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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 71, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{10mm}}
\begin{document}
\title[\hfilneg EJDE-2005/71\hfil On Sylvester operator equations]
{On Sylvester operator equations, complete trajectories,
regular admissibility, \\ and stability of $C_0$-semigroups}
\author[E. Immonen\hfil EJDE-2005/71\hfilneg]
{Eero Immonen}
\address{Institute of Mathematics \\ Tampere University of Technology \\
PL 553, 33101 Tampere, Finland}
\email{Eero.Immonen@tut.fi}
\date{}
\thanks{Submitted April 17, 2005. Published June 30, 2005.}
\subjclass[2000]{47D03}
\keywords{Sylvester operator equation; regularly admissible space; \hfill\break\indent
complete nontrivial trajectory; $C_0$-semigroup; exponential stability;
strong stability; \hfill\break\indent exponential dichotomy}
\begin{abstract}
We show that the existence of a nontrivial bounded uniformly
continuous (BUC) complete trajectory for a $C_0$-semigroup $T_A(t)$
generated by an operator $A$ in a Banach space $X$ is equivalent
to the existence of a solution $\Pi = \delta_0$ to the homogenous
operator equation $\Pi S|_\mathcal{M} = A\Pi$. Here $S|_\mathcal{M}$
generates the shift $C_0$-group $T_S(t)|_\mathcal{M}$ in a closed
translation-invariant subspace $\mathcal{M}$ of $BUC(\mathbb{R},X)$,
and $\delta_0$ is the point evaluation at the origin. If, in addition,
$\mathcal{M}$ is operator-invariant and
$0 \neq \Pi \in \mathcal{L}(\mathcal{M},X)$
is any solution of $\Pi S|_\mathcal{M} = A\Pi$, then all functions
$t \to \Pi T_S(t)|_\mathcal{M}f$, $f \in \mathcal{M}$, are complete
trajectories for $T_A(t)$ in $\mathcal{M}$. We connect these results
to the study of regular admissibility of Banach function spaces for
$T_A(t)$; among the new results are perturbation theorems for regular
admissibility and complete trajectories.
Finally, we show how strong stability of a $C_0$-semigroup can be
characterized by the nonexistence of nontrivial bounded complete
trajectories for the sun-dual semigroup, and by the surjective solvability
of an operator equation $\Pi S|_\mathcal{M} = A\Pi$.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{rem}[thm]{Remark}
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\section{Introduction}
Consider the abstract Cauchy problem
\begin{equation} \label{acp} \dot{x}(t) =
Ax(t), \quad t \geq 0, \quad x(0)=x_0 \in X
\end{equation}
where $A$ generates a $C_0$-semigroup $T_A(t)$ in some Banach space $X$. It
is well known that a unique mild solution $x(t) = T_A(t)x_0$, $t
\geq 0$, of \eqref{acp} always exists. However, sometimes there
also exist so-called complete trajectories for $T_A(t)$. A
complete trajectory for $T_A(t)$ is a continuous function $x : \mathbb{R}
\to X$ such that $x(t) = T_A(t-s)x(s)$ for each $t,s \in \mathbb{R}$ for
which $t\geq s$, and $x(0) = x_0$. Such a trajectory is nontrivial
if it is not identically zero. Bounded nontrivial complete
trajectories for $T_A(t)$ are important e.g. in the study of
equations \eqref{acp} on the whole real line \cite{pazy, vu1993};
Vu has studied their existence and construction in \cite{vu1993}.
His main result asserts that if $T_A(t)$ is uniformly bounded and
sun-reflexive, and its sun-dual semigroup $T_A^\odot(t)$ (see
Subsection \ref{preli}) is not strongly stable\footnote{A $C_0$-semigroup $T(t)$ in a Banach space $Z$ is strongly stable if
$\lim_{t\to\infty}T(t)z = 0$ for every $z \in Z$},
then there
exist nontrivial bounded complete trajectories provided one of the
following conditions holds: $i\mathbb{R} \nsubseteq \sigma(A)$ or
$\mathop{\rm ran}(T_A^\odot(t_0))$ is dense in $X^\odot$ for some $t_0 > 0$.
Vu also shows in \cite{vu1993} that if the intersection of the
approximate point spectrum of $A$ and the imaginary axis is
countable, then every bounded uniformly continuous complete
trajectory for $T_A(t)$ is almost periodic provided $X$ does not
contain an isomorphic copy of $c_0$, the Banach space of sequences
convergent to $0$, or the trajectory itself is weakly compact.
A related problem for the inhomogenous abstract Cauchy problem
\begin{equation} \label{iacp} \dot{x}(t) = Ax(t)+f(t), \quad t \in \mathbb{R} \end{equation} in
$X$ is the following \cite{levitanzhikov, vuschuler}. Let
$\mathcal{M}$ be a closed translation-invariant operator-invariant
(i.e. CTO, see Definition \ref{ctospace}) subspace of $BUC(\mathbb{R},X)$,
the space of bounded uniformly continuous $X$-valued functions. We
say that $\mathcal{M}$ is regularly admissible for $T_A(t)$ if for
each $f \in \mathcal{M}$ there exists a unique mild solution $x
\in \mathcal{M}$ of \eqref{iacp}, i.e. for which
\begin{equation}
x(t) =T_A(t-s)x(s) + \int_s^t T_A(t-\tau)f(\tau)d\tau \quad \forall t
\geq s, \quad t,s \in \mathbb{R}
\end{equation}
Vu and Sch\"uler \cite{vuschuler}
showed, among other things, that $\mathcal{M}$ is regularly
admissible for $T_A(t)$ if and only if the operator equation $\Pi
S|_\mathcal{M} = A\Pi + \delta_0$, where $S|_\mathcal{M} =
\frac{d}{dx}|_\mathcal{M}$ and $\delta_0$ is the point evaluation
operator in $\mathcal{M}$ centered at the origin, has a unique
solution $\Pi \in \mathcal{L}(\mathcal{M},X)$ (see Section \ref{soln}).
The main purpose of the present article is to interconnect the
results in \cite{vu1993} and \cite{vuschuler}. To avoid repetition
we shall assume the reader to have access to these papers. Our
main results are the following. We show that the existence of a
nontrivial complete trajectory $x \in BUC(\mathbb{R},X)$ for $T_A(t)$ is
equivalent to the existence of a solution $\Pi = \delta_0$ to the
homogenous operator equation $\Pi S|_\mathcal{M} = A\Pi$ for some
closed translation-invariant subspace $\mathcal{M}$ of
$BUC(\mathbb{R},X)$. If, in addition, $\mathcal{M}$ is operator-invariant
and $0 \neq \Pi \in \mathcal{L}(\mathcal{M},X)$ is any solution of $\Pi
S|_\mathcal{M} = A\Pi$, then all functions $t \to \Pi
T_S(t)|_\mathcal{M}f$, $f \in \mathcal{M}$ are complete
trajectories for $T_A(t)$ in $\mathcal{M}$. There are three
remarkable features in these results. First of all, we do not need
to assume e.g. the uniform boundedness of $T_A(t)$ or restrict
$\sigma(A)\cap i\mathbb{R}$ in any explicit way to obtain nontrivial
bounded complete trajectories. Secondly, the complete trajectories
are known to be in $\mathcal{M}$ -- hence we can conclude more
than just boundedness of the trajectory. For example $\mathcal{M}$
could be the space $AP(\mathbb{R},X)$ of $X$-valued almost periodic
functions. Finally, these results also provide a way to construct
bounded complete trajectories for $T_A(t)$ via the solution
operators $\Pi$.
