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\markboth{\hfil A spectral mapping theorem \hfil EJDE--2002/70}
{EJDE--2002/70\hfil Constantin Bu\c se \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 70, pp. 1--11. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\vspace{\bigskipamount} \\
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A spectral mapping theorem for evolution semigroups on
asymptotically almost periodic functions defined on the
half line
%
\thanks{ {\em Mathematics Subject Classifications}: 47G10, 47D03, 47A63.
\hfil\break\indent
{\em Key words}: periodic families, almost periodic functions,
exponential stability.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted April 02, 2002. Published July 25, 2002.} }
\date{}
%
\author{Constantin Bu\c se}
\maketitle
\begin{abstract}
We prove that the evolution semigroup on $AAP_0(\mathbb{R}_+, X)$
is strongly continuous. Then we prove some properties of the generator
of this evolution semigroup and show some applications in the
theory of inequalities.
\end{abstract}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\numberwithin{equation}{section}
\section{Introduction}
Let $X$ be a complex Banach space and ${\cal L}(X)$ the Banach
algebra of all linear and bounded operators acting on $X$. The norms in $X$ and
in ${\cal L}(X)$ will be denoted by $\|\cdot\|$.
Let $A$ be a linear and bounded operator acting on $X$. We consider the system
\begin{equation}
\dot u(t)=A u(t) \quad t\geq 0 \label{A}
\end{equation}
and the Cauchy problem
\begin{equation} \begin{gathered}
\dot u(t) =Au(t)+e^{i \mu t}x \quad t\geq 0\\
u(0)=0
\end{gathered} \label{Amux}
\end{equation}
where $\mu\in\mathbb{R}$ and $x\in X$. It is well-known
\cite{DK,Ba} that the system \eqref{A} is exponentially stable; that
is, there exist the constants $N>0$ and $\nu >0$ such that
$$
\|e^{tA}\|\leq Ne^{-\nu t} \quad \mbox{for all } t\geq 0,
$$
if and only if the solution of the Cauchy problem \eqref{Amux} is
bounded for every $\mu\in\mathbb{R}$ and any $x\in X$, i.e., if and
only if
$$
\sup_{t> 0} \|\int^t_0 e^{-i \mu \xi} e^{\xi A} x d\xi\|< \infty, \quad
\forall \mu\in\mathbb{R}\mbox{ and } \forall x\in X.
$$
For unbounded linear operators, the above result is false,
see e.g. \cite[Example 3.1]{RB}. However some weaker results, described
as follows, hold.
Let ${\bf T}= \{ T(t): t\geq 0\}\subset {\cal L}(X)$ be
a strongly continuous semigroup on $X$ and $A:D(A)\subset X\to X$
its infinitesimal generator. It is well-known that the Cauchy
problem
\begin{equation} \begin{gathered}
\dot u(t) = Au(t)\\
u(0)=x\in X \end{gathered} \label{1}
\end{equation}
is well-posed and the mild solution of \eqref{1} is defined by
\begin{equation}
u(t)=T(t)x, \quad t\geq 0. \label{2}
\end{equation}
For well-posedness of equations we refer the reader to \cite{S1,S2}
and the references therein. The mild solution
of the non-homogeneous Cauchy problem
\begin{equation} \begin{gathered}
\dot u(t) =Au(t) + f(t) \quad t\geq 0\\
u(0)=x
\end{gathered} \label{3}
\end{equation}
is
\begin{equation}
u_{f}(t)=T(t)x+\int^t_0 T(t-\xi) f(\xi)d \xi, \quad t\geq 0. \label{4}
\end{equation}
Particularly for $x=0$, $y\in X$, $\mu\in\mathbb{R}$ and
$f(t):= e^{i \mu t}y$, the solution $u_{f}(\cdot)$ can be written as
$$
u_{\mu y}(t)=\int^t_0 T(t- \xi)e^{i \mu \xi}y d \xi = e^{i \mu t}
\int^t_0 e^{-i \mu \xi} T(\xi) y d \xi.
$$
In \cite{RB}, it is shown that if $u_{\mu y}(\cdot)$ is bounded
on $\mathbb{R}_+$
for every $\mu\in\mathbb{R}$ and all $y\in X$ then
\begin{equation}
\sigma(A)\subset\{\lambda\in{\bf C}: Re(\lambda)<0\}. \label{5}
\end{equation}
Conversely if \eqref{5} holds and ${\bf T}$ is uniformly bounded (i.e.
$\sup_{t\ge 0}\|T(t)\|<\infty$) then $u_{\mu y}(\cdot)$ is bounded
on $\mathbb{R}_+$ for every $\mu\in\mathbb{R}$ and all $y\in X$.
