Spectra of composition operators on Paley-Wiener spaces and some consequences

TL;DR

This paper thoroughly investigates the properties of bounded composition operators on Paley-Wiener spaces, including their spectrum, chaos, and stability, revealing detailed operator behavior in this functional analysis context.

Contribution

It provides comprehensive characterizations of composition operators on Paley-Wiener spaces regarding compactness, spectrum, chaos, and other dynamical properties, filling gaps in the understanding of these operators.

Findings

Complete characterization of compactness and spectrum

Identification of conditions for Li-Yorke chaos and positive expansivity

Results on positive shadowing property and Cesàro boundedness

Abstract

Bounded composition operators in Paley-Wiener spaces have simple forms, and they are just operators composed through affine mappings of the complex plane. The purpose of this article is to explore some notions about bounded operators and linear dynamics and provide complete answers for composition operators in Paley-Wiener spaces concerning compactness, spectrum, spectral radius, Li-Yorke chaos, positive expansivity, positive shadowing property, and absolute Ces\`aro boundedness.

Equations

$$C_{\varphi }f=f\circ \varphi ,f\in \mathcal{H}.$$

$$⟨f,g⟩:={\int }_{-\infty }^{\infty }f(t)\overline{g(t)}\mathrm{d}t.$$

$$k_w(z)=\frac{\sin a(z-\overline{w})}{\pi (z-\overline{w})},z\in \mathbb{C}.$$

$$\varphi (z)=cz+d\text{where}c\in \mathbb{R}\text{with}0<|c|\le 1\text{and}d\in \mathbb{C}.$$

$${\varphi }^{[n]}(z)=\{\begin{array}{ll}z+nd,& \text{if}c=1,\\ c^{n}z+\frac{\left(1-c^n\right)}{1-c}d,& \text{if}c\ne 1.\end{array}$$

$$({\hat{C}}_{\varphi }F)(t)=\frac{1}{|c|}{\chi }_{(-|c|a,|c|a)}(t)e^{\frac{\mathrm{i}dt}{c}}F\left(\frac{t}{c}\right)$$

$$\underset{n\to \infty }{\mathrm{liminf}}\Vert T^{n}x\Vert =0\text{and}\underset{n\to \infty }{\mathrm{limsup}}\Vert T^{n}x\Vert =\infty .$$

$$\underset{n\in \mathbb{N}}{\sup }\left\{\frac{1}{n}\sum _{j=1}^n\Vert T^{j}x\Vert \right\}\le L\Vert x\Vert ,$$

$$\Vert k_w{\Vert }^2=\frac{\sin a(w-\overline{w})}{\pi (w-\overline{w})}=\frac{\sin 2\mathrm{i}a\mathrm{Im}(w)}{2\mathrm{i}\pi \mathrm{Im}(w)}=\frac{\sinh 2a\mathrm{Im}(w)}{2\pi \mathrm{Im}(w)}$$

$$\Vert C_{\varphi }^{\ast }{K}_n{\Vert }^2$$

$$\varphi =\psi \circ \phi$$

$$\varphi =\tau \circ \phi \circ {\tau }^{-1}$$

$$\Vert C_{\phi }f{\Vert }^2={\int }_{-\infty }^{\infty }|f(ct){|}^2\mathrm{d}t=c^{-1}{\int }_{\infty }^{\infty }|f(t){|}^2\mathrm{d}t=c^{-1}\Vert f{\Vert }^2$$

$$\Vert C_{\phi }f{\Vert }^2=\Vert C_{\tilde{\phi }}{C}_{\psi }f\Vert =(-c{)}^{-1}\Vert C_{\psi }f{\Vert }^2=\Vert f{\Vert }^2,$$

$$|c{|}^{-1/2}\le \Vert C_{\varphi }\Vert \le e^{|\mathrm{Im}(d)|a}|c{|}^{-1/2}$$

$$|c{|}^{-1/2}\le \Vert C_{\varphi }^n{\Vert }^{\frac{1}{n}}\le e^{\frac{(1-c^n)|\mathrm{Im}(d)|\sigma }{n(1-c)}}|c{|}^{-1/2}.$$

$$\Vert C_{\varphi }^{n}f{\Vert }^2$$

$$\ge {\ell }^2{\int }_{A_{\ell }}{e}^{-2\mathrm{Im}(d)nt}\mathrm{d}t\ge {\ell }^2{e}^{2|\mathrm{Im}(d)|na}{\int }_{A_{\ell }}\mathrm{d}t.$$

$${\phi }_n(z)=c^{n}z\text{and}{\psi }_n(z)=z+\frac{1-c^n}{1-c}d$$

$$\Vert C_{{\psi }_n}f\Vert \Vert k_w\Vert \ge |⟨C_{{\psi }_n}f,k_w⟩|=|⟨f,C_{{\psi }_n}^{\ast }{k}_w⟩|=|⟨f,k_{{\psi }_n(w)}⟩|=|f({\psi }_n(w))|$$