By combining our main results with those in
\cite{vu1993,vuschuler} we obtain several useful corollaries. For
example, we immediately see that if $\mathcal{M}$ is regularly
admissible for $T_A(t)$, then there cannot be complete nontrivial
trajectories for $T_A(t)$ in $\mathcal{M}$. Since all CTO
subspaces $\mathcal{M} \subset BUC(\mathbb{R},X)$ are regularly admissible
for an exponentially dichotomous semigroup $T_A(t)$
\cite{vuschuler}, exponentially dichotomous $C_0$-semigroups
cannot have bounded uniformly continuous complete trajectories.
Consequently the same is true for exponentially stable
$C_0$-semigroups.
In Section \ref{pert} we shall show that the existence of
nontrivial bounded complete trajectories for $T_A(t)$ is a
fragile property; arbitrarily small bounded additive perturbations
to the generator $A$ may destroy it. On the other hand, we shall
show that the \emph{nonexistence} of such trajectories may be a
stable property even under certain unbounded additive
perturbations to $A$. We also show that regular admissibility of
$\mathcal{M}$ for $T_A(t)$ may sustain some unbounded additive
perturbations to $A$. Hence we have another situation in which the
nonexistence of bounded complete trajectories in $\mathcal{M}$ is
not affected by perturbations to $A$.
We conclude this article with some new characterizations for
strong stability of a $C_0$-semigroup $T_A(t)$. We shall show
that if $T_A(t)$ is uniformly bounded and $\sigma_A(A) \cap i\mathbb{R}$
is countable, then $T_A(t)$ is \emph{not} strongly stable if and
only if the sun-dual semigroup $T_A^\odot(t)$ has a nontrivial
bounded complete trajectory. We also show that strong stability of
$T_A(t)$ is equivalent to the existence of a surjective solution
to the operator equation $\Pi S|_\mathcal{M} = A\Pi$ for a closed
translation-invariant subspace $\mathcal{M} \subset C_0(\mathbb{R}_+,X)$.
\subsection{Preliminaries} \label{preli}
As in the above, let $X$ be a Banach space and consider a
$C_0$-semigroup $T_A(t)$ in $X$ generated by $A$. The spectrum,
point spectrum, approximate point spectrum and resolvent set of
$A$ are denoted by $\sigma(A)$, $\sigma_P(A)$, $\sigma_A(A)$ and
$\rho(A)$ respectively. $A^*$ denotes the adjoint operator of $A$
and for every $\lambda \in \rho(A)$ we denote by $R(\lambda,A)$
the resolvent operator of $A$. A linear operator $\Delta_A :
\mathcal{D}(\Delta_A) \subset X \to X$ is called $A$-bounded if $\mathcal{D}(A)
\subset \mathcal{D}(\Delta_A)$ and for some nonnegative constants $a,b$
we have \begin{equation} \label{araj} \norm{\Delta_A x} \leq a \norm{x} + b
\norm{Ax} \quad \forall x \in \mathcal{D}(A) \end{equation} If the Banach space $X$
is not reflexive, then the adjoint semigroup $T_A^*(t)$ is not
necessarily strongly continuous. However, the subspace \begin{equation}
X^\odot = \setm{\phi \in X^*}{T_A^*(t)\phi \text{ is strongly
continuous}} \end{equation} is closed in $X^*$ and invariant for $T_A^*(t)$.
Additionally, $X^\odot = \overline{\mathcal{D}(A^*)}$ and the restriction
$T_A^*(t)|_{X^\odot}$ defines a strongly continuous semigroup in
$X^\odot$, the so-called sun-dual semigroup $T_A^\odot(t)$
\cite{engelnagel, vu1993}.
We denote the Banach space (with sup-norm) of bounded uniformly
continuous functions $t \to X$ by $BUC(\mathbb{R},X)$. The shift
operators $T_S(t)$, $t \in \mathbb{R}$, are defined for each $f \in
BUC(\mathbb{R},X)$ as $T_S(t)f = f(\cdot + t)$. It is clear that $T_S(t)$
constitutes a strongly continuous group in $BUC(\mathbb{R},X)$. Its
infinitesimal generator is the differential operator
$S=\frac{d}{dx}$ with a suitable domain of definition. Clearly the
restrictions $T_S(t)|_\mathcal{M}$ of the shift group to closed
(in the $\sup$-norm) translation-invariant subspaces $\mathcal{M}
\subset BUC(\mathbb{R},X)$ are also strongly continuous. The infinitesimal
generator of such a restriction $T_S(t)|_\mathcal{M}$ is denoted
by $S|_\mathcal{M}$. Of special interest are the so-called CTO
(closed translation-invariant operator-invariant) subspaces
$\mathcal{M}$ of $BUC(\mathbb{R},X)$:
\begin{defn} \label{ctospace}
A $\sup$-norm closed translation-invariant function space
$\mathcal{M} \subset BUC(\mathbb{R},X)$ is operator-invariant if for each
$C \in \mathcal{L}(\mathcal{M},X)$ and every $f \in \mathcal{M}$ the
function $t \to CT_S(t)f$ is in $\mathcal{M}$.
\end{defn}
Several interesting function spaces are CTO. For example:
Continuous $p$-periodic $X$-valued functions, almost periodic
functions $\mathbb{R} \to X$ and functions in $BUC(\mathbb{R},X)$ whose Carleman
spectrum is contained in a given closed subset $\Lambda$ of $i\mathbb{R}$,
the imaginary axis. Recall that almost periodic functions are
those which can be uniformly approximated by trigonometric
polynomials \cite{abhn}, and that the Carleman spectrum $sp(f)$ of
a function $f \in BUC(\mathbb{R},X)$ is defined as the set of
singularities of its Carleman transform
\begin{equation}
\widetilde{f}(\lambda) = \begin{cases}
\int_0^\infty e^{-\lambda t}f(t)dt, & \Re(\lambda) > 0 \\
-\int_{-\infty}^0 e^{-\lambda t}f(t)dt, & \Re(\lambda) < 0
\end{cases}
\end{equation}
on $i\mathbb{R}$. The reader is referred to \cite{abhn, sininen,
vuschuler} for more details.
In this article we shall use the well known fact that for
every closed translation-invariant subspace $\mathcal{M} \subset
BUC(\mathbb{R},X)$ there exists a sequence $(\mathcal{M}_n)_{n \in \mathbb{N}}
\subset \mathcal{M}$ of closed translation-invariant subspaces
with the following properties \cite{lyubich,vuschuler}:
\begin{enumerate}
\item $\mathcal{M}_n \subset \mathcal{M}_{n+1}$ for every $n \in \mathbb{N}$.
\item $S_n = S|_{\mathcal{M}_n}$ is a bounded operator for every $n \in \mathbb{N}$.
\item $\sigma(S_n) \subset \sigma(S|_\mathcal{M})$ for every $n \in \mathbb{N}$.
