\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 175, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2012/175\hfil Spectral mapping theorem]
{Spectral mapping theorem for an evolution semigroup on a space
of vector-valued almost-periodic functions}
\author[O. Saierli \hfil EJDE-2012/175\hfilneg]
{Olivia Saierli} % in alphabetical order
\address{Olivia Saierli \newline
West University of Timisoara, Department of Mathematics,
Bd. V. Parvan No. 4, 300223 - Timisoara, Rom\^ania.\newline
Tibiscus University of Timisoara, Department of Computer Sciences,
Str. Lasc\u ar Catargiu, No. 4-6, 300559 - Timisoara, Rom\^ania}
\email{saierli\_olivia@yahoo.com}
\thanks{Submitted May 11, 2012. Published October 12, 2012.}
\subjclass[2000]{47A05, 47A30, 47D06, 47A10, 35B15, 35B10}
\keywords{Periodic evolution families; uniform exponential stability;
\hfill\break\indent boundedness; evolution semigroup;
almost periodic functions}
\begin{abstract}
We give some characterizations for exponential stability of a periodic
evolution family of bounded linear operators acting on a Banach
space in terms of evolution semigroups acting on a special space of
almost periodic functions. As a consequence, a spectral mapping
theorem is stated.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction}
In the recent article \cite{BLNS12}, some connections between exponential
stability of a $q$-periodic evolution family of bounded linear operators
acting on a Banach space and spectral properties of the infinitesimal
generator of the evolution semigroup associated to the evolution family,
was established. There we cannot close the chain of equivalences, as
in Theorem \ref{Theorem 1-5} below, because the state space of functions
where the evolution semigroup acts, is not rich enough.
The aim of this article is to enlarge the state space of functions, used there,
such that the chain to be closed. As consequence, a spectral mapping
theorem for the evolution semigroup is obtained.
This article is organized as follows.
The next section contains the necessary definitions for the paper to
be self-contained. In the third section we introduce the evolution
semigroup associated with the periodic evolution family. Section $4$
is devoted to prove the main result, while the last section deals with
a spectral mapping theorem for the evolution semigroup, which is a consequence
of the theoretical result established in the previous section.
\section{Notation and preliminary results}
Throughout this article $X$ stands for a Banach space and $\mathcal{L}(X)$
denotes the Banach algebra of all linear and bounded operators acting on $X$.
The norms in $X$ and in $\mathcal{L}(X)$ are denoted by the same symbol,
namely with $\|\cdot\|$.
Let $q>0$. Recall that a family
$\mathcal{U}=\{U(t,s):t\geq s\geq0\}\subset\mathcal{L}(X)$
is a strongly continuous and $q$-periodic evolution family on $X$ if:
\begin{enumerate}
\item $U(t,s)U(s,r)=U(t,r)$ for all $t\geq s\geq r\ge 0$.
\item $U(t,t)=I$ for $t\ge 0$, where $I$ is the identity operator
of $\mathcal{L}(X)$.
\item For each $x\in X$, the map
$$
(t,s)\mapsto U(t,s)x:\{(t,s)\in\mathbb{R}^2:t\geq s\ge 0\}\to X
$$
is continuous.
\item $U(t+q, s+q)=U(t, s)$ for all pairs $(t, s)$ with $t\ge s\ge 0$.
\end{enumerate}
Clearly, any $q$-periodic evolution family $\mathcal{U}=\{U(t, s)\}$ defined
for the pairs $(t, s)$ with $t\ge s\ge 0$ could be extended to a
$q$-periodic evolution family for all pairs $(t, s)$ with $t\ge s\in\mathbb{R}$,
by setting $U(t, s)=U(t+kq, s+kq)$, where $k$ is the smallest positive integer
number for which $s+kq\ge 0$. We say that the evolution family
$\mathcal{U}$ {\it has exponential growth} if there exist the constants $M\geq 1$
and $\omega\in\mathbb{R}$
such that $\|U(t,s)\|\leq Me^{\omega(t-s)}$, for all $t\geq s$.
Every strongly continuous and $q$-periodic evolution family acting on a
Banach space has an exponential growth, \cite{BP}.
Recall that a one parameter family ${\bf T}=\{T(t)\}_{t\geq0}$ is a strongly
continuous semigroup if $T(t+s)=T(t)\circ T(s)$ for all $t\ge s\ge 0$, $T(0)=I$
and for each $x\in X$ the map $t\mapsto T(t)x$ is
continuous. If a strongly continuous evolution family
$\mathcal{U}=\{U(t, s)\}_{t\ge s\ge 0}$, verifies the convolution
condition, $U(t, s)=U(t-s, 0)$, for every pair $(t, s)$ with $t\ge s\ge 0$,
then the one parameter family, $\{T(t)\}_{t\ge 0}$, defined
by $T(t):=U(t, 0)$, is a strongly continuous semigroup.
Each strongly continuous semigroup $\mathbf{T}$ has an infinitesimal generator
$A: D(A)\subset X\to X$, defined by $Ax:=\frac{d}{dt}T(t)x|_{t=0}$.
It is well-known that $A$ is linear, densely defined and closed operator.
The domain $D(A)$ consists by all $x\in X$ for which the map $t\mapsto T(t)x$
is differentiable at $t=0$. By $\rho(A)$ is denoted the resolvent set
of $A$, i.e. the set of all complex scalars $z$ for which $zI-A$ is an
invertible operator in $\mathcal{L}(X)$.
The set $\sigma(A):=\mathbb{C}\setminus\rho(A)$
is the spectrum of the operator $A$ and the set
$s(A):=\sup\{Re(\lambda):\lambda\in\sigma(A)\}$ is the spectral bound of $A$.