This last result is proven in \cite[Proposition 2]{BB}.
Another result of this type is due to Arendt and Batty in \cite{AB}.
For $x\in X$, Let $\omega(x)$ the infimum of all $\omega \in \mathbb{R}$
for which there exists $M_{\omega}> 0$ such that
$\|T(t)x\|\leq M_{\omega}e^{\omega t}$ for all $t\geq 0$.
Let $\omega _1({\bf T})$ the supremum of all $\omega (x)$ with $x\in D(A)$.
Frank Neubrander \cite{Neu} proved that $\omega_1({\bf T})$ is the
infimum of all $\omega \in \mathbb{R}$ with the property that
$$
\{ \mathop{\rm Re}(\lambda) > \omega\}\subset \rho
(A) \quad \mbox{and there is} \quad R(\lambda, A)x=\lim_{t\to \infty}
\int^t_0 e^{- \lambda s} T(s) x ds
$$
for every $\lambda \in {\bf C}$ with $Re(\lambda) > \omega$ and
any $x\in X$.
Neerven \cite{Ne1,Ne2} has shown that if
\begin{equation}
\sup_{\mu \in\mathbb{R}}\sup_{t>0}
\|\int^t_0 e^{- i \mu \xi} T(\xi)y d \xi \| = M(x) <
\infty,\quad \forall \mu \in \mathbb{R}
\mbox{ and } \forall x\in X \label{6}
\end{equation}
then $\omega_1({\bf T}) < 0$; that is, if \eqref{6} holds then every solution
of the system \eqref{A}, starting in $D(A)$, is exponentially stable.
However, there can be solutions of the system \eqref{A} starting in
$X\setminus D(A)$ which are not exponentially stable, even if \eqref{6}
holds, see e.g. \cite[Example 2]{B1}. Moreover, in
\cite[Corollary 5 and the proof of Theorem 4]{Ne1}
it is shown that if \eqref{6} holds then the
operator resolvent $R(\lambda, A)$ exists and the function
$\lambda\mapsto R(\lambda, A)$ is uniformly bounded
on $\{ Re(\lambda)>0\}$. Combining this fact with the
Gearhart's famous stability theorem \cite{Ge}
(see also Herbst \cite{He}, Howland \cite{Ho}, Huang \cite{Hu},
Pr\"uss \cite{P} Weiss \cite{W})
follows that if $X$ is a complex Hilbert space and \eqref{6} holds then
$$
\omega_0 ({\bf T}) :=\lim_{t\to\infty} \frac{\ln \|T(t)\|}{t}
$$
is negative, i.e. in these conditions every solution of the system
\eqref{A} is exponentially stable. This and related results are explicitly
presented in a very recent paper of Phong \cite{Vu}.
It seems that the last stability result, having \eqref{6} as hypothesis,
cannot be extended for periodic evolution families, but a weaker
result holds, see Theorem \ref{thm4} below.
For a well-posed, non-autonomous Cauchy problem
\begin{equation} \begin{gathered}
\dot u(t)=A(t)u(t) \quad t\geq 0\\
u(0)=x\in X \end{gathered} \label{7}
\end{equation}
with (possibly unbounded) linear operators $A(t)$, the mild solutions
lead to an evolution family on
$\mathbb{R}_+$, ${\cal U}=\{U(t,s): t\geq s\geq 0\} \subset {\cal L}(X)$;
that is:
\begin{itemize}
\item[(e1)] $U(t, r)=U(t, s)U(s, r)$ for all $t\ge s\ge r\ge 0$ and
$U(t,t)=I$ for any $t\ge 0$, (I is the identity operator in ${\cal L}(X))$
\item[(e2)] The maps $(t,s)\mapsto U(t,s)x:\{(t,s):t\geq s\geq
0\}\to X$ are continuous for each $x\in X$.
\end{itemize}
An evolution family is {\it exponentially bounded} if there exist
$\omega\in \mathbb{R}$ and $M_{\omega}>0$ such that
\begin{equation}
\|U(t,s)\|\leq M_{\omega}e^{\omega (t-s)}, \quad \forall t\geq s\geq 0. \label{8}
\end{equation}
An evolution family is {\it exponentially stable} if \eqref{8} holds with
some negative $\omega$.
If the evolution family ${\cal U}$ verifies the condition
\begin{itemize}
\item[(e3)] $ U(t,s)=U(t-s,0)$ for all $t\geq s\geq 0$,
\end{itemize}
then the family ${\bf T}=\{U(t,0): t\geq
0\}\subset{\cal L}(X)$ is a strongly continuous semigroup on
$X$. In this case the estimate \eqref{8} holds automatically.