\item $\cup_{n \in \mathbb{N}} \mathcal{M}_n$ is dense in $\mathcal{M}$.
\end{enumerate}
\section{Mild and Strong Solutions of $\Pi S|_\mathcal{M} = A \Pi + \Delta$}
\label{soln}
Let $\mathcal{M} \subset BUC(\mathbb{R},X)$ be a closed
translation-invariant function space and let $\Delta \in
\mathcal{L}(\mathcal{M},X)$. As before, we assume that $A$ generates the
$C_0$-semigroup $T_A(t)$ in $X$. In this section we shall study
the operator equation
\begin{equation}
\label{homog} \Pi S|_\mathcal{M} = A\Pi + \Delta
\end{equation}
which will play a prominent role throughout this
article. Equation \eqref{homog} is a special instance of general
linear Sylvester type operator equations. Such equations have a
long history: For classical finite-dimensional results the reader
is referred to \cite{gantmacher} and to the excellent survey
article \cite{bhatiarosenthal}. The treatment of Bhatia and
Rosenthal \cite{bhatiarosenthal} actually also covers the case of
bounded linear operators in infinite-dimensional spaces. Many of
these results can be generalized for unbounded operators which may
or may not generate $C_0$-semigroups. Such results can be found
e.g. in \cite{ars, tanhlan2001, vu, vuschuler}.
Vu and Sch\"uler \cite{vuschuler} concentrated on the unique
solvability of \eqref{homog} for each $\Delta$. They showed that
it is equivalent to the regular admissibility of $\mathcal{M}$ for
$T_A(t)$. It turns out, however, that also nonunique solutions of
\eqref{homog} have importance. We shall see in the next section
that the existence of a nontrivial solution $\Pi = \delta_0$ to
the homogenous equation $\Pi S|_\mathcal{M} = A \Pi$
--- which implies nonuniqueness of solutions of \eqref{homog} --- is equivalent to the existence of nontrivial
bounded uniformly continuous complete trajectories for $T_A(t)$.
In order to establish this result we consider two types of
solutions for \eqref{homog}:
\begin{defn} \label{strong} \rm
An operator $\Pi \in \mathcal{L}(\mathcal{M},X)$ is called a strong
solution of \eqref{homog} if $\Pi(\mathcal{D}(S|_\mathcal{M}))\subset
\mathcal{D}(A)$ and $\Pi S|_\mathcal{M} f = A \Pi f + \Delta f$ for every
$f \in \mathcal{D}(S|_\mathcal{M})$.
\end{defn}
\begin{defn} \label{mild} \rm
An operator $\Pi \in \mathcal{L}(\mathcal{M},X)$ is called a mild
solution of \eqref{homog} if \begin{equation} \label{miehtaa} \Pi
T_S(t)|_\mathcal{M} f = T_A(t) \Pi f + \int_0^t T_A(t-s)\Delta
T_S(s)|_\mathcal{M}fds \end{equation} for every $f \in \mathcal{M}$ and
every $t \geq 0$.
\end{defn}
The main result of this section shows that mild and strong
solutions of \eqref{homog} coincide. Hence we may refer to them
as just solutions of \eqref{homog}.
\begin{thm} \label{weak=strong}
An operator $\Pi \in \mathcal{L}(\mathcal{M},X)$ is a mild solution of
\eqref{homog} if and only if it is a strong solution of
\eqref{homog}.
\end{thm}
\begin{proof}
Assume first that $\Pi \in \mathcal{L}(\mathcal{M},X)$ is a strong
solution of the \eqref{homog}. Let $f \in \mathcal{D}(S|_\mathcal{M})$ be
arbitrary. Then since $\Pi(\mathcal{D}(S|_\mathcal{M})) \subset \mathcal{D}(A)$,
we have for every $t \geq 0$ that
\begin{align}
\Pi T_S(t)|_\mathcal{M}f - T_A(t)\Pi f & = \big|_{\tau = 0}^t T_A(t-\tau)
\Pi T_S(\tau)|_\mathcal{M}f d\tau \\
& = \int_0^t \frac{d}{d\tau} T_A(t-\tau)\Pi T_S(\tau)|_\mathcal{M}f d\tau \\
& = \int_0^t T_A(t-\tau)[\Pi S|_\mathcal{M} - A\Pi]T_S(\tau)|_\mathcal{M}f
d\tau \\
& = \int_0^t T_A(t-\tau)\Delta T_S(\tau)|_\mathcal{M}f d\tau
\end{align}
because $T_S(\tau)|_\mathcal{M}f \in \mathcal{D}(S|_\mathcal{M})$ for every
$\tau \geq 0$. Since $\mathcal{D}(S|_\mathcal{M})$ is dense in $\mathcal{M}$,
we must have that
\begin{equation}
\Pi T_S(t)|_\mathcal{M} f = T_A(t) \Pi f + \int_0^t T_A(t-\tau)\Delta T_S(\tau)|_\mathcal{M}f d\tau \quad \forall f \in \mathcal{M} \quad \forall t \geq 0
\end{equation}
In other words $\Pi$ is a mild solution of \eqref{homog}.
Assume then that $\Pi \in \mathcal{L}(\mathcal{M},X)$ is a mild solution
of \eqref{homog}. We first show that $\Pi(\mathcal{D}(S|_\mathcal{M}))
\subset \mathcal{D}(A)$. Let $f \in \mathcal{D}(S|_\mathcal{M})$. Then for every
$h > 0$
\begin{align}
\frac{T_A(h)\Pi f - \Pi f}{h}
& = \frac{T_A(h)\Pi f - \Pi T_S(h)|_\mathcal{M} f}{h}
+ \frac{\Pi T_S(h)|_\mathcal{M}f-\Pi f}{h} \\
& = -\frac{\int_0^h T_A(h-\tau)\Delta T_S(\tau)|_\mathcal{M}fd\tau}{h}
+ \frac{\Pi T_S(h)|_\mathcal{M}f-\Pi f}{h} \label{conv}
\end{align}
which by the boundedness of $\Pi$ shows that $\Pi f \in \mathcal{D}(A)$; also
observe that the function $t \to \Delta T_S(t)|_\mathcal{M}f$ is
continuously differentiable so that the convolution in \eqref{conv}
is differentiable. Moreover, we see that
$A\Pi f = -\Delta f + \Pi S|_\mathcal{M} f$ for each
$f \in \mathcal{D}(S|_\mathcal{M})$. Consequently $\Pi$ is a strong solution of \eqref{homog}.
