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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} \\
 %
  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 s<t.
\end{array}
\right. \label{13}
\end{equation}

\section{Results}

\begin{lemma} \label{lm1}
Let ${\bf T}=\{T(t): t\ge 0\}$ be a strongly continuous
semigroup and $A: D(A)\subset X\to X$ its infinitesimal generator.
If ${\bf T}$ is uniformly stable, i.e. there exists a positive constant
$M$ such that $\sup_{t\ge 0}\|T(t)\|=M<\infty$, then
$$\|Ax\|^2\le 4M^2\|A^2x\|\|x\|,\quad\mbox{ for all } x\in D(A^2).$$
\end{lemma}
The proof of this lemma can be found in \cite{KR}.

\begin{lemma} \label{lm2}
The semigroup ${\bf T}=\{T(t): t\ge 0\}$ described in \eqref{13}
 is defined on $AAP_0(\mathbb{R}_+, X)$ and is strongly continuous.
This semigroup is called evolution semigroup associated to ${\cal U}$
on the space  $AAP_0(\mathbb{R}_+, X)$.
\end{lemma}

\paragraph{Proof.}
 Let $h=f+g$ with $f\in C_0(\mathbb{R}_+, X)$ and
$g\in AP(\mathbb{R}_+, X)$ such that $h(0)=0$ and let
$F\in C_0(\mathbb{R}, X)$ and $G\in AP(\mathbb{R}, X)$ such that
$F(s)=f(s)$ and $G(s)=g(s)$ for all $s\ge 0$. It is easy to see that for
each $t\ge 0$, we have
$$
T(t)h=(1_{[0, \infty)}){\cal S}(t)G)+(1_{[t, \infty)}T(t)f-1_{[0, t)}
{\cal S}(t)G).
$$
Here $\{{\cal S}(t)\}_{t\ge 0}$ is the evolution semigroup on
$AP(\mathbb{R}, X)$
given in \eqref{12} and
$1_J$ is the characteristic function of the interval $J$. If we put
 $g_1:=1_{[0, \infty)}{\cal S}(t)G$ and $f_1:=1_{[t, \infty)}T(t)f-
1_{[0, t]}{\cal S}(t)G$ then $f_1\in C_0(\mathbb{R}_+, X)$,
$g_1\in AP(\mathbb{R}_+, X)$ and $(f_1+g_1)(0)=0$, so $T(t)$ is defined on
$AAP_0(\mathbb{R}_+, X)$ for every $t\ge 0$. Moreover, for all
 $h\in AAP_0(\mathbb{R}_+, X)$, we have:
\begin{align*}
\sup_{s\ge 0}\|T(t)h-h\|  \le &
 \sup_{s\ge t}\|(T(t)h-h)(s)\|+\sup_{s\in [0, t]}\|(T(t)h-h)(s)\|\\
\le& \sup_{s\ge t}\|({\cal S}(t)G-G)(s)\|+\sup_{s\ge t}
 \|({\cal T}(t)F-F)(s)\|+\sup_{s\in [0, t]}\|h(s)\|\\
\le & \|{\cal S}(t)G-G\|_{AP(\mathbb{R}, X)}+\|{\cal T}(t)F-F\|_{C_0(\mathbb{R}, X)}+
  \sup_{s\in [0, t]}\|h(s)\|.
\end{align*}
Thus $\|T(t)h-h\|_{AAP_0(\mathbb{R}_+, X)}$ tends to $0$ as $t\to 0$;
i.e., the semigroup ${\bf T}$ is strongly continuous. \hfill$\square$

\begin{lemma} \label{lm3}
Let ${\cal U}=\{U(t, s): (t, s)\in \Delta\}$ be a $q$-periodic
evolution family of bounded linear operators on $X$,
 ${\bf T}=\{T(t): t\ge 0\}$
 the evolution semigroup associated to ${\cal U}$ on the space
 $AAP_0(\mathbb{R}_+, X)$, given in \eqref{13}, and
$(A, D(A))$ its infinitesimal generator.
Let $u, f$ in $AAP_0(\mathbb{R}_+, X)$. The following two
statements are equivalent.
\begin{enumerate}
\item $u\in D(A)$ and $Au=-f$
\item $u(t)=\int_0^t U(t, s)f(s)ds$ for all $t\ge 0$.
\end{enumerate}
\end{lemma}
The proof of this lemma follows from lemma \ref{lm2} using an argument
given in  \cite[lemma 1.1]{MRS}. \smallskip

