\begin{prop}\label{adjoint formula}
If $\phi(z)=az+b$ where $a\in \mathbb{R}$ with $0<\abs{a}\leq 1$, $b\in \mathbb{C}$, then $C_{\phi}$ on $ B^2_{\sigma}$ is unitarily equivalent to a weighted composition operator $\widehat{C}_{\phi}$ on $L^2[-\sigma,\sigma]$ defined by
\[
(\widehat{C}_{\phi}F)(t)=\frac{1}{\abs{a}}\chi_{(-\abs{a}\sigma, \abs{a}\sigma)}(t)e^{\frac{ibt}{a}}F\parens[\Bigg]{\frac{t}{a}}
\]
where $\chi_{(c,d)}$ denotes a characteristic function. Moreover we have $(\widehat{C}_{\phi}^*F)(t)=\overline{e^{ibt}}F(at)$.
\end{prop}

\begin{proof}
We first assume $0<a\leq 1$. For each $f\in B^2_{\sigma}$, we have
\[
\begin{split}
(C_{\phi}f)(z)
	& = f(az+b)
\\	& = \frac{1}{\sqrt{2\pi}}\int^\sigma_{-\sigma}\widehat{f}(t)e ^{i(az+b)t}\,\dd t
\\	& = \frac{1}{\sqrt{2\pi}}\int^\sigma_{-\sigma}\widehat{f}(t)e ^{iazt}e^{ibt}\,\dd t
\\	& = \frac{1}{\sqrt{2\pi}}\int^{a\sigma}_{-a\sigma}\frac{1}{a}\widehat{f}\parens[\Bigg]{\frac{s}{a}}e ^{isz}e^{ib\parens*{\frac{s}{a}}}\,\dd s
\\	& = \frac{1}{\sqrt{2\pi}}\int^{\sigma}_{-\sigma}\frac{1}{a}\chi_{(-a\sigma, a\sigma)}(t)e^{\frac{ibs}{a}}\widehat{f}\parens[\Bigg]{\frac{s}{a}}e ^{isz}\,\dd s
\\	& = \frac{1}{\sqrt{2\pi}}\int^{\sigma}_{-\sigma}(\widehat{C}_{\phi}\widehat{f})(s)e ^{isz}\,\dd s
\\	& = (\mathfrak{F}^{-1}\widehat{C}_\phi\mathfrak{F}f)(z)
\end{split}
\]
which gives $\mathfrak{F}C_{\phi}= \widehat{C}_{\phi}\mathfrak{F}$. For $-1\leq a <0$, first consider the symbol $\eta(z)=-z$. Then the simple change of variables $s=-t$ again gives
\begin{equation}\label{eqc3}
(C_{\eta}f)(z) =f(-z)=\frac{1}{\sqrt{2\pi}}\int^{\sigma}_{-\sigma}(\widehat{C}_{\eta}\widehat{f})(s)e ^{isz}\,\dd s=(\mathfrak{F}^{-1}\widehat{C}_\eta\mathfrak{F}f)(z)
\end{equation}
and hence $\mathfrak{F}C_{\eta}= \widehat{C}_{\eta}\mathfrak{F}$. Now let $\psi(z)=-az+b$ and note that $C_{\phi}=C_{\eta}C_{\psi}$. Therefore
\[
\widehat{C}_{\eta}\widehat{C}_{\psi}=(\mathfrak{F}C_{\eta}\mathfrak{F}^{-1})(\mathfrak{F}C_{\psi}\mathfrak{F}^{-1})=\mathfrak{F}C_{\phi}\mathfrak{F}^{-1}
\]
and to verify that indeed $\widehat{C}_\phi=\widehat{C}_{\eta}\widehat{C}_{\psi}$, one easily sees that
\[
(\widehat{C}_{\eta}\widehat{C}_{\psi}F)(t) = \frac{1}{-a}e^{-\frac{ibt}{-a}}\chi_{(a\sigma, -a\sigma)}(-t)F\parens[\Bigg]{\frac{-t}{-a}} =(\widehat{C}_{\phi}F)(t)
\]
for all $F\in L^2[-\sigma,\sigma]$. Therefore $\mathfrak{F}C_{\phi}= \widehat{C}_{\phi}\mathfrak{F}$ for all $-1\leq a<0$ as well. Since the Fourier transform $\mathfrak{F}\colon B^2_\sigma\to L^2[-\sigma,\sigma]$ is an isometric isomorphism, the proof is complete. The adjoint formula follows by the change of variables $t=s/a$ as follows:
\[
\angles{\widehat{C}_{\phi}^*F, G}=\int^\sigma_{-\sigma}\overline{e^{ibt}}F(at)\overline{G(t)}\,\dd t=\int^{\abs{a}\sigma}_{-\abs{a}\sigma}F(s)\overline{\frac{1}{\abs{a}}e^{\frac{ibs}{a}}G\parens[\Bigg]{\frac{s}{a}}}\,\dd s=\angles{F, \widehat{C}_{\phi}G}
\]
for all $F,G\in L^2[-\sigma,\sigma]$. This completes the proof of the result.
\end{proof}