If the Cauchy problem \eqref{7} is $q$-periodic, i.e. $A(t+q)=A(t)$ for
$t\geq 0$, then the corresponding evolution family ${\cal U}$
is $q$-periodic, that is,
\begin{itemize}
\item[(e4)] $U(t+q, s+q) = U(t,s)$ for all $t\geq s\geq 0$.
\end{itemize}
Every $q$-periodic evolution family is exponentially bounded
\cite[Lemma 4.1]{BP}. For a locally Bochner integrable function
$f:\mathbb{R}_+\to X$, the mild solution of the well-posed, inhomogeneous
Cauchy problem
\begin{equation}\begin{gathered}
\dot u(t)=A(t)u(t)+f(t),\quad t\geq 0\\
u(0)=x \end{gathered} \label{9}
\end{equation}
is
\begin{equation}
u_{f}(t,x) : = U(t,0)x+\int^t_0 U(t,
\tau)f(\tau)d\tau, (t\ge 0). \label{10}
\end{equation}
We also consider evolution families on the line. We shall use the same
notation as in the case of evolution families on $\mathbb{R}_+$ with the
mention that variables $s$ and $t$ can take any value in $\mathbb{R}$.
For more details about the strongly continuous
semigroups and evolution families we refer to \cite{EN}. \smallskip
We recall the notion of evolution semigroup. For more details we refer
the reader to \cite{CL,CLMR} and references therein.
Let us consider the following spaces:
\begin{itemize}
\item $BUC(\mathbb{R}, X)$ is the space of all $X$-valued, bounded and
uniformly continuous functions on the real line endowed with the sup-norm.
\item $C_{0}(\mathbb{R}, X)$ is the subspace of $BUC(\mathbb{R}, X)$ consisting of
all functions $f$ such that $\lim_{|t|\to\infty} f(t)=0$. \item $AP(\mathbb{R}, X)$ is the space of all almost periodic functions, that is,
the smallest closed subspace of $BUC(\mathbb{R}, X)$
containing the functions of the form, \cite{LZ},
$$t\mapsto e^{i\mu t}x,\quad\mu\in\mathbb{R} \mbox{ and } x\in X\,.
$$
\end{itemize}
Let ${\cal U}=\{U(t, s): t\ge s\in\mathbb{R}\}$ be a strongly continuous and exponentially
bounded evolution family of bounded linear operators on $X$. For every
$t\ge 0$ and each $F\in C_0(\mathbb{R}, X)$ the function
\begin{equation}
s\mapsto({\cal T}(t)F)(s):=U(s, s-t)F(s-t):\mathbb{R}\to X \label{11}
\end{equation}
belongs to $C_0(\mathbb{R}, X)$ and the family ${\cal T}=\{{\cal T}(t): t\ge 0\}$
is a strongly continuous semigroup on $C_0(\mathbb{R}, X)$, \cite{LM}.
If ${\cal U}=\{U(t, s): t\ge s\in\mathbb{R}\}$
is a $q$-periodic evolution family, $t\ge 0$, and $G\in AP(\mathbb{R}, X)$
then the function given by
\begin{equation}
s\mapsto ({\cal S}(t)G)(s):=U(s, s-t)G(s-t):\mathbb{R}\to X, \label{12}
\end{equation}
belongs to $AP(\mathbb{R}, X)$ and the one-parameter family
${\cal S}=\{{\cal S}(t): t\ge 0\}$ is a strongly continuous semigroup on
$AP(\mathbb{R}, X)$, \cite{NM}. ${\cal T}$ and ${\cal S}$ are called
evolution semigroups on $C_0(\mathbb{R}, X)$ and $AP(\mathbb{R}, X)$, respectively. In the following we will
consider spaces consisting of functions defined on $\mathbb{R}_+$. $AP(\mathbb{R}_+, X)$ and $C_0(\mathbb{R}_+, X)$ are the spaces consisting of all
functions $g:\mathbb{R}_+\to X$ for which there exists $G\in AP(\mathbb{R}, X)$, respectively $G\in C_0(\mathbb{R}, X)$, such that $G(s)=g(s)$ for all $s\ge 0$. $C_{00}(\mathbb{R}_+, X)$ is the subspace of $C_0(\mathbb{R}_+, X)$ consisting of
all functions $f$ for which $f(0)=0$, and
$AAP_0(\mathbb{R}_+, X)$ is the space of all $X$-valued functions $h$
such that $h(0)=0$ and there exist $f\in C_0(\mathbb{R}_+, X)$ and
$g\in AP(\mathbb{R}_+, X)$ such that $h=f+g$. For each
$h\in AAP_0(\mathbb{R}_+, X)$ and every $t\ge 0$ consider the function $T(t)h$
given by
\begin{equation}
[T(t)h](s)=\left\{\begin{array}{ll}
U(s, s-t)h(s-t), & s\ge t\\
0, & 0\le s0.$$
\end{theorem}
The proof of this theorem follows from Theorem \ref{thm4} using an argument
given in \cite[Corollary 2.4]{MRS}.