For further details concerning the theory
of strongly continuous semigroups we refer to the monographs
\cite{EN,ABHN}.
\begin{proposition}\label{prop1}
Let $\mathcal{U}=\{U(t,s): t\geq s\geq 0\}$ be a strongly
continuous and $q$-periodic evolution family acting on the Banach
space $X$. The following four statements are equivalent:
\begin{enumerate}
\item The family $\mathcal{U}$ is uniformly exponentially stable.
\item There exist two positive constants $N$ and $\nu$ such that
$$
\|U(t,0)\|\leq N e^{-\nu t}\mbox{, for all }t\geq 0.
$$
\item The spectral radius of $U(q,0)$ is less than one; i.e.,
$$
r(U(q, 0)):=\sup\{|\lambda|, \lambda\in\sigma(U(q, 0))\}
=\lim_{n\to\infty}\|U(q, 0)^n\|^{\frac{1}{n}}<1.
$$
\item For each $\mu\in\mathbb{R}$, one has
$$
\sup_{n\geq 1}\|\sum_{k=1}^{n}e^{-i\mu k}U(q, 0)^{k}\|:=M(\mu)<\infty.
$$
\end{enumerate}
\end{proposition}
The proof of the implications $(1)\Rightarrow
(2)\Rightarrow (3)\Rightarrow (4)$ is obvious. The proof of
$(4)\Rightarrow (1)$ can be found in \cite[Lemma~1]{BCDS05}.
\section{An evolution semigroup}
In this section we consider a space of $X$-valued functions and define an
evolution semigroup acting on it. For this purpose, we need the following spaces:
\begin{itemize}
\item $BUC(\mathbb{R}, X)$ which is the space of all $X$-valued bounded
uniformly continuous functions defined on $\mathbb{R}$, endowed with the ``sup"
norm $\|f\|_\infty:=\sup_{t\in\mathbb{R}}\|f\|$.
\item $P_q(\mathbb{R}, X)$ which is the subspace of $BUC(\mathbb{R}, X)$
consisting of all functions $F$ such that $F(t+q)=F(t)$ for all $t\in\mathbb{R}$.
\item $AP_1(\mathbb{R}, X)$ which is the space of all $X$-valued functions
defined on $\mathbb{R}$ representable in the form
$f(t)=\sum_{k=-\infty}^{k=\infty}e^{i\mu_k t}c_k(f)$ for all $t\in\mathbb{R}$,
where $\mu_k\in\mathbb{R}$, $c_k(f)\in X$ and
$$
\|f\|_1:=\sum_{k=-\infty}^{k=\infty}\|c_k(f)\|<\infty.
$$
Further details about the space
$AP_1(\mathbb{R}, X)$ can be found in \cite{Cordu09}.
\end{itemize}
For an arbitrary $t\geq 0$, we denote by $\mathcal{A}_t$ the set of all
$X$-valued functions defined on $\mathbb{R}$ such that
there exists a function $F$ in $P_q(\mathbb{R}, X)\cap AP_1(\mathbb{R}, X)$
with $F(t)=0$, $f=F_{|_{[t,\infty)}}$ and $f(s)=0$,
for all $s0\\
u(0)=0,
\end{gather*}
is bounded on $\mathbb{R}_+$.
\end{corollary}
\section{Applications}
An immediate consequence of Theorem \ref{Theorem 1-5} is the spectral mapping
theorem for the evolution semigroup
$\mathcal{T}$ on $\widetilde{E}(\mathbb{R},X)$. Similar results can be found
in \cite[Theorem~2.5]{Bu02}, \cite[Theorem~3.5]{BKRT12}, \cite[Theorem~3.6]{BL03},
\cite[Theorem~3.1]{BJ03}, \cite[Corollary~2.4]{MRS} for evolution semigroups
acting on other spaces.
\begin{theorem}
Let $\mathcal{U}$ be a strongly continuous and $q$-periodic evolution family
acting on $X$ and let $\mathcal{T}$ be its associated evolution semigroup
on $\widetilde{E}(\mathbb{R},X)$. Let denote by $G$ the infinitesimal generator
of $\mathcal{T}$. Suppose that there is a dense subset $D$ of $X$ such that
for each $x\in D$ the map $s\mapsto U(s,0)x:\mathbb{R}_+\to X$ satisfy a
Lipschitz condition on $\mathbb{R}_+$. Then
$$
\sigma(G)=\{z\in\mathbb{C}:Re(z)\leq s(G)\}.
$$
\end{theorem}
\begin{proof}
It is well-known that $\rho(G)\supseteq\{z\in\mathbb{C}:Re(z)>s(G)\}$.
To establish the converse inclusion, let $\lambda\in\rho(G)$ and
$\mu\in\mathbb{C}$ with $Re(\mu)\geq Re(\lambda)$. We prove that $\mu\in\rho(G)$.
Consider the evolution family $U_\lambda(t,s):=e^{-\lambda(t-s)}U(t,s)$,
$t\geq s\geq 0$ whose associated evolution semigroup is
$\mathcal{T}_\lambda(t):=e^{-\lambda t}\mathcal{T}(t)$.
Obviously, $\lambda I-G$ is the infinitesimal generator of $\mathcal{T}_\lambda$.
Because $\lambda I-G$ is invertible and applying Theorem \ref{Theorem 1-5},
$\mathcal{T}_\lambda$ (and then $\mathcal{T}_\mu$)is
uniformly exponentially stable. Therefore, by applying again
Theorem \ref{Theorem 1-5}, $\mu\in\rho(G)$.
\end{proof}
\subsection*{Acknowledgements}
The author would like to thank the anonymous referee for his/her valuable
recommendations that helped to improve the article. Also, the author would
like to thank Prof. Constantin Bu\c se for the useful discussions
and suggestions during the course of this paper.
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\end{document}