\documentclass[twoside]{article} \usepackage{amssymb,amsmath} % font used for R in Real numbers \pagestyle{myheadings} \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} \\ % 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 $$\dot u(t)=A u(t) \quad t\geq 0 \label{A}$$ and the Cauchy problem $$\begin{gathered} \dot u(t) =Au(t)+e^{i \mu t}x \quad t\geq 0\\ u(0)=0 \end{gathered} \label{Amux}$$ 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 $$\begin{gathered} \dot u(t) = Au(t)\\ u(0)=x\in X \end{gathered} \label{1}$$ is well-posed and the mild solution of \eqref{1} is defined by $$u(t)=T(t)x, \quad t\geq 0. \label{2}$$ 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 $$\begin{gathered} \dot u(t) =Au(t) + f(t) \quad t\geq 0\\ u(0)=x \end{gathered} \label{3}$$ is $$u_{f}(t)=T(t)x+\int^t_0 T(t-\xi) f(\xi)d \xi, \quad t\geq 0. \label{4}$$ 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 $$\sigma(A)\subset\{\lambda\in{\bf C}: Re(\lambda)<0\}. \label{5}$$ 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 $$\sup_{\mu \in\mathbb{R}}\sup_{t>0} \|\int^t_0 e^{- i \mu \xi} T(\xi)y d \xi \| = M(x) < \infty,\quad \forall \mu \in \mathbb{R} \mbox{ and } \forall x\in X \label{6}$$ 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 $$\begin{gathered} \dot u(t)=A(t)u(t) \quad t\geq 0\\ u(0)=x\in X \end{gathered} \label{7}$$ 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 $$\|U(t,s)\|\leq M_{\omega}e^{\omega (t-s)}, \quad \forall t\geq s\geq 0. \label{8}$$ 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 $$\begin{gathered} \dot u(t)=A(t)u(t)+f(t),\quad t\geq 0\\ u(0)=x \end{gathered} \label{9}$$ is $$u_{f}(t,x) : = U(t,0)x+\int^t_0 U(t, \tau)f(\tau)d\tau, (t\ge 0). \label{10}$$ 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 $$s\mapsto({\cal T}(t)F)(s):=U(s, s-t)F(s-t):\mathbb{R}\to X \label{11}$$ 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 $$s\mapsto ({\cal S}(t)G)(s):=U(s, s-t)G(s-t):\mathbb{R}\to X, \label{12}$$ 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 [T(t)h](s)=\left\{\begin{array}{ll} U(s, s-t)h(s-t), & s\ge t\\ 0, & 0\le s0.$$\end{theorem} The proof of this theorem follows from Theorem \ref{thm4} using an argument given in \cite[Corollary 2.4]{MRS}. Another application of Theorem \ref{thm4} is the following inequality of Landau's type. For more details about theorems of this form, see \cite{BD}. \begin{theorem} \label{thm6} Let {\cal U}=\{U(t, s): t\ge s\ge 0\} be a q-periodic evolution family of bounded linear operators acting on X and let f\in{\cal X}:=AAP_0(\mathbb{R}_+, X). Suppose that the following two conditions are satisfied: \begin{itemize} \item[(i)] u_f(\cdot, 0)=\int_0^{\cdot}U(\cdot, s)f(s)ds belongs to {\cal X} \item[(ii)] v_f(\cdot):=\int_0^{\cdot}(\cdot-s)U(\cdot, s)f(s)ds belongs to {\cal X}. \end{itemize} If \sup\{\|U(t, s)\|: t\ge s\ge 0\}=M<\infty then $$\|u_f(\cdot, 0)\|_{\cal X}^2\le 4M^2\|f\|_{\cal X}\cdot\|v_f(\cdot)\|_{\cal X}. \label{15}$$ \end{theorem} \paragraph{Proof.} Let {\bf T} the evolution semigroup associated to {\cal U} on the space {\cal X} and (A, D(A)) its infinitesimal generator. From Lemma \ref{lm3} results that u_f(\cdot, 0) belongs to D(A) and Au_f(\cdot, 0)=-f. Using Fubini's theorem it is easy to see that v_f(t)=\int_0^tU(t, r)u_f(r, 0)dr for every t\ge 0. Then from Lemma \ref{lm3} follows that v_f(\cdot)\in D(A^2) and A^2v_f(\cdot)=f. Now the inequality \eqref{15} can be easily obtained from Lemma \ref{lm1}. \hfill\square \smallskip For U(t, s)=I, Theorem \ref{thm6} can be generalized in the following sense. \begin{proposition} \label{prop7} Let f be a X-valued, locally Bochner integrable function on \mathbb{R}_+ and g, h the mappings on \mathbb{R}_+ given by$$g(t):=\int_0^tf(s)ds \quad\mbox{and}\quad h(t)=\int_0^t(t-s)f(s)ds. $$If \sup\{|f(t)|: t\ge 0\}=M_1<\infty and \sup\{|h(t)|: t\ge 0\}=M_3<\infty then $$|g(r)|^2\le 4M_1M_3, \quad\forall r\ge 0. \label{16}$$ \end{proposition} \paragraph{Proof} For every t\ge 0 and any X-valued function F on \mathbb{R}_+ let us consider the function F_t given by$$ F_t(s)=\left\{\begin{array}{ll} F(s-t), & s\ge t\\ 0, & 0\le s0. \label{18} If $M_1=0$ or $M_3=0$ then $g=0$ and \eqref{16} holds with equality. 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