$$f_n=\frac{\delta }{\Vert C_{\varphi }f\Vert }\sum _{j=0}^{n-1}{C}_{\varphi }^{n-j}f$$

$$\Vert C_{\varphi }^{n}g-f_n\Vert \Vert k_{\alpha }\Vert \ge |⟨C_{\varphi }^{n}g-f_n,k_{\alpha }⟩|=|g(\alpha )-\frac{n\delta f(\alpha )}{\Vert C_{\varphi }f\Vert }|\ge \frac{n\delta |f(\alpha )|}{\Vert C_{\varphi }f\Vert }-|g(\alpha )|$$

$$\underset{n\in \mathbb{N}}{\sup }\left\{\frac{1}{n}\sum _{j=1}^n\Vert C_{\varphi }^{j}f\Vert \right\}\le e^{|\mathrm{Im}(d)|a}\Vert f\Vert .$$

$$\frac{1}{n}\sum _{j=1}^n\Vert C_{\varphi }^j{k}_1\Vert \ge \frac{1}{n}\Vert C_{\varphi }^n{k}_1\Vert \ge \frac{\delta }{n\sqrt{|c{|}^n}}\Vert k_1\Vert$$

$$\underset{n\in \mathbb{N}}{\sup }\left\{\frac{1}{n}\sum _{j=1}^n\Vert C_{\varphi }^j{k}_1\Vert \right\}=\underset{n\in \mathbb{N}}{\sup }\left\{\frac{\delta }{n\sqrt{|c{|}^n}}\right\}\Vert k_1\Vert =\infty .$$

Full text

O.R. Severiano is a postdoctoral fellow at the Programa de Matemática and is supported by UNICAMP (Programa de Pesquisador de Pós-Doutorado PPPD)

1. Introduction

Let $\mathcal{H}$ be a complex Hilbert space of analytic functions on a subset $\Omega$ of the complex plane $\mathbb{C}$. If $\phi$ is an analytic self-map of $\Omega$, then the composition operator $C_{\phi}$ with symbol $\phi$ is defined by

[TABLE]

Operators of this type have been studied on a variety of spaces, such as Bergman, Dirichlet series, Hardy, Fock and Newton spaces [13, 15, 19, 21, 24, 26], and with special interest on the Hardy space of the open unit disk [15, 29]. As is well known in the literature, there are several basic questions in the study of composition operators: boundedness [15, 18], compactness [30], norms [11, 13, 15], spectra [15, 19], spectral radius [15, 18], normality [15, 19], complex symmetry [22, 23, 27, 28, 31], cyclicity [10, 20, 22, 28, 29], hypercyclicity [10, 20, 29], supercyclicity ( [14, 20], and more recently notions such as expansivity (and its variants), Li-Yorke chaos and the positive shadowing property [2, 7]. In many cases, it is difficult to solve these problems completely, and some remain open [16, 27].

For $a>0,$ the Paley-Wiener space $\mathscr{PW}_{a}$ consists of all entire functions of exponential type less than or equal to $a,$ whose restriction to $\mathbb{R}$ belongs to $L^{2}(\mathbb{R}).$ Each space $\mathscr{PW}_{a}$ is a Hilbert space when endowed with the inner product

[TABLE]

Moreover, $\mathscr{PW}_{a}$ is a reproducing kernel Hilbert space which means the point evaluations of functions on $\mathscr{PW}_{a}$ are bounded linear functionals. Indeed, if $w\in\mathbb{C}$ and $f\in\mathscr{PW}_{a}$ then $f(w)=\langle f,k_{w}\rangle$ where the reproducing kernel function $k_{w}$ is given by

[TABLE]

The sequence of kernels $(k_{\frac{\pi n}{a}})_{n\in\mathbb{Z}}$ constitute a complete orthonormal set on $\mathscr{PW}_{a}.$ For more information about these spaces, see the monographs [9, 12, 25, 32].

In [14], Chacón, Chacón, and Giménez initiated the study of composition operators on the classical Paley–Wiener space $\mathscr{PW}_{\pi}$. They proved that $C_{\phi}$ is bounded on $\mathscr{PW}_{\pi}$ if and only if $\phi(z)=cz+d$ where $c\in\mathbb{R}$ with $|c|\in(0,1]$ and $d\in\mathbb{C}$, and as a consequence, showed that such operators are neither compact nor supercyclic. Similarly, on $\mathscr{PW}_{a}$, the only bounded composition operators are those induced by symbols of the form $\phi(z)=cz+d$ where $c\in\mathbb{R}$ with $|c|\in(0,1]$ and $d\in\mathbb{C}$. In [22], Hai, Noor, and Severiano studied bounded composition operators on the full range of Paley-Wiener spaces $\mathscr{PW}_{a}$, completely characterizing which of these operators are self-adjoint, unitary, normal, and complex symmetric [22, Proposition 3]. They also studied properties concerning linear dynamics such as cyclicity and supercyclicity, for instance, they extended [14, Theorem 2.7], showing that no Paley–Wiener space $\mathscr{PW}_{a}$ supports supercyclic bounded composition operators. While, the cyclicity of $C_{\phi}$ occurs if and only if $c=1$ and $d\in\mathbb{C}\setminus\mathbb{R}$ or $0<|d|\leq\pi/a$ with $d\in\mathbb{R}$ [22, Theorem 4].