In~\cite[p.~2209]{chacon2007composition} it is claimed that if $\phi=az+b$ with $0<a<1$, then the adjoint of $C_\phi$ is
\[
(C_{\phi}^*f)(z)=\frac{1}{a}f\parens[\Bigg]{\frac{z-\bar{b}}{a}}, \quad f\in B^2_{\pi}.
\]
However this cannot be correct since if $f$ has exponential type $\pi$ then $f(z/a)$ must have type $\pi/a>\pi$. Hence $C_\phi^*$ is not well-defined on $B^2_{\pi}$. This unfortunately leads to an incomplete description of normal composition operators (compare~\cite[Proposition~2.6]{chacon2007composition} with Table~\ref{tab:example}).

\vspace{.2\baselineskip}

\section{Cyclicity and complex symmetry of \texorpdfstring{$C_{\phi}$}{lg}}\label{s6}

\vspace{.1\baselineskip}

One of the main leitmotifs of this article is to show how results about cyclicity can lead to results about complex symmetry and vice versa. This hinges on a simple observation. If $TJ=JT^*$ for some operator $T$ and conjugation $J$, then $T$ is cyclic if and only if $T^*$ is cyclic. The conjugation $J$ serves as a bijection between the cyclic vectors of $T$ and $T^*$. We demonstrate this by first dealing with the case when $0<\abs{a}<1$.

\begin{prop}\label{case a<1}
Let $\phi(z)=az+b$ where $a\in \mathbb{R}$, $0<\abs{a}<1$ and $b\in\mathbb{C}$. Then $C_\phi$ is not cyclic where\-as~$C_\phi^*$ is cyclic on $B^2_\sigma$ for $\sigma>0$. Hence $C_\phi $ is not complex symmetric.
\end{prop}