\end{proof}
\begin{rem} \rm
As mentioned in the introductory section, the special case $\Delta
= \delta_0 \in \mathcal{L}(\mathcal{M},X)$ has turned out to be
particularly important in the qualitative theory of differential
equations. Theorem \ref{weak=strong} immediately reveals why this
is so. Clearly $f(t) = \delta_0 T_S(t)|_\mathcal{M}f$ for every $f
\in \mathcal{M}$ and $t\in \mathbb{R}$ and hence if $\Pi \in
\mathcal{L}(\mathcal{M},X)$ is a solution of the operator equation $\Pi
S|_\mathcal{M} = A\Pi +\delta_0$, then \eqref{miehtaa} reads \begin{equation}
\label{mieh} \Pi T_S(t)|_\mathcal{M} f = T_A(t) \Pi f + \int_0^t
T_A(t-s)f(s)ds, \quad t \geq 0 \end{equation} so that for $x(0) = \Pi f$ the
right hand side of \eqref{mieh} is the mild solution of the
inhomogenous differential equation $\dot{x}(t) = Ax(t) + f(t)$, $t
\geq 0$. If in addition, $\mathcal{M}$ is a CTO subspace of
$BUC(\mathbb{R},X)$, then this mild solution $t \to \Pi
T_S(t)|_\mathcal{M}f$ is in $\mathcal{M}$ for every $f \in
\mathcal{M}$. Consequently we may deduce e.g. the existence of
periodic mild solutions from solvability of the operator equation
$\Pi S|_\mathcal{M} = A\Pi +\delta_0$ in a suitable space
$\mathcal{M}$. We shall not pursue this discussion any further;
the interested reader is referred to \cite{inheriteddynamics,
vuschuler} for a related discussion.
\end{rem}
The operator equation \eqref{homog} has also been studied as an operator
equation $\tau_{A,S|_\mathcal{M}} \Pi = \Delta$ in the literature \cite{ars}.
Here $\tau_{A,S|_\mathcal{M}}$ is an (unbounded) operator on
$\mathcal{L}(\mathcal{M},X)$ defined as follows.
\begin{subequations} \label{tau}
\begin{gather}
\begin{aligned}
\mathcal{D}(\tau_{A,S|_\mathcal{M}})
= \big\{&X \in \mathcal{L}(\mathcal{M},X): X(\mathcal{D}(S|_\mathcal{M}))
\subset \mathcal{D}(A), \, \exists Y \in \mathcal{L}(\mathcal{M},X) :\\
&Yu=XS|_\mathcal{M}u-AXu \: \forall u \in \mathcal{D}(S|_\mathcal{M}) \big\} \\
\end{aligned}\\
\tau_{A,S|_\mathcal{M}}X = Y
\end{gather}
\end{subequations}
It can be shown that $\tau_{A,S|_\mathcal{M}}$ is a closed operator on $\mathcal{L}(\mathcal{M},X)$ \cite{ars}. The following result is then evident.
\begin{prop} \label{tauta}
Equation \eqref{homog} has a unique solution for every
$\Delta \in \mathcal{L}(\mathcal{M},X)$ if and only if
$0 \in \rho(\tau_{A,S|_\mathcal{M}})$. The homogenous equation
$\Pi S|_\mathcal{M} = A\Pi$ has a nontrivial solution if and only if
$0 \in \sigma_P(\tau_{A,S|_\mathcal{M}})$.
\end{prop}
Proposition \ref{tauta} is particularly useful if $T_A(t)$ is a
holomorphic semigroup or if $S|_\mathcal{M}$ is bounded. By the
results of Arendt, R\"abiger and Sourour \cite{ars}, in both cases
$\sigma(\tau_{A,S|_\mathcal{M}}) = \sigma(A) +
\sigma(S|_\mathcal{M})$. We shall, however, use Proposition
\ref{tauta} in a different context in Section \ref{pert}: We make
use of the well known fact that bounded invertibility of a closed
operator is preserved under small (but possibly unbounded)
additive perturbations.
\section{Complete Trajectories, Regular Admissibility and
$\Pi S|_\mathcal{M} = A\Pi$} \label{trajectories}
The main results of this article are Theorem \ref{bct0} and
Theorem \ref{bct} below. They connect the existence of nontrivial
bounded uniformly continuous complete trajectories for $T_A(t)$ to
the nonunique solvability of the homogenous operator equation $\Pi
S|_\mathcal{M} = A\Pi$. Consequently they provide the link between
the articles \cite{vu1993} and \cite{vuschuler} mentioned in the
introductory section.
\begin{thm} \label{bct0}
Let $A$ generate a $C_0$-semigroup $T_A(t)$ in $X$. Then the
following are equivalent.
\begin{enumerate}
\item There exists a nontrivial bounded uniformly continuous complete
trajectory $x(t)$ for $T_A(t)$.
\item There exists a nontrivial closed translation-invariant subspace
$\mathcal{M}$ of \break $BUC(\mathbb{R},X)$ in which $\delta_0$ solves the operator
equation $\Pi S|_\mathcal{M} = A\Pi$.
\item There exists a nontrivial closed translation-invariant subspace
$\mathcal{M}$ of \break $BUC(\mathbb{R},X)$ for which every $x \in \mathcal{M}$
is a bounded uniformly continuous complete trajectory for $T_A(t)$.
\end{enumerate}
\end{thm}
\begin{proof}
Since by Theorem \ref{weak=strong} mild and strong solutions of
the operator equation \eqref{homog} coincide, we may restrict our
attention to mild solutions. We show $1 \implies 2 \implies 3 \implies 1$.
\begin{enumerate}
\item[$1 \implies 2$ :]
Assume that $x \in BUC(\mathbb{R},X)$ is a nontrivial bounded complete trajectory
for $T_A(t)$. Let $\mathcal{M} = \overline{\mathop{\rm span}}\setm{x(\cdot+t)}{t\in\mathbb{R}}$
where closure is taken in the $\sup$-norm. Then $\mathcal{M} \neq 0$ is a
closed translation invariant subspace of $BUC(\mathbb{R},X)$ and clearly
$\delta_0 \in \mathcal{L}(\mathcal{M},X)$. Moreover
$x(t) = \delta_0 T_S(t)|_\mathcal{M} x$ for each $t \in \mathbb{R}$. Furthermore,
for any $\tau \geq 0$ and $s \in \mathbb{R}$ we have
\begin{equation}
x(\tau+s) = \delta_0 T_S(\tau)|_\mathcal{M}T_S(s)|_\mathcal{M}x
= T_A(\tau)x(s) = T_A(\tau)\delta_0 T_S(s)|_\mathcal{M}x
\end{equation}
since $x$ is a complete trajectory for $T_A(t)$. This shows that
$\delta_0 T_S(\tau)|_\mathcal{M}x(\cdot+s) = T_A(\tau) \delta_0 x(\cdot+s)$
for each $\tau \geq 0$ and $s \in \mathbb{R}$ because $T_S(s)|_\mathcal{M}x = x(\cdot+s)$. In other words $\delta_0$ is a mild solution of the operator equation $\Pi S|_\mathcal{M} = A\Pi$ in the set $\setm{x(\cdot+s)}{s \in \mathbb{R}}$. Upon extensions by linearity and continuity we immediately have that for $\mathcal{M}$ as in the above, the equation $\Pi S|_\mathcal{M} = A\Pi$ has a nontrivial mild solution $\Pi = \delta_0$.
\item[$2 \implies 3$ :] Assume that the homogenous equation
$\Pi S|_\mathcal{M} = A\Pi$ has a mild solution
$\delta_0 \in \mathcal{L}(\mathcal{M},X)$. Let $f \in \mathcal{M}$.
Then $f(t) = \delta_0 T_S(t)|_\mathcal{M}f$ for every $t \in \mathbb{R}$.