In this article, we continue the study of composition operators on Paley-Wiener spaces $\mathscr{PW}_{a}$. We show that certain properties of these operators, regarding boundedness and linear dynamics can be completely characterized. More precisely, we compute the spectrum and the spectral radius of each bounded composition operator on $\mathscr{PW}_{a}$. Moreover, we completely characterize which of these operators are Li–Yorke chaotic, positively expansive, absolutely Cesàro bounded, and possess the positive shadowing property.

2. Preliminaries

2.1. Bounded composition operators on $\mathscr{PW}_{a}$

The only bounded composition operators $C_{\phi}$ on $\mathscr{PW}_{a}$ are those induced by affine symbols of the form

[TABLE]

In this case, the adjoint of $C_{\phi}$ acts on reproducing kernels as follows $C_{\phi}^{*}k_{w}=k_{\phi(w)}$ for $w\in\mathbb{C}.$ Since $|c|\in(0,1],$ the only invertible bounded composition operators on $\mathscr{PW}_{a}$ occur when $c=\pm 1.$ Moreover, Hai, Noor and Severiano [22, Proposition 3] showed that $C_{\phi}$ is normal on $\mathscr{PW}_{a}$ if and only if $c=1$ or $c=-1$ and $d\in\mathbb{R}.$ In particular, $C_{\phi}$ is unitary if and only if $c=\pm 1$ and $d\in\mathbb{R}.$

Let us denote by $\phi^{[n]}$ the $n$-th iterate of $\phi,$ where $\phi$ is as in (2.1), then a simple computation gives

[TABLE]

and $C_{\phi}^{n}=C_{\phi^{[n]}}.$ Thus, we see from (2.2) that if $|c|<1$ and $\alpha:=d/(1-c),$ then $\alpha$ is an attractive fixed point of $\phi$, that is, $\phi^{[n]}(z)\rightarrow\alpha$ pointwise as $n\rightarrow\infty$ for all $z\in\mathbb{C}.$

2.2. Spectra of linear operators

Now we recall some basic definitions and facts from general spectral theory that will be used in the subsequent section. Let $\mathcal{B}(X)$ denote the space of all bounded linear operators on a Banach space $X.$ The spectrum and spectral radius of $T\in\mathcal{B}(X)$ are denoted by $\sigma(T)$ and $r(T),$ respectively. It is well known that $r(T)$ is determined by the spectral radius formula $r(T)=\displaystyle\lim\|T^{n}\|^{1/n}.$ And as usual, $\mathbb{T}$ denotes the unit circle $\{z\in\mathbb{C}:|z|=1\}$

If the operators $T_{1}\in\mathcal{B}(X_{1})$ and $T_{2}\in\mathcal{B}(X_{2})$ are similar (resp. isometrically similar), i.e. there exists an invertible (resp. isometric invertible) operator $U:X_{1}\rightarrow X_{2}$ such that $T_{1}=U^{-1}T_{2}U,$ then their spectra coincide. This is very useful for our purposes, since if $\phi(z)=cz+d$ is as in (2.1) with $c\neq 1$ then we can write $\phi=\tau\circ\varphi\circ\tau^{-1}$ where $\varphi(z)=cz$ and $\tau(z)=z+d/(1-c).$ This allows us to obtain $C_{\tau^{-1}}C_{\phi}C_{\tau}=C_{\varphi},$ and hence $\sigma(C_{\phi})=\sigma(C_{\varphi}).$

The next result was first proved in [14, page 2208] (and corrected in [22, Proposition 2.2]) and it shows that each bounded composition operator on $\mathscr{PW}_{a}$ is isometrically similar to a bounded weighted composition operator on $L^{2}[-a,a].$

Proposition 2.1**.**

If $\phi(z)=cz+d$ where $c\in\mathbb{R}$ with $|c|\in(0,1]$ and $d\in\mathbb{C},$ then the composition operator $C_{\phi}$ on $\mathscr{PW}_{a}$ is isometrically similar to the weighted composition operator $\widehat{C}_{\phi}$ on $L^{2}[-a,a]$ defined by

[TABLE]

where $\chi_{(p,q)}$ denotes a characteristic function. Moreover, we have $(\widehat{C}_{\phi}^{*}F)(t)=\overline{e^{idt}}F(ct).$