Another application of Theorem \ref{thm4} is the following inequality of Landau's type.
For more details about theorems of this form, see \cite{BD}.
\begin{theorem} \label{thm6}
Let ${\cal U}=\{U(t, s): t\ge s\ge 0\}$ be a $q$-periodic
evolution family of bounded linear operators acting on $X$ and let
$f\in{\cal X}:=AAP_0(\mathbb{R}_+, X)$. Suppose that the following two
conditions are satisfied:
\begin{itemize}
\item[(i)] $u_f(\cdot, 0)=\int_0^{\cdot}U(\cdot, s)f(s)ds$ belongs to
${\cal X}$
\item[(ii)] $v_f(\cdot):=\int_0^{\cdot}(\cdot-s)U(\cdot, s)f(s)ds$ belongs
to ${\cal X}$.
\end{itemize}
If $\sup\{\|U(t, s)\|: t\ge s\ge 0\}=M<\infty$ then
\begin{equation}
\|u_f(\cdot, 0)\|_{\cal X}^2\le 4M^2\|f\|_{\cal X}\cdot\|v_f(\cdot)\|_{\cal X}.
\label{15}
\end{equation}
\end{theorem}
\paragraph{Proof.}
Let ${\bf T}$ the evolution semigroup associated to
${\cal U}$ on the space ${\cal X}$ and $(A, D(A))$ its infinitesimal generator.
From Lemma \ref{lm3} results that $u_f(\cdot, 0)$ belongs to $D(A)$ and
$Au_f(\cdot, 0)=-f$. Using Fubini's theorem it is easy to see that
$v_f(t)=\int_0^tU(t, r)u_f(r, 0)dr$ for every $t\ge 0$.
Then from Lemma \ref{lm3} follows that $v_f(\cdot)\in D(A^2)$ and
$A^2v_f(\cdot)=f$. Now the inequality \eqref{15}
can be easily obtained from Lemma \ref{lm1}.
\hfill$\square$ \smallskip
For $U(t, s)=I$, Theorem \ref{thm6} can be generalized in the
following sense.
\begin{proposition} \label{prop7}
Let $f$ be a $X$-valued, locally Bochner integrable function on
$\mathbb{R}_+$ and $g, h$ the mappings on $\mathbb{R}_+$ given by
$$g(t):=\int_0^tf(s)ds \quad\mbox{and}\quad h(t)=\int_0^t(t-s)f(s)ds.
$$
If $\sup\{|f(t)|: t\ge 0\}=M_1<\infty$ and $\sup\{|h(t)|: t\ge 0\}=M_3<\infty$
then
\begin{equation}
|g(r)|^2\le 4M_1M_3, \quad\forall r\ge 0. \label{16}
\end{equation}
\end{proposition}
\paragraph{Proof}
For every $t\ge 0$ and any $X$-valued function $F$
on $\mathbb{R}_+$ let us consider the function $F_t$ given by
$$ F_t(s)=\left\{\begin{array}{ll}
F(s-t), & s\ge t\\
0, & 0\le s0. \label{18}
\end{equation}
If $M_1=0$ or $M_3=0$ then $g=0$ and \eqref{16} holds with equality. If $M_1>0$
and $M_3>0$ then \eqref{16} can be obtained from \eqref{18} with
$t=\sqrt{4M_3/M_1}$. \hfill$\square$
\paragraph{Remark.}
If $f$ is a continuous function then Proposition \ref{prop7}
follows directly and easily by \cite{KR}, because $g'(t)=f(t)$ and
$h'(t)=g(t)$ for all $t\ge 0$. The author thanks to the referee who
brought to the author's attention about this fact.
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\noindent\textsc{Constantin Bu\c se}\\
Department of Mathematics\\
West University of Timi\c soara\\
Bd. V. P\^arvan No. 4\\
1900-Timi\c soara, Rom\^ania \\
e-mail buse@hilbert.math.uvt.ro
\end{document}