By $P_q^0(\mathbb{R}_+, X)$ we will denote the space of all
$q$-periodic, $X$-valued functions $f$ on $\mathbb{R}_+$,  such that $f(0)=0$.
\begin{theorem} \label{thm4}
Let ${\cal U}, {\bf T}$ and $(A, D(A))$ as in Lemma \ref{lm3}. The
following five statements are equivalent.
\begin{itemize}
\item[(i)] ${\cal U}$ is uniformly exponentially stable
\item[(ii)] $A$ is an invertible operator
\item[(iii)] For every $f\in AAP_0(\mathbb{R}_+, X)$ the function
$t\mapsto u_f(t, 0)=\int_0^t U(t, s)f(s)ds$ belongs to
 $AAP_0(\mathbb{R}_+, X)$
\item[(iv)]  For every $f\in AAP_0(\mathbb{R}_+, X)$ the function $u_f(\cdot, 0)$
is bounded on $\mathbb{R}_+$
\item[(v)]  For every $f\in P_q^0(\mathbb{R}_+, X)$ and  $\mu\in\mathbb{R}$
 the function
 $t\mapsto \int_0^t U(t, s)e^{-i\mu s}f(s)ds$ is bounded on
$\mathbb{R}_+$.
\end{itemize}
\end{theorem}

\paragraph{Proof.}
$(i)\Rightarrow (ii)$
 Let ${\cal X}:=AAP_0(\mathbb{R}_+, X)$. Then
\begin{eqnarray*}
\|T(t)\|_{{\cal L}({\cal X})} & =& \sup\{\sup_{s\ge t}
\|U(s, s-t)h(s-t)\|: \|h\|_{{\cal X}}=1\}\\
&\le&  \sup\{M_{\omega}e^{\omega t}\sup_{s\ge t}\|h(s-t)\|:
 \|h\|_{{\cal X}}=1\}
\le  M_{\omega}e^{\omega t},
\end{eqnarray*}
for all $t\ge 0$.
Thus $\omega_0({\bf T}):=\lim_{t\to\infty}\frac{\ln(\|T(t)\|)}{t}\le \omega<0$
and by the general theory of linear semigroups \cite[p. 4-5]{Pa}
it follows that $0\in\rho(A)$; that is,
$A$ is an invertible operator.

\noindent$(ii)\Rightarrow (iii)$ follows from Lemma \ref{lm3}.

\noindent$(iii)\Rightarrow (iv)$ and $(iv)\Rightarrow (v)$ are obvious.

\noindent$(v)\Rightarrow (i)$ follows as in \cite[Theorem 4]{B};
see also \cite[Theorem 2.1]{BJP}, or \cite{NM}. \hfill$\square$

\paragraph{Remarks:}
{\bf 1.} Let $A(\cdot)$ be a $q$-periodic operator-valued function on
$\mathbb{R}_+$ and $\{U(t, s): t\ge s\ge 0\}$ the $q$-periodic evolution family
 associated to it. Since the
function $t\mapsto\int_0^tU(t, s)f(s)ds$ is the mild solution of the
abstract Cauchy problem
\begin{equation} \begin{gathered}
\dot u(t)=A(t)u(t)+f(t)\quad (t\ge 0)\\
u(0)=0, \end{gathered} \label{14}
\end{equation}
the equivalence between {\bf (i)} and {\bf (iii)} from Theorem \ref{thm4} can be
 interpreted  in the following way:
\begin{quote}
The system $\dot u(t)=A(t)u(t)$ is exponentially stable if and only
 if for every input $f\in AAP_0(\mathbb{R}_+, X)$ the mild solution of the
Cauchy problem \eqref{14}, belongs to $AAP_0(\mathbb{R}_+, X)$.
\end{quote}

\noindent{\bf 2.} A related result on the individual stability, where the space
$ AAP(\mathbb{R}_+, X)$ is also involved, can be found in
\cite[Proposition 2.9]{BHR}.