Hence, if $\psi(z)=z+d$ with $d\in\mathbb{C}$ then $C_{\psi}$ is isometrically similar to the multiplication operator $M_{e^{\mathrm{i}dt}}.$ Therefore, $\sigma(C_{\psi})=\sigma(M_{e^{\mathrm{i}dt}})$ and $\|C_{\psi}\|=\|M_{e^{\mathrm{i}dt}}\|_{L^{2}[-a,a]}.$ From the multiplication operator theory, we know that if $\Phi:\Omega\rightarrow\mathbb{C}$ is a continuous function essentially bounded then $M_{\Phi}$ is a bounded operator on $L^{2}(\Omega),$ $\|M_{\Phi}\|_{L^{2}(\Omega)}=\sup\{|\Phi(t)|:t\in\Omega\}$ and $\sigma(M_{\Phi})=\{\Phi(t):t\in\Omega\}.$ This allows us to determine the spectrum and the norm of $C_{\psi}$ on each $\mathscr{PW}_{a},$ namely, $\sigma(C_{\psi})=\{e^{\mathrm{i}dt}:t\in[-a,a]\}$ and $\|C_{\psi}\|=e^{|{\rm Im}\,(d)|a}$ where $\mathrm{Im}(d)$ denotes the imaginary part of the complex number $d.$

2.3. Notions in linear dynamics

Let us introduce some notions concerning linear dynamics that are relevant to our work. Such notions will be studied in subsequent sections.

Definition 2.2**.**

An operator $T\in\mathcal{B}(X)$ is said to be hyperbolic if $\sigma(T)\cap\mathbb{T}=\emptyset.$

The definition of a Li-Yorke chaotic operator that we use here is obtained from the equivalent statements of [4, Theorem 5].

Definition 2.3**.**

An operator $T\in\mathcal{B}(X)$ is said to be Li-Yorke chaotic if there exists $x\in X$ such that

[TABLE]

In this case, such vector $x$ is called an irregular vector for $T.$

In [4], Bermúdez et al. showed that if $T\in\mathcal{B}(X)$ is a Li-Yorke operator then $T$ is neither normal nor hyperbolic.

Definition 2.4**.**

An operator $T\in\mathcal{B}(X)$ is said to be positively expansive if for each $x\in X$ with $\|x\|=1,$ there exists $n\in\mathbb{N}$ such that $\|T^{n}x\|\geq 2.$

An alternative characterization for positively expansive operators was given by Bernardes et al. [6, Proposition 19], namely, $T$ is positively expansive if and only if for every non-zero vector $x\in X$ the set $\{\|T^{n}x\|:n\in\mathbb{N}\}$ is unbounded.

Definition 2.5**.**

Let $T\in\mathcal{B}(X).$

1. (i)

*For $\delta>0,$ we say that a sequence $(x_{n})_{n\in\mathbb{N}}$ in $X$ is a *$\delta-$pseudotrajectory of $T,$ if $\|Tx_{n}-x_{n+1}\|\leq\delta$ for all $n\in\mathbb{N}.$ 2. (ii)

*We say that $T$ has the positive shadowing property if for every $\epsilon>0$ there exists $\delta>0$ such that every *$\delta-$pseudotrajectory $(x_{n})_{n\in\mathbb{N}}$ of $T$ is $\epsilon$-shadowed by an $x\in X,$ that is, there is $x\in X$ such that $\|T^{n}x-x_{n}\|\leq\epsilon$ for all $n\in\mathbb{N}.$

Bernardes et al. proved the following criterion for deciding whether a normal operator has the positive shadowing property [6, Theorem 30], let $T\in\mathcal{B}(X)$ be a normal operator, then $T$ has the positive shadowing property if and only if $T$ is hyperbolic.

Definition 2.6**.**

An operator $T\in\mathcal{B}(X)$ is said to be absolutely Cesàro bounded if there exists a constant $L>0$ such

[TABLE]

for all $x\in X.$

We refer to [1, 3, 4, 5, 6, 7] for more details on Li-Yorke chaotic, positively expansive (and its variants), absolutely Cesàro operator and to have the positive shadowing property.

3. Main results

3.1. Compactness

We know from [14, Corollary 2.5] that there no compact composition operator on $\mathscr{PW}_{\pi}.$ Here we extend this result by showing that no Paley-Wiener space $\mathscr{PW}_{a}$ supports compact composition operators.

For $w\in\mathbb{C},$ we have $\|k_{w}\|^{2}=2a/\pi$ when $w\in\mathbb{R}$ while

[TABLE]

when $w\notin\mathbb{R}.$ This helps us to show the following result.

Proposition 3.1**.**

There are no compact composition operators on $\mathscr{PW}_{a}.$

Proof.