\begin{proof}
We know that $C_\phi$ on $B^2_\sigma$ is unitarily equivalent to $\widehat{C}_\phi$ on $L^2[-\sigma,\sigma]$ by Proposition~\ref{adjoint formula}. So we show that $\widehat{C}_\phi$ is not cyclic. For any $F\in L^2[-\sigma,\sigma]$ we have $\widehat{C}_\phi F=\chi_{(-\abs{a}\sigma, \abs{a}\sigma)}G$ for some $G\in L^2[-\sigma,\sigma]$. So every element in $\vspan\braces[\big]{\widehat{C}_\phi^n F:n=0,1,\dotsc}$ must have the form $cF+H$ where $H$ vanishes ouside $(-\abs{a}\sigma, \abs{a}\sigma)$. In other words, the closure of this span consists only of functions that coincide with a constant multiple of $F$ outside $(-\abs{a}\sigma, \abs{a}\sigma)$. This clearly makes the cyclicity of $\widehat{C}_\phi$ and hence of $C_\phi$ impossible. For $C^*_\phi$ we show that every reproducing kernel $k_w$ is a cyclic vector. This is because $(C^{*}_\phi)^nk_w=C^*_{\phi^{[n]}}k_w=k_{\phi^{[n]}(w)}$ and $\phi^{[n]}(w)$ converges to the attractive fixed point $\alpha\coloneqq \frac{b}{1-a}$ of $\phi$. So any $f\in B^2_\sigma$ orthogonal to the orbit of $k_w$ under $C^*_\phi$ must vanish on a sequence with a limit point and hence $f\equiv 0$. Therefore the span of the orbit of every $k_w$ must be dense in $B^2_\sigma$ and hence $C_\phi^*$ is cyclic. It then follows that $C_\phi$ is not complex symmetric by the discussion above.
\end{proof}

By Proposition~\ref{case a<1} we need only consider the cases $a=\pm1$ for the following result. Note that in both cases ($J_af)(z)=\overline{f(-a\bar{z})}$ defines a conjugation on $B^2_\sigma$ for $\sigma>0$.

\begin{prop}\label{normal}
Let\/ $\phi(z)=az+b$ where $a=\pm1$ and $b\in \mathbb{C}$. Then $C_{\phi}$ on $B^2_{\sigma}$ is:
\begin{enumerate}
\item \label{normal_1} always complex symmetric;
\item \label{normal_2} normal if and only if $a=1$, $b\in\mathbb{C}$ or $a=-1, b\in \mathbb{R}$;
\item \label{normal_3} self-adjoint if and only if $a=1,b\in i\mathbb{R}$ or $a=-1,b\in \mathbb{R}$;
\item \label{normal_4} unitary if and only if $b\in \mathbb{R}$.
\end{enumerate}
Moreover $C_\phi$ is $J_a$-symmetric on $B^2_{\sigma}$.
\end{prop}

\begin{proof}
We first note that when $a=-1$ we have $C^2_\phi=I$ and hence $C_\phi$ is complex symmetric since every operator that is algebraic of order $2$ is complex symmetric by~\cite[Theorem~2]{Garcia_1}. In general by Proposition~\ref{adjoint formula} we have
\[
(\widehat{C}_\phi F)(t) =e^{\frac{ibt}{a}}F(t/a) \qquad \text{and} \qquad (\widehat{C}^*_\phi F)(t) =e^{\overline{ibt}}F(at)
\]
for $F\in L^2[-\sigma,\sigma]$. So when $a=1$ we see that $\widehat{C}_\phi=M_{e^{ibt}}$ is a multiplication operator which is normal. This shows that $C_\phi$ is always complex symmetric and gives~\eqref{normal_1}. If $a=-1$, then $\widehat{C}_\phi \widehat{C}_\phi^*=\widehat{C}_\phi ^*\widehat{C}_\phi$ precisely when $e^{-ibt}e^{-\overline{ibt}}=e^{\overline{ibt}}e^{ibt}$ or when $e^{-i2\Imag(b)t}=e^{i2\Imag(b)t}$, and this holds only when $b\in\mathbb{R}$. This gives~\eqref{normal_2} and one easily obtains~\eqref{normal_3} and~\eqref{normal_4} similarly. Finally we show that $C_\phi$ is $J_a$-symmetric by first noting that
\[
(J_a C_\phi J_af)(z)=\overline{(C_\phi J_af)(-a\bar{z})}=\overline{(J_af)(-\bar{z}-a\bar{b})}=f(az-a\bar{b})=(C_\psi f)(z)
\]
where $\psi(z)=az-a\bar{b}$. We claim that $C_\psi=C_\phi^*$ and this follows by Proposition~\ref{adjoint formula} because
\[
(\widehat{C}_{\psi}F)(t)=e^{-i\bar{b}t}F(t/a)=\overline{e^{ibt}}F(at)=(\widehat{C}_{\phi}^*F)(t)
\]
for all $F\in L^2[-\sigma,\sigma]$. Therefore $J_a C_\phi J_a=C_\phi^*$ and completes the proof of the result.
\end{proof}