Furthermore for every $t, s \in \mathbb{R}$ such that $t \geq s$ we have
\begin{align*}
T_A(t-s)f(s) & = T_A(t-s)\delta_0 T_S(s)|_\mathcal{M} f\\
&= \delta_0 T_S(t-s)|_\mathcal{M}T_S(s)|_\mathcal{M}f\\
&= \delta_0 T_S(t)|_\mathcal{M}f = f(t)
\end{align*}
This shows that every $f \in \mathcal{M}$ is a complete nontrivial
trajectory for $T_A(t)$.
\item[$3 \implies 1$ :] This is trivial.
\end{enumerate}
\end{proof}
We state the following corollary to emphasize that in parts $2$ and $3$ of
Theorem \ref{bct0} the closed translation invariant spaces are equal.
\begin{cor} \label{bct1}
Let $T_A(t)$ be a $C_0$-semigroup in $X$ generated by $A$, and let
$\mathcal{M} \subset BUC(\mathbb{R},X)$ be a closed and translation-invariant subspace.
Then every $x \in \mathcal{M}$ is a complete trajectory for $T_A(t)$ if and
only if $\delta_0$ is a solution of the operator equation
$\Pi S|_\mathcal{M} = A\Pi$.
\end{cor}
\begin{proof}
Assume that every $x \in \mathcal{M}$ is a bounded complete trajectory for
$T_A(t)$. Then for any $\tau \geq 0$ and $s \in \mathbb{R}$ we have
$x(\tau+s) = \delta_0 T_S(\tau)|_\mathcal{M}T_S(s)|_\mathcal{M}x
= T_A(\tau)x(s) = T_A(\tau)\delta_0 T_S(s)|_\mathcal{M}x$ for each
$x \in \mathcal{M}$, because every $x \in \mathcal{M}$ is a complete
trajectory for $T_A(t)$. Consequently $\delta_0$ is a mild solution of the
operator equation $\Pi S|_\mathcal{M} = A\Pi$ in the set
$\setm{x(\cdot+s)}{s \in \mathbb{R}}$ for each $x \in \mathcal{M}$. Since $\mathcal{M}$
is translation-invariant, we have
$\mathcal{M} = \cup_{x \in \mathcal{M}} \setm{x(\cdot+s)}{s \in \mathbb{R}}$.
This shows that $\delta_0$ is a mild solution of the operator equation
$\Pi S|_\mathcal{M} = A\Pi$. The converse claim is contained in the proof of
Theorem \ref{bct0}.
\end{proof}
In the above results we assumed that $\mathcal{M}$ is a closed and
translation-invariant subspace of $BUC(\mathbb{R},X)$. If $\mathcal{M}$ is in
addition CTO, then also other nontrivial solutions of the homogenous
operator equation $\Pi S|_\mathcal{M} = A\Pi$ yield nontrivial bounded complete
trajectories for $T_A(t)$:
\begin{thm} \label{bct}
Let $T_A(t)$ be a $C_0$-semigroup in $X$ generated by $A$. Then the following
assertions are equivalent for a given CTO space
$0 \neq \mathcal{M} \subset BUC(\mathbb{R},X)$.
\begin{enumerate}
\item There exists a nonzero operator $\Pi \in \mathcal{L}(\mathcal{M},X)$ such that
for every $f \in \mathcal{M}$, the function $t \to \Pi
T_S(t)|_\mathcal{M}f$ is a complete trajectory for $T_A(t)$ in
$\mathcal{M}$.
\item The homogenous operator equation $\Pi S|_\mathcal{M} = A\Pi$ has a
nontrivial solution $\Pi \in \mathcal{L}(\mathcal{M},X)$.
\item There exists an operator $\Delta \in \mathcal{L}(\mathcal{M},X)$ such that
the operator equation $\Pi S|_\mathcal{M} = A\Pi + \Delta$ has at least two
distinct solutions.
\item The operator $\tau_{A,S|_\mathcal{M}}$ defined in \eqref{tau} has $0$
as its eigenvalue.
\end{enumerate}
\end{thm}
\begin{proof}
We show $1 \Longleftrightarrow 2 \Longleftrightarrow 3$ and
$2 \Longleftrightarrow 4$:
\begin{itemize}
\item[ 1 $\Longleftrightarrow$ 2 :] First assume that for every
$f \in \mathcal{M}$ the functions $t \to x_f(t) = \Pi T_S(t)|_\mathcal{M}f$
are complete trajectories for $T_A(t)$ in $\mathcal{M}$. Hence for each
$f \in \mathcal{M}$ and $\tau \geq 0$ and $s \in \mathbb{R}$ we have
\begin{align*}
x_f(\tau+s) &= \Pi T_S(\tau+s)|_\mathcal{M} f \\
&= \Pi T_S(\tau)|_\mathcal{M}T_S(s) |_\mathcal{M}f \\
&= T_A(\tau)x_f(s) \\
&= T_A(\tau)\Pi T_S(s)|_\mathcal{M}f
\end{align*}
This shows that $\Pi T_S(\tau)|_\mathcal{M}f(\cdot+s) = T_A(\tau) \Pi f(\cdot+s)$
for each $f \in \mathcal{M}$, $\tau \geq 0$ and $s \in \mathbb{R}$. As we let $s = 0$
we see that $\Pi$ satisfies the operator equation $\Pi
S|_\mathcal{M} = A\Pi$.
Conversely assume that the operator equation $\Pi S|_\mathcal{M} = A\Pi$ has
a nonzero mild solution $\Pi \in \mathcal{L}(\mathcal{M},X)$. Let $f \in \mathcal{M}$
and define the function $x_f : \mathbb{R} \to X$ such that
$x(t) = \Pi T_S(t)|_\mathcal{M} f$ for each $t \in \mathbb{R}$. Since
$\mathcal{M}$ is CTO, $x_f \in \mathcal{M}$. Furthermore for every
$t, s \in \mathbb{R}$ such that $t \geq s$ we have
\begin{align*}
T_A(t-s)x_f(s) & = T_A(t-s)\Pi T_S(s)|_\mathcal{M} f\\
&= \Pi T_S(t-s)|_\mathcal{M}T_S(s)|_\mathcal{M}f\\
&= \Pi T_S(t)|_\mathcal{M}f = x_f(t)
\end{align*}
because $T_S(t)|_\mathcal{M}f = f(\cdot+t) \in \mathcal{M}$ for each $t \in \mathbb{R}$.
This shows that for every $f \in \mathcal{M}$ the function $x_f$ is a complete
nontrivial trajectory for $T_A(t)$ in $\mathcal{M}$.
\item[$2 \Longleftrightarrow 3:$] This is trivial.
\item[$2 \Longleftrightarrow 4:$] This is contained in Proposition \ref{tauta}.
\end{itemize}
\end{proof}
\begin{rem} \rm
Vu \cite{vu1993} studied bounded uniformly continuous and almost
periodic complete nontrivial trajectories for $T_A(t)$. Theorem
\ref{bct0} and Theorem \ref{bct} provide more flexibility. For
example, in Theorem \ref{bct} one may look for $p$-periodic
continuous complete trajectories or complete trajectories $x \in
BUC(\mathbb{R},X)$ such that the Carleman spectrum $sp(x)$ of $x$ is
contained in some closed set $\Lambda \subset i\mathbb{R}$.