If $\phi(z)=cz+d$ where $c\in\mathbb{R}$ with $|c|\in(0,1]$ and $d\in\mathbb{C},$ then ${\rm Im}\,\left(\phi\left(n\pi/a\right)\right)={\rm Im}\,(d)$ for all $n\in\mathbb{N}.$ Let $K_{n}$ denote the normalized reproducing kernel $k_{\frac{n\pi}{a}},$ by (3.1) we get

[TABLE]

Since $(K_{n})_{n\in\mathbb{N}}$ tends weakly to zero and $C_{\phi}^{*}K_{n}$ does not converge to zero in norm, it follows that $C_{\phi}^{*}$ is not compact. Therefore, $C_{\phi}$ also is not compact. ∎

3.2. Spectra and spectral radius

Let us consider the symbol $\phi(z)=cz+d$ where $c\in\mathbb{R}$ with $|c|\in(0,1]$ and $d\in\mathbb{C},$ then $\phi$ can be written as follows

[TABLE]

where $\varphi(z)=cz$ and $\psi(z)=z+b.$ This allows us to obtain $C_{\phi}=C_{\varphi}C_{\psi}.$ If $c\neq 1,$ we also have

[TABLE]

where $\tau(z)=z+b/(1-c),$ which gives us $C_{\phi}=C_{\tau^{-1}}C_{\varphi}C_{\tau}.$ Since $C_{\phi}$ and $C_{\varphi}$ are similar, this allows us to obtain properties of $C_{\phi}$ from $C_{\varphi}$.

Proposition 3.2**.**

If $\varphi(z)=cz$ where $c\in\mathbb{R}$ with $|c|\in(0,1],$ then $|c|^{1/2}C_{\varphi}$ is an isometry on $\mathscr{PW}_{a}.$ Moreover, the spectrum of $C_{\varphi}$ acting on $\mathscr{PW}_{a}$ is $\sigma(C_{\varphi})=\{\lambda\in\mathbb{C}:|\lambda|\leq|c|^{-1/2}\}$ when $|c|<1.$

Proof.

Let $f\in\mathscr{PW}_{a}.$ If $c\in(0,1]$ we have

[TABLE]

which shows that $c^{1/2}C_{\varphi}$ is an isometry. If $c\in[-1,0),$ we consider the auxiliary symbols $\tilde{\varphi}(z)=-cz$ and $\psi(z)=-z$ then $\varphi=\psi\circ\tilde{\varphi}.$ For what we have done so far and since $C_{\psi}$ is unitary, we obtain

[TABLE]

which also shows that $(-c)^{1/2}C_{\varphi}$ is an isometry.

Since $C_{\varphi}$ is not invertible when $|c|<1$ and the spectrum of an isometry non-invertible operator is the closed unit disk, we obtain $\sigma(C_{\varphi})=\{\lambda\in\mathbb{C}:|\lambda|\leq|c|^{-1/2}\}.$ ∎

As a consequence we deal with the problem of identifying the bounded composition operators on $\mathscr{PW}_{a}$ with closed range and their spectra.

Corollary 3.3**.**

Each bounded composition operator on $\mathscr{PW}_{a}$ has closed range.

Proof.

Let $\phi(z)=cz+d$ where $c\in\mathbb{R}$ with $|c|\in(0,1]$ and $d\in\mathbb{C}.$ If $c=1$ then $C_{\phi}$ is invertible and hence has closed range. If $c\neq 1,$ then $C_{\phi}$ is similar to $C_{\varphi}$ where $\varphi(z)=cz.$ Since closed range is preserved under similarity and $C_{\varphi}$ has closed range on $\mathscr{PW}_{a},$ it follows that $C_{\phi}$ also has closed range. ∎

Corollary 3.4**.**

Let $\phi(z)=cz+d$ where $c\in\mathbb{R}$ with $|c|\in(0,1]$ and $d\in\mathbb{C}.$ Then the spectrum of $C_{\phi}$ acting on $\mathscr{PW}_{a}$ is

1. (i)

$\sigma(C_{\phi})=\{-1,1\},$* when $c=-1.$* 2. (ii)

$\sigma(C_{\phi})=\left\{\lambda\in\mathbb{C}:|\lambda|\leq|c|^{-1/2}\right\},$* when $|c|\in(0,1).$* 3. (iii)

$\sigma(C_{\phi})=\{e^{\mathrm{i}dt}:t\in[-a,a]\},$* when $c=1.$*

Proof.