\noindent{\bf 3.} If every solution of the $q$-periodic homogeneous system
$\dot v(t)=A(t)v(t)$, $t\in\mathbb{R}_+$ is exponentially stable then for each
$f\in P_q^0(\mathbb{R}_+, X)$ and every $\mu\in\mathbb{R}$ there is a mild solution
$u(\cdot)$ of the Cauchy problem
\begin{gather*}
\dot v(t)=A(t)v(t)+e^{i\mu t}f(t), \quad (t\ge 0)\\
v(0)=x
\end{gather*}
such that the function $t\mapsto e^{-i\mu t}u(t)$ is $q$-periodic on
$\mathbb{R}_+$.
Indeed the family ${\cal U}=\{U(t, s): t\ge s\ge 0\}$ is exponentially stable,
so the resolvent set $\rho(V)$ contains all complex number $z$ with $|z|\ge 1$. Here $V:=U(q, 0)$ denotes the monodromy operator associated to ${\cal U}$. Then for each $\mu\in\mathbb{R}$ we have that $e^{i\mu q}\in\rho(V)$. Let $y:=\int_0^q U(q, \tau)e^{i\mu\tau}$ and $x=(e^{i\mu q}-V)^{-1}(y)$. Using \eqref{10} follows that
$$u(t)=U(t, 0)x+\int_0^tU(t, \tau)e^{i\mu\tau}f(\tau)d\tau.$$
Thus
$$
u(t+q)=U(t, 0)Vx+U(t, 0)y+e^{i\mu q}\int_q^tU(t, \tau)e^{i\mu\tau}f(\tau)d\tau.
$$
In the end we obtain that $e^{-i\mu q}u(t+q)=u(t)$ for every $t\ge 0$, and now it is easy to see that the function $e^{-i\mu\cdot}u(\cdot)$ is
$q$-periodic.

\noindent{\bf 4.}
The proof of Theorem \ref{thm4}  depends of Lemma \ref{lm3}
which also depends of the strongly continuity of the evolution semigroup
${\bf T}$, defined in \eqref{13}. Thus the condition, $h(0)=0$, which appears in the definition of
the space $AP_0(\mathbb{R}_+, X)$, is essentially in the proof of
Theorem \ref{thm4}, because
it is involved in the proof of Lemma \ref{lm2}. \hfill$\square$ \smallskip

An immediate consequence of Theorem \ref{thm4} is the spectral mapping theorem for the
evolution semigroup ${\bf T}$ on $AAP_0(\mathbb{R}_+, X)$. Similar results, but for
the evolution semigroup on $C_{00}(\mathbb{R}_+, X)$, can be found in
\cite[Theorem 2.2, Corollary 2.4]{MRS}.
  Recall that $\sigma(L)$ denotes the {\it spectrum} of the linear operator
$L$ acting on $X$, and
$\rho(L):={\bf C}\setminus\sigma(L)$ is the resolvent set of $L$. The
{\it spectral radius} of $L$ is $r(L):=\sup\{|\lambda|: \lambda\in\sigma(L)\}$
and the {\it spectral bound} is
 $s(L):=\sup\{ Re(\lambda): \lambda\in\sigma(L)\}$.

\begin{theorem} \label{thm5}
Let ${\cal U}$ be a $q$-periodic evolution family of bounded
 linear operators on the Banach space $X$. Then the evolution semigroup
 ${\bf T}$ on $AAP_0(\mathbb{R}_+, X)$ satisfies the spectral mapping theorem,
 as follows:
$$e^{t\sigma(A)}=\sigma(T(t))\setminus\{0\}, \quad t\ge 0.$$
Moreover,
$\sigma(A)=\{\lambda\in{\bf C}: Re(\lambda)\le s(A)\}$
and
$$ \sigma(T(t))=\{\lambda\in{\bf C}: |\lambda|\le r(T(t))\}, \mbox{ for all }
t>0.$$
\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
$$ F_t(s)=\left\{\begin{array}{ll}
F(s-t),  & s\ge t\\
 0,  & 0\le s<t.
\end{array}\right.$$
With this notation, we have
\begin{equation}
h_t(r)-h(r)+tg(r)=\int_0^t(t-s)f_s(r)ds,\quad\forall t\ge 0,
 \mbox{ and }\forall r\ge 0. \label{17}
\end{equation}
Passing to the norm in this equation, we obtain
\begin{equation}
\|g(r)\|\le\frac{2M_3}{t}+\frac{tM_1}{2},\quad\forall t>0. \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

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