\end{rem}
\begin{rem} \label{konstruktio} \rm
Theorem \ref{bct} also provides a way to construct nontrivial
complete trajectories in $\mathcal{M} \subset BUC(\mathbb{R},X)$ for
$T_A(t)$ via nontrivial solutions of the homogenous operator
equation $\Pi S|_\mathcal{M} = A\Pi$.
\end{rem}
The following result is of fundamental importance, since it
provides a simple necessary condition for the existence of a
nontrivial bounded complete trajectory for $T_A(t)$, and since
this condition allows us to combine our results with the regular
admissibility theory of Vu and Sch\"uler \cite{vuschuler}. Because
of its importance we choose to give two separate proofs for this
result.
\begin{thm} \label{nonexistence}
Let $\mathcal{M}$ be a nontrivial closed translation-invariant
subspace of $BUC(\mathbb{R},X)$ and assume that $A$ generates a
$C_0$-semigroup $T_A(t)$ in $X$. If $\sigma(S|_\mathcal{M}) \cap
\sigma(A) = \emptyset$, then there are no nontrivial complete
trajectories for $T_A(t)$ in $\mathcal{M}$.
\end{thm}
\begin{proof}[Proof 1] Assume, conversely, that there exists a nontrivial
complete trajectory $x$ for $T_A(t)$ in $\mathcal{M}$. Then by Proposition 3.5
in \cite{vu1993} $sp(x) = \sigma(S_x)$ where $S_x$ is the restriction
of $S|_\mathcal{M}$ to the space $\overline{\mathop{\rm span}}\setm{x(\cdot+t)}{t \in \mathbb{R}}$.
Consequently $sp(x) \subset \sigma(S|_\mathcal{M})$, and
$sp(x) \cap \sigma(A) = \emptyset$. But by Proposition 3.7 in \cite{vu1993}
$sp(x) \subset \sigma_A(A)$ which implies $sp(x) = \emptyset$. According to
Wiener's Tauberian Theorem \cite{vu1993} this is possible only if
$x$ is identically zero --- a contradiction.
\end{proof}
\begin{proof}[Proof 2] Assume again, conversely, that there exists a
nontrivial complete trajectory $x$ for $T_A(t)$ in $\mathcal{M}$.
By Theorem \ref{bct0} there exists a nontrivial closed
translation-invariant subspace $\mathcal{N} \subset \mathcal{M}$
in which the operator equation $\Pi S|_\mathcal{N} = A\Pi$ has a
nontrivial solution. Then by a result stated in Subsection
\ref{preli} there exists another nontrivial closed
translation-invariant subspace $\mathcal{N}_0 \subset \mathcal{N}$
in which the restriction $S|_{\mathcal{N}_0}$ is a nonzero bounded
operator. Moreover the operator equation
$\Pi S|_{\mathcal{N}_0}=A\Pi$ also has a nontrivial solution.
But this is impossible
since $\sigma(S|_{\mathcal{N}_0}) \cap \sigma(A) \subset
\sigma(S|_\mathcal{M}) \cap \sigma(A) = \emptyset$ and the
boundedness of $S|_{\mathcal{N}_0}$ imply that the only solution
of $\Pi S|_{\mathcal{N}_0} = A\Pi$ is the zero operator (see
Section 2 in \cite{vuschuler}).
\end{proof}
Throughout the following corollaries $A$ generates a $C_0$-semigroup
$T_A(t)$ in $X$.
\begin{cor} \label{regadm}
If a given CTO space $\mathcal{M} \subset BUC(\mathbb{R},X)$ is regularly admissible
for $T_A(t)$, then there cannot be complete nontrivial trajectories
for $T_A(t)$ in $\mathcal{M}$.
\end{cor}
\begin{proof}
By Corollary 3.2 in \cite{vuschuler} we have
$\sigma(S|_\mathcal{M}) \cap \sigma(A) = \emptyset$. By
Theorem \ref{nonexistence} there cannot be complete nontrivial trajectories
in $\mathcal{M}$.
\end{proof}
\begin{cor}
Let $\mathcal{M}$ be a CTO subspace of $BUC(\mathbb{R},X)$ and suppose that\break
$\sigma(T_A(1)) \cap \sigma(T_S(1)|_\mathcal{M}) = \emptyset$. Then there
cannot be complete nontrivial trajectories for $T_A(t)$ in $\mathcal{M}$.
\end{cor}
\begin{proof}
By Corollary 2.4 and Theorem 3.1 in \cite{vuschuler} $\mathcal{M}$ is regularly
admissible for $T_A(t)$. By Corollary \ref{regadm} there cannot be complete
nontrivial trajectories for $T_A(t)$ in $\mathcal{M}$.
\end{proof}
\begin{cor}
Assume that there are no complete trajectories for $T_A(t)$ in a CTO subspace
$\mathcal{M}$ of $BUC(\mathbb{R},X)$ and that the operator equation
$\Pi S|_\mathcal{M} = A\Pi + \delta_0$ has a solution
$\Pi \in \mathcal{L}(\mathcal{M},X)$. Then $\mathcal{M}$ is regularly admissible.
\end{cor}
\begin{proof}
By Theorem \ref{bct}, $\Pi$ must be the unique solution of the operator
equation $\Pi S|_\mathcal{M} = A\Pi + \delta_0$. The result follows by
Theorem 3.1 in \cite{vuschuler}.
\end{proof}
Recall that $T_A(t)$ is exponentially dichotomous if there exists a bounded
projection operator $P$ on $X$ and positive constants $M, \omega$ such that
\begin{enumerate}
\item $PT_A(t) = T_A(t)P$ for all $t \geq 0$.
\item $\norm{T_A(t)x_0} \leq M e^{-\omega t}\norm{x_0}$ for all
$x_0 \in \mathop{\rm ran}(P)$ and all $t \geq 0$.
\item The restriction $T_A(t)|_{\ker(P)}$ extends to a $C_0$-group and
$\norm{T_A(-t)|_{\ker(P)}x_0} \leq Me^{-\omega t}\norm{x_0}$ for all
$x_0 \in \ker(P)$ and all $t \geq 0$.
\end{enumerate}
Clearly if $T_A(t)$ is exponentially stable, then it is also exponentially
dichotomous. Vu (\cite{vu1993}, Example 2.7) showed that there are no
complete bounded trajectories for the diffusion semigroup on $C_0(\mathbb{R})$.
The following result implies that the same is in fact true for all
exponentially stable semigroups.
\begin{cor} \label{diko}
Let $T_A(t)$ be exponentially dichotomous. Then there cannot exist
nontrivial bounded uniformly continuous complete trajectories for $T_A(t)$.
\end{cor}
\begin{proof}
By Theorem 4.1 in \cite{vuschuler} the space $BUC(\mathbb{R},X)$ is regularly
admissible for $T_A(t)$. The result follows by Corollary \ref{regadm}.