If $c=-1,$ we have that $\phi^{[2]}$ is the identity map and hence $C_{\phi}^{2}=I.$ Hence for the polynomial $p(t)=t^{2}-1,$ we obtain $\sigma(p(C_{\phi}))=\{0\}.$ Moreover, by spectral mapping theorem [17, Corollary 2.37], we have $p(\sigma(C_{\phi}))=\sigma(p(C_{\phi})).$ Thus $\lambda\in\sigma(C_{\phi})$ if and only if $p(\lambda)=0$ or equivalently if $\lambda=\pm 1.$ If $|c|\in(0,1),$ then $C_{\phi}$ is similar to $C_{\varphi}$ (see (3.3)) and hence $\sigma(C_{\phi})=\sigma(C_{\varphi}).$ By Proposition 3.2, the result follows. If $c=1,$ then $C_{\phi}$ is similar to the multiplication operator $\widehat{C}_{\phi}=M_{e^{\mathrm{i}dt}}$ on $L^{2}[-a,a]$ and hence $\sigma(C_{\phi})=\{e^{\mathrm{i}dt}:t\in[-a,a]\}.$ ∎

As a consequence of computing the spectrum, we obtain the following estimates.

Lemma 3.5**.**

If $\phi(z)=cz+d$ where $c\in\mathbb{R}$ with $|c|\in(0,1]$ and $d\in\mathbb{C},$ then

[TABLE]

on $\mathscr{PW}_{a}$. In particular, if $d\in\mathbb{R}$ then (3.4) is an equality.

Proof.

Since $\phi=\psi\circ\varphi$ where $\varphi(z)=cz$ and $\psi(z)=z+d$ (see (3.3)), $\|C_{\psi}\|=e^{|{\rm Im}\,(d)|a}$ and $\|C_{\varphi}\|=|c|^{-1/2},$ we obtain $\|C_{\phi}\|\leq\|C_{\varphi}\|\|C_{\psi}\|=e^{|{\rm Im}\,(d)|a}|c|^{-1/2}$ which gives the upper estimate. The lower estimate follows directly from Corollary 3.4. ∎

Proposition 3.6**.**

If $\phi(z)=cz+d$ where $c\in\mathbb{R}$ with $|c|\in(0,1]$ and $d\in\mathbb{C},$ then the spectral radius of $C_{\phi}$ on $\mathscr{PW}_{a}$ is

1. (i)

$r(C_{\phi})=|c|^{-1/2},$* when $c\neq 1;$* 2. (ii)

$r(C_{\phi})=e^{|{\rm Im}\,(d)|a},$* when $c=1.$*

Proof.

If $c=1,$ then $C_{\phi}$ is normal and hence $r(C_{\phi})=\|C_{\phi}\|=e^{|{\rm Im}\,(d)|a}.$ Now assume $c\neq 1.$ By Lemma 3.5 and the formula of the $n$-iterate established in (2.2), we obtain

[TABLE]

Letting $n$ go to infinity in (3.5), we get the equality $r(C_{\phi})=|c|^{-1/2}.$ ∎

3.3. Positive expansivity

Now we deal with the problem of identifying the bounded composition operators in $\mathscr{PW}_{a}$ that are positively expansive. Before proceeding, we need the following auxiliary results.

Lemma 3.7**.**

Let $\phi(z)=z+d$ where $d\in\mathbb{C}\setminus\mathbb{R}.$ If $f\in\mathscr{PW}_{a}$ is a non-zero function then there exists $\delta>0$ such that $\|C_{\phi}^{n}f\|\geq\delta e^{|{\rm Im}\,(d)|na}$ for all $n.$

Proof.

By Proposition 2.1, there exists an isometric isomorphism $\mathcal{U}:\mathscr{PW}_{a}\rightarrow L^{2}[-a,a]$ such that $C_{\phi}=\mathcal{U}^{-1}M_{e^{\mathrm{i}dt}}\mathcal{U}.$ Let $f\in\mathscr{PW}_{a}$ be a non-zero function. Then $\mathcal{U}f$ also is non-zero function on $L^{2}[-a,a].$ Hence there exists a real number $\ell>0$ such that $A_{\ell}=\{t\in[-a,a]:|\mathcal{U}f(t)|\geq\ell\}$ has positive measure. Now observe that

[TABLE]

If $\delta=\ell\left(\int_{A_{\ell}}\mathrm{d}t\right)^{1/2},$ we obtain $\|C_{\phi}^{n}f\|\geq\delta e^{|{\rm Im}\,(b)|na}$ for all $n.$ ∎

The following proof mimics the steps of the proof from [8, Lemma 3.2] , though we enclose it here for the sake of completeness.

Lemma 3.8**.**

If $\phi(z)=cz+d$ where $c\in\mathbb{R}$ with $|c|\in(0,1)$ and $d\in\mathbb{C},$ then for each $f\in\mathscr{PW}_{a}$ there exists $\delta>0$ such that $\|C_{\phi}^{n}f\|\geq(\delta/\sqrt{|c|^{n}})\|f\|$ for all sufficiently large $n.$

Proof.