Our main result characterizes the cyclicity of $C_\phi$ and $C_\phi^*$ simultaneously using complex symmetry, and then shows that no $C_\phi$ is supercyclic on $B^2_\sigma$ for all $\sigma>0$ via normality. The latter was proved for the case $\sigma=\pi$ in~\cite[Theorem~2.7]{chacon2007composition} using a lengthier argument.

\begin{theo}\label{cyclic}
Let $\phi(z)=az+b$ where $a\in\mathbb{R}$, $0<\abs{a}\leq 1$ and $b\in \mathbb{C}$. Then $C_{\phi}$ on $B^2_\sigma$ is:
\begin{enumerate}
\item \label{cyclic_1} cyclic if and only if $a=1$ with $b\in\mathbb{C}\setminus\mathbb{R}$ or $0<\abs{b}\leq\pi/\sigma$, $b\in\mathbb{R}$;
\item \label{cyclic_2} never supercyclic.
\end{enumerate}
Moreover $C_\phi^*$ is cyclic if and only if\/ $0<\abs{a}<1$ or when $C_\phi$ is cyclic.
\end{theo}

\begin{proof}
The case $0<\abs{a}<1$ has been dealt with in Proposition~\ref{case a<1} showing that $C_\phi$ is not cyclic or supercyclic. So first note that when $a=-1$ then $C_\phi^2f=f$ and hence $C_\phi$ cannot be cyclic or supercyclic as the orbit of any $f\in B^2_\sigma$ contains at most two elements $f$ and $f\circ\phi$. So let $a=1$. Then $C_\phi$ is normal and normal operators are never supercyclic~\cite[p.~564]{Hilden}. Therefore this proves~\eqref{cyclic_2}. For cyclicity we first observe that $\widehat{C}_\phi=M_{e^{ibt}}$ is a multiplication operator on $L^2[-\sigma,\sigma]$. A consequence of the spectral theory for normal operators is that a multiplication operator $M_\Phi$ on $L^2(\mu)$ (where $\Phi\in L^\infty(\mu)$ and $\mu$ is a compactly supported measure on $\mathbb{C}$) is cyclic if and only if $\Phi$ is injective on a set of full measure (see~\cite[Theorem~1.1 and Proposition~1.3]{Ross}). Therefore we need to determine when $\Phi(t)=e^{ibt}$ is injective almost everywhere on $[-\sigma,\sigma]$. For $b\in\mathbb{C}\setminus\mathbb{R}$ we see that $\abs[\big]{\Phi(t)}=e^{-\Imag(b)t}$ which is clearly injective on all of $[-\sigma,\sigma]$ and hence so is $\Phi$. For the case when $b\in\mathbb{R}$ note that $\Phi$ is $2\pi/\abs{b}$ periodic. It follows that $\Phi$ is injective on $(-\sigma,\sigma)$ (hence a.e.\ on $[-\sigma,\sigma]$) precisely when the period is greater than or equal to the length of the interval, that is, precisely when
\[
\frac{2\pi}{\abs{b}}\geq 2\sigma \qquad \text{equivalently} \qquad 0<\abs{b}\leq\frac{\pi}{\sigma}.
\]
The assertion about cyclicity of $C_\phi^*$ follows by Proposition~\ref{case a<1} and complex symmetry.
\end{proof}

Finally we ask the following question: \emph{are any of the reproducing kernels $(k_w)_{w\in\mathbb{C}}$ cyclic vectors for $C_\phi$}? The answer reveals an interesting dichotomy and a connection with the question of completeness of exponential sequences in $L^2[-\sigma,\sigma]$.