\end{proof}
The last corollary of Theorem \ref{bct} provides a sufficient condition for
the almost periodicity of a nontrivial complete trajectory for $T_A(t)$.
\begin{cor}
Let $\sigma_A(A) \cap i\mathbb{R}$ be countable and assume that the space $X$ does
not contain a subspace which is isomorphic to $c_0$ (the Banach space of
numerical sequences which converge to zero). Let $\mathcal{M}$ be a CTO
subspace of $BUC(\mathbb{R},X)$. If the operator equation $\Pi S|_\mathcal{M} = A \Pi$
has a nontrivial solution $\Pi \in \mathcal{L}(\mathcal{M},X)$, then
$x_f(t) = \Pi T_S(t) f$ is an almost periodic complete trajectory for $T_A(t)$
for each $f \in \mathcal{M}$.
\end{cor}
\begin{proof}
By Theorem 3.10 in \cite{vu1993} all bounded uniformly continuous bounded
trajectories are almost periodic. By Theorem \ref{bct}, the function
$t \to \Pi T_S(t) f$ is a complete trajectory in
$\mathcal{M} \subset BUC(\mathbb{R},X)$ for every $f \in \mathcal{M}$.
\end{proof}
\section{Some Perturbation Results} \label{pert}
Consider again a closed translation-invariant subspace
$\mathcal{M}$ of $BUC(\mathbb{R},X)$. Clear\-ly for every $f \in \mathcal{M}$ the
trajectory $T_S(t)|_\mathcal{M}f$ of the left shift group is bounded and
complete, and it is in $\mathcal{M}$. However, for every $\epsilon > 0$ the
semigroup $T_{S-\epsilon I}(t)$ generated by $S-\epsilon I$ in $\mathcal{M}$
is exponentially stable. By Corollary \ref{diko} there are no nontrivial
bounded complete trajectories for $T_{S-\epsilon I}(t)$ in $\mathcal{M}$,
and hence the existence of nontrivial bounded complete trajectories for a
semigroup is a fragile property; arbitrarily small bounded additive
perturbations to the generator may destroy it. On the other hand, in this
section we shall provide conditions under which the \emph{nonexistence} of
nontrivial bounded complete trajectories is not destroyed by small unbounded
(but possibly structured) additive perturbations to the generator $A$.
\begin{prop}
Let $A$ generate a $C_0$-semigroup $T_A(t)$ in $X$. Let $\mathcal{M}$ be a
closed translation-invariant subspace of $BUC(\mathbb{R},X)$ and let
$\sigma(A) \cap \sigma(S|_\mathcal{M}) = \emptyset$. Let
$\Delta_A : \mathcal{D}(\Delta_A) \subset X \to X$ be a linear $A$-bounded operator
such that
\begin{enumerate}
\item $A+\Delta_A$ with domain $\mathcal{D}(A)$ generates a $C_0$-semigroup
$T_{A+\Delta_A}(t)$ in $X$.
\item The $A$-boundedness constants $a,b$ in \eqref{araj} satisfy
\begin{equation}
\sup_{i\omega \in \sigma(S|_\mathcal{M})} a \norm{R(i\omega,A)}+b\norm{AR(i\omega,A)} < 1
\end{equation}
\end{enumerate}
Then there are no nontrivial complete trajectories in $\mathcal{M}$ for
$T_A(t)$ and the same holds for the perturbed $C_0$-semigroup $T_{A+\Delta_A}(t)$.
\end{prop}
\begin{proof}
By Theorem IV.3.17 in \cite{kato},
$\sigma(S|_\mathcal{M}) \subset \rho(A+\Delta_A)$. The result then follows
by Theorem \ref{nonexistence}.
\end{proof}
It is well known that if $A$ generates an analytic or contractive
$C_0$-semigroup, then so does $A+\Delta_A$ under rather mild additional
conditions for the $A$-bounded perturbation $\Delta_A$ \cite{engelnagel}.
We next prove that regular admissibility of $\mathcal{M}$ for $T_A(t)$ is
also preserved under certain additive perturbations to $A$. According to
Corollary \ref{regadm} we then have another situation in which the nonexistence
of bounded complete trajectories in $\mathcal{M}$ is not affected by such
perturbations. In order to establish this result, we need some notation.
Let $\mathcal{M}$ be a CTO subspace of $BUC(\mathbb{R},X)$. Let
$\Delta_A : \mathcal{D}(\Delta_A) \subset X \to X$ be a closed linear operator
such that $\mathcal{D}(A) \subset \mathcal{D}(\Delta_A)$ and such that $A-\Delta_A$
(with domain $\mathcal{D}(A)$) generates a $C_0$-semigroup in $X$. Define another
linear operator
$\underline{\Delta_A} : \mathcal{D}(\underline{\Delta_A}) \subset \mathcal{L}(\mathcal{M},X)
\to \mathcal{L}(\mathcal{M},X)$ such that
\begin{subequations}
\begin{align}
\mathcal{D}(\underline{\Delta_A}) & = \setm{X \in \mathcal{L}(\mathcal{M},X)}{A_\Delta X \in \mathcal{L}(\mathcal{M},X)} \\
\underline{\Delta_A}X & = A_\Delta X \quad \forall X \in \mathcal{D}(\underline{\Delta_A})
\end{align}
\end{subequations}
\begin{prop} \label{pertua}
In the above notation assume that $\mathcal{M}$ is regularly admissible for
$T_A(t)$. Let
\begin{equation}
M = \sup \setm{\norm{\Pi}}{\Pi S|_\mathcal{M} = A\Pi + \Delta, \norm{\Delta} = 1}
\end{equation}
If $\underline{\Delta_A}$ is $\tau_{A,S|_\mathcal{M}}$-bounded with the
boundedness constants $a,b$ in \eqref{araj} satisfying $aM + b < 1$, then
$\mathcal{M}$ is regularly admissible for $T_{A-\Delta_A}(t)$.
\end{prop}
\begin{proof}
First observe that by Theorem 3.1 in \cite{vuschuler} regular admissibility
of $\mathcal{M}$ for $T_A(t)$ is equivalent to the unique solvability of the
operator equation $\Pi S|_\mathcal{M} = A\Pi + \Delta$ for every
$\Delta \in \mathcal{L}(\mathcal{M},X)$. Consequently by Proposition \ref{tauta} we
have $0 \in \rho(\tau_{A,S|_\mathcal{M}})$ and
$\Pi S|_\mathcal{M} = A\Pi + \Delta$ if and only if
$\Pi = \tau_{A,S|_\mathcal{M}}^{-1} \Delta$. Hence
\begin{equation}
\norm{\tau_{A,S|_\mathcal{M}}^{-1}} = \sup_{\norm{\Delta} = 1}
\norm{\tau_{A,S|_\mathcal{M}}^{-1} \Delta} = \sup_{\norm{\Delta} = 1}
\setm{\norm{\Pi}}{\Pi S|_\mathcal{M} = A\Pi + \Delta} = M
\end{equation}
By our assumptions $\underline{\Delta_A}$ is $\tau_{A,S|_\mathcal{M}}$-bounded,
with the boundedness constants $a,b$ in \eqref{araj} satisfying
$a\norm{\tau_{A,S|_\mathcal{M}}^{-1}} + b < 1$. Theorem IV.1.16 in \cite{kato}
then implies that the operator $\tau_{A,S|_\mathcal{M}} + \underline{\Delta_A}$
with domain $\mathcal{D}(\tau_{A,S|_\mathcal{M}})$ is also boundedly invertible.