For each $n,$ we consider the auxiliary symbols $\varphi_{n}$ and $\psi_{n}$ defined as follows

[TABLE]

then $\phi^{[n]}=\psi_{n}\circ\varphi_{n}.$ Let $f\in\mathscr{PW}_{a},$ if $f$ is the zero function then the result is immediate. So we can assume that $f$ is not the zero function. By Proposition 3.2, we get $\|C_{\phi}^{n}f\|=\|C_{\varphi_{n}}C_{\psi_{n}}f\|=(1/\sqrt{|c|^{n}})\|C_{\psi_{n}}f\|.$ Now we obtain a lower estimate for $\|C_{\psi_{n}}f\|.$ For each $w\in\mathbb{C},$ the Cauchy Schwarz inequality gives

[TABLE]

for all $n.$ Since $\psi_{n}(w)\rightarrow w+d/(1-c)$ as $n\rightarrow\infty,$ we obtain $|f(\psi_{n}(w))|\rightarrow|f(w+d/(1-c))|$ as $n\rightarrow\infty.$ Let $w_{0}\in\mathbb{C}$ such that $f(w_{1})\neq 0,$ where $w_{1}=w_{0}+d/(1-c).$ Then for all sufficiently large $n,$ we get $|f(\psi_{n}(w_{0}))|>|f(w_{1})|/2.$ If $\delta=|f(w_{1})|/2\|k_{w_{0}}\|,$ we obtain $\|C_{\phi}^{n}f\|>(\delta/\sqrt{|c|^{n}})\|f\|$ for all sufficiently large $n.$ ∎

Proposition 3.9**.**

Let $\phi(z)=cz+d$ where $c\in\mathbb{R}$ with $|c|\in(0,1]$ and $d\in\mathbb{C}$. Then $C_{\phi}$ is positively expansive on $\mathscr{PW}_{a}$ if and only if $|c|\in(0,1)$ or $c=1$ and $d\notin\mathbb{R}.$

Proof.

We first assume $c=-1.$ In this case, $\phi^{[2]}$ is the identity map, and hence $C_{\phi}^{2n}=I$ ($I$ denotes the identity operator) and $C_{\phi}^{2n+1}=C_{\phi}$ for all $n\in\mathbb{N}.$ This allows us to obtain $\|C_{\phi}^{n}f\|\leq e^{|{\rm Im}\,(d)\|a}\|f\|$ for all $n\in\mathbb{N}$ and all $f\in\mathscr{PW}_{a}$ (see Lemma 3.5). Since $\{\|C_{\phi}^{n}f\|:n\in\mathbb{N}\}$ is bounded for each $f\in\mathscr{PW}_{a}$, it follows from [6, Proposition 19] that $C_{\phi}$ is not positively expansive. If $|c|\in(0,1)$ then Lemma 3.8 shows that for each $f\in\mathscr{PW}_{a}$ there exists $\delta>0$ such that $\|C_{\phi}^{n}f\|\geq(\delta/\sqrt{|c|^{n}})\|f\|$ for all sufficiently large $n.$ Since $f$ is not the zero function and $1/\sqrt{|c|^{n}}\rightarrow\infty$ as $n\rightarrow\infty,$ it follows that there exists $n$ sufficiently large such that $\|C_{\phi}^{n}f\|\geq 2.$ Hence $C_{\phi}$ is positively expansive. Finally let $c=1.$ If $d\in\mathbb{R}$ then $C_{\phi}$ is a unitary operator and hence $C_{\phi}$ is not positively expansive. If $d\notin\mathbb{R}$ then by Lemma 3.7 for each non-zero function $f\in PW_{a}$ there exists $\delta>0$ such that $\|C_{\phi}^{n}f\|\geq\delta e^{|{\rm Im}\,(d)|na}$ for all $n.$ Since $e^{|{\rm Im}\,(d)|na}\rightarrow\infty$ as $n\rightarrow\infty,$ it follows that there exists $n$ sufficiently large such that $\|C_{\phi}^{n}f\|\geq 2$ and hence $C_{\phi}$ is positively expansive. ∎

3.4. Li-Yorke chaos

Now we study the existence of irregular vectors for bounded composition operators in $\mathscr{PW}_{a}$ and consequently the problem of identifying which theses operators are Li-Yorke chaotic.

Proposition 3.10**.**

No bounded composition on $\mathscr{PW}_{a}$ is Li-Yorke chaotic.

Proof.

Let $\phi(z)=cz+d$ where $c\in\mathbb{R}$ with $|c|\in(0,1]$ and $d\in\mathbb{C}.$ We have already seen that if $c=-1$ then $\|C_{\phi}^{n}f\|\leq e^{|{\rm Im}\,(d)|a}\|f\|$ for all $n$ and all $f\in\mathscr{PW}_{a}$ (see proof of Proposition 3.9). In this case, $\displaystyle\limsup_{n\to\infty}\|C_{\phi}^{n}f\|\leq e^{|{\rm Im}\,(d)|a}\|f\|,$ which implies that $C_{\phi}$ does not admit an irregular vector. Now assume $|c|\in(0,1).$ By Lemma 3.8, for each $f\in\mathscr{PW}_{a}$ there exists $\delta>0$ such that $\|C_{\phi}^{n}f\|>(\delta/\sqrt{|c|^{n}})\|f\|.$ Since $1/\sqrt{|c^{n}|}\rightarrow\infty$ as $n\rightarrow\infty$ and $f$ is not the zero function, it follows that $\|C_{\phi}^{n}f\|\geq 2$ for all sufficiently large $n.$ In this case, $\displaystyle\liminf_{n\to\infty}\|C_{\phi}^{n}f\|\geq 2$ and hence $C_{\phi}$ does not admit an irregular vector. If $c=1,$ then $C_{\phi}$ is normal on $\mathscr{PW}_{a}$ and hence it is not Li-Yorke chaotic [4, Corollary 6]. ∎