\begin{theo}\label{cyclic vector}
Let $\phi(z)=z+b$ where $b\in \mathbb{C}$. Then the kernels $(k_w)_{w\in\mathbb{C}}$ in $B^2_\sigma$ are:
\begin{enumerate}
\item all cyclic vectors for $C_\phi$ when $b\in\mathbb{C}\setminus\mathbb{R}$ or $0<\abs{b}<\pi/\sigma$, $b\in\mathbb{R}$;
\item never cyclic for $C_\phi$ when $\abs{b}=\pi/\sigma$.
\end{enumerate}
\end{theo}

\begin{proof}
We first consider the case when $b\in\mathbb{C}\setminus\mathbb{R}$. Note that since $\phi^{[n]}(z)=z+nb$, one easily sees that \smash{$C_{\phi}^nk_w=k_{w-n\bar{b}}$} for $w\in\mathbb{C}$. So any function $f\in B^2_\sigma$ orthogonal to the span of the orbit of $k_w$ must vanish at $w_n\coloneqq w-n\bar{b}$ for all $n\geq 0$. But $(w_n)_{n\geq 0}$ does not satisfy the Blaschke-type condition required for zero sets of functions in $B^2_\sigma$, that is
\[
\sum_{n\geq 0}\frac{\abs[\big]{\Imag(w_n)}}{1+\abs{w_n}^2}\simeq \sum_{n\geq 0}\frac{n}{1+n^2}=\infty.
\]
Hence $f\equiv 0$ and $\vspan(C_{\phi}^nk_w)_{n\geq 0}$ is dense in $B^2_\sigma$. So now let $b\in\mathbb{R}$. Since $(\mathfrak{F}k_w)(t)=e^{-i\bar{w}t}$ for $w\in\mathbb{C}$, it is enough to show that every exponential $e^{iwt}$ is a cyclic vector for $\widehat{C}_\phi=M_{e^{ibt}}$. This is equivalent to the completeness of the sequence $\parens{e^{i(bn+w)t}}_{n\geq 0}$ in $L^2[-\sigma,\sigma]$. But this sequence is the image of $(e^{ibnt})_{n\geq 0}$ under $M_{e^{iwt}}$ which has dense range in $L^2[-\sigma,\sigma]$. Therefore it is sufficient to prove the completeness of $(e^{ibnt})_{n\geq 0}$ in $L^2[-\sigma,\sigma]$. For this we use a classical result of Carleman (see~\cite[p.~97]{young}): if $(\lambda_n)_{n\geq 0}$ is a sequence of positive real numbers and
\[
\liminf_{n\to\infty}\frac{n}{\lambda_n}>\frac{\sigma}{\pi},
\]
then $(e^{i\lambda_nt})_{n\geq0}$ is complete in $\mathcal{C}[-\sigma,\sigma]$ (space of continuous functions). This condition is clearly satisfied for $\lambda_n\coloneqq bn$ when $0<b<\pi/\sigma$. The case $-\pi/\sigma<b<0$ follows since now $(e^{-ibnt})_{n\geq0}$ is complete and then conjugating. If $\abs{b}=\pi/\sigma$, then the exponentials $(e^{ibnt})_{n\geq0}$ are $2\sigma$-periodic as we saw in the proof of Theorem~\ref{cyclic}. It follows that $(e^{ibnt})_{n\in\mathbb{Z}}$ is an orthogonal basis for $L^2[-\sigma,\sigma]$. So $(e^{ibnt})_{n\geq0}$ in not complete in $L^2[-\sigma,\sigma]$ and neither is its image under the injective op\-er\-a\-tor~$M_{e^{iwt}}$. This completes the proof.
\end{proof}

The last result hints at the possibility of finding deeper connections between composition operators on model subspaces $K_\Theta$ of $H^2(\mathbb{C}_+)$ and non-harmonic Fourier analysis.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

\printbibliography

\end{document}