3.5. Positive shadowing property

In order to characterize which bounded composition operators $C_{\phi}$ on $\mathscr{PW}_{a}$ have the positive shadowing property, we analyze the value of $c.$ If $c=1,$ then $C_{\phi}$ is normal and hence we can decide whether $C_{\phi}$ has the positive shadowing property from $\sigma(C_{\phi}).$ If $c\neq 1,$ then $\phi$ has a fixed point in $\mathbb{C}.$ Then we can proceed as the in proof of [2, Proposition 1] to decide whether $C_{\phi}$ has the positive shadowing property property.

Proposition 3.11**.**

No bounded composition operator on $\mathscr{PW}_{a}$ has the positive shadowing property.

Proof.

Let $\phi(z)=cz+d$ where $c\in\mathbb{R}$ with $|c|\in(0,1]$ and $d\in\mathbb{C}.$ If $c=1$ then $C_{\phi}$ is a normal operator on $\mathscr{PW}_{a}.$ In this case, [6, Theorem 30] ensures that $C_{\phi}$ has the positive shadowing property if and only if $C_{\phi}$ is hyperbolic. By Corollary 3.4, $C_{\phi}$ is not hyperbolic. Therefore, $C_{\phi}$ does not have the positive shadowing property. Now assume $c\neq 1,$ then $\alpha=d/(1-c)$ is the fixed point of $\phi$ in $\mathbb{C}.$ Choose $f\in\mathscr{PW}_{a}$ such that $f(\alpha)\neq 0.$ For each $\delta>0,$ the sequence $(f_{n})_{n\in\mathbb{N}}$ defined as follows

[TABLE]

is a $\delta-$pseudotrajectory of $C_{\phi}.$ Moreover, $f_{n}(\alpha)=\frac{n\delta f(\alpha)}{\|C_{\phi}f\|}$ for all $n\in\mathbb{N}.$ By Cauchy-Schwarz inequality, we get

[TABLE]

for every $g\in\mathscr{PW}_{a}.$ Taking $n\rightarrow\infty$ in (3.6), we obtain $\|C_{\phi}^{n}g-f_{n}\|\rightarrow\infty$ which shows that $(f_{n})_{n\in\mathbb{N}}$ cannot be $\epsilon$-shadowed for any $\epsilon>0.$ ∎

3.6. Absolutely Cesàro bounded

Finally, we study the notion of absolutely Cesàro bounded. As we will see below, the key to our studies are the estimates for norms obtained in Lemma 3.5 and the inequalities from Lemmas 3.7 and3.8.

Proposition 3.12**.**

Let $\phi(z)=cz+d$ where $c\in\mathbb{R}$ with $|c|\in(0,1]$ and $d\in\mathbb{C}$. Then $C_{\phi}$ is absolutely Cesàro bounded on $\mathscr{PW}_{a}$ if and only if $c=-1$ or $c=1$ and $d\in\mathbb{R}.$

Proof.

If $c=-1$ or $c=1$ and $d\in\mathbb{R},$ then by Lemma 3.5 we obtain $\|C_{\phi}^{n}f\|\leq e^{|{\rm Im}\,(d)|a}\|f\|$ for all $n\in\mathbb{N}$ and all $f\in\mathscr{PW}_{a}.$ In both cases we obtain

[TABLE]

Now we consider $|c|\in(0,1).$ In this case, the sequence $(1/n\sqrt{|c|^{n}})_{n\in\mathbb{N}}$ is unbounded. Moreover, Lemma 3.8 ensures that there exists $\delta>0$ such that $\|C^{n}_{\phi}k_{1}\|\geq(\delta/\sqrt{|c|^{n}})\|k_{1}\|$ for all sufficiently large $n\in\mathbb{N}.$ Taking sufficiently large $n,$ we get

[TABLE]

which implies

[TABLE]

Therefore, $C_{\phi}$ is not an absolutely Cesàro bounded operator. If $c=1$ and $d\notin\mathbb{R},$ we use Lemma 3.7 and we argue as in the case $|c|\in(0,1)$ to conclude that $C_{\phi}$ also is not an absolutely Cesàro bounded operator. ∎