But for each $X \in \mathcal{D}(\tau_{A,S|_\mathcal{M}})$ and
$u \in \mathcal{D}(S|_\mathcal{M})$ we have
\begin{align*}
[\tau_{A,S|_\mathcal{M}} + \underline{\Delta_A}]Xu
&= XS|_\mathcal{M}u-AXu + \underline{\Delta_A}Xu \\
&= XS|_\mathcal{M}u-AXu + \Delta_AXu \\
&= XS|_\mathcal{M}u-(A-\Delta_A)Xu
\end{align*}
which shows that for every $\Delta \in \mathcal{L}(\mathcal{M},X)$ the operator
equation $XS|_\mathcal{M}-(A-\Delta_A)X = \Delta$ has a unique solution
$X=\Pi_\Delta \in \mathcal{L}(\mathcal{M},X)$. By Theorem 3.1 in \cite{vuschuler}
this implies regular admissibility of $\mathcal{M}$ for $T_{A-\Delta_A}(t)$.
\end{proof}
\begin{rem} \rm
For bounded additive perturbations $\Delta_A \in \mathcal{L}(X)$ to $A$ the content
of Proposition \ref{pertua} may be formulated in a much simpler way:
There exists $\epsilon > 0$ such that whenever $\norm{\Delta_A} < \epsilon$,
the space $\mathcal{M}$ is regularly admissible for $T_{A+\Delta_A}(t)$.
\end{rem}
\begin{rem} \rm
It follows from Theorem 5.1 in \cite{vuschuler} that regular admissibility of
a space $\mathcal{M}$ is not destroyed by certain sufficiently continuous and
small nonlinear perturbations to $A$. Theorem \ref{pertua} is, however, not
entirely contained in this result of Vu and Sch\"uler, because we allow for
a degree of unboundedness in the additive perturbation operator $\Delta_A$.
Furthermore, their proof relies on a fixed point argument, and consequently
it is rather different from ours.
\end{rem}
\section{On Strong Stability of $C_0$-semigroups} \label{stability}
Exponential stability of a $C_0$-semigroup can be completely
characterized in many equivalent ways: There are the well-known
conditions of the Datko Theorem \cite{abhn}, and a condition of Vu
and Sch\"uler \cite{vuschuler} according to which exponential
stability of a $C_0$-semigroup $T_A(t)$ is equivalent to the
uniform boundedness of $T_A(t)$ and the unique solvability of the
operator equation $\Pi S = A\Pi + \delta_0$. On the other hand, it
has turned out that strong stability of a $C_0$-semigroup is
considerably more difficult to characterize. Since the pioneering
work of Arendt, Batty, Lyubich and Vu \cite{arendtbatty,
lyubichvu} this question has received much attention in the
literature; the reader is referred to \cite{abhn, batty,
battyphong, bnr1, bnr2} and the references therein. It is obvious
that a strongly stable $C_0$-semigroup $T_A(t)$ is uniformly
bounded and that $\sigma_P(A^*) \cap i\mathbb{R} = \emptyset$. On the
other hand, the ABLV Theorem states that if $T_A(t)$ is uniformly
bounded, $\sigma_P(A^*) \cap i\mathbb{R} = \emptyset$ \emph{and}
$\sigma(A)\cap i\mathbb{R}$ is countable, then $T_A(t)$ is strongly
stable.
We next present new characterizations for strong stability of a
$C_0$-semigroup $T_A(t)$ in terms of nontrivial bounded complete
trajectories for the sun-dual semigroup $T_A^\odot(t)$ and
nontrivial solvability of an operator equation $\Pi S|_\mathcal{M}
= A\Pi$.
\begin{thm}
Assume that $\sigma_A(A) \cap i\mathbb{R}$ is countable and that $T_A(t)$
is a uniformly bounded $C_0$-semigroup in $X$ generated by $A$.
Then there exists a nontrivial bounded complete trajectory for the
sun-dual semigroup $T_A^\odot(t)$ if and only if $T_A(t)$ is not
strongly stable.
\end{thm}
\begin{proof}
Assume first that $T_A(t)$ is not strongly stable. Then
$\sigma(A)\cap i\mathbb{R} = \sigma_A(A)\cap i\mathbb{R} \neq i\mathbb{R}$, which by
Theorem 2.3 in \cite{vu1993} immediately shows that there exists a
nontrivial bounded complete trajectory for the sun-dual semigroup
$T_A^\odot(t)$.
For the converse, suppose that there exists a nontrivial bounded
complete trajectory $f$ for the sun-dual semigroup $T_A^\odot(t)$.
Since $T_A(t)$ is uniformly bounded, the sun-dual semigroup
$T_A^\odot(t)$ is uniformly bounded, and hence $f \in
BUC(\mathbb{R},X^\odot)$. Then $sp(f) \subset \sigma(A^\odot)\cap i\mathbb{R}
\subset \sigma(A)\cap i\mathbb{R}$ by Proposition 3.7 in \cite{vu1993}
and Proposition IV.2.18 in \cite{engelnagel}. This shows that
$sp(f)$ is a closed countable subset of the imaginary axis, and
so it must contain an isolated point. Consider the closed
translation-invariant subspace $\mathcal{M}_f =
\overline{\mathop{\rm span}}\setm{f(\cdot+t)}{t\in\mathbb{R}}$ of $BUC(\mathbb{R},X^\odot)$
and the restriction $T_S(t)|_{\mathcal{M}_f}$ of the translation
group $T_S(t)$ to $\mathcal{M}_f$. By Theorem \ref{bct0} and
Corollary \ref{bct1} every $g \in \mathcal{M}_f$ is a complete
trajectory for $T_A^\odot(t)$. Furthermore, the generator $S_f$ of
this restriction $T_S(t)|_{\mathcal{M}_f}$ has an isolated point
$i\lambda \in i\mathbb{R}$ in its spectrum because $\sigma(S_f) = sp(f)$
by Proposition 3.5 in \cite{vu1993}. It then follows from
Gelfand's Theorem (cf. \cite{abhn} Corollary 4.4.9) that
$i\lambda$ must be an eigenvalue of $S_f$. Hence there exists a
nonzero $g \in \mathcal{M}_f$ such that $T_S(t)|_{\mathcal{M}_f}g
= e^{i\lambda t} g$ for each $t \in \mathbb{R}$. Now the function $t \to
\delta_0 T_S(t)|_{\mathcal{M}_f}g = g(t) = g(0)e^{i\lambda t}$ is
a (nontrivial) complete trajectory for $T_A^\odot(t)$ in
$\mathcal{M}_f$. It is easy to see that this implies $i\lambda \in
\sigma_P(A^\odot) \cap i\mathbb{R} = \sigma_P(A^*) \cap i\mathbb{R}$. Consequently
$T_A(t)$ cannot be strongly stable.
\end{proof}
In the following theorem we shall characterize strongly stable
semigroups by the solvability of an operator equation $\Pi
S|_\mathcal{M} = A\Pi$. However, in contrast to the previous
sections, here $\mathcal{M}$ is a closed translation-invariant
subspace of $C_0(\mathbb{R}_+,X) = \setm{f \in BUC([0,\infty),X)}{\lim_{t
\to \infty} f(t) = 0}$, and $S|_\mathcal{M}$ generates the
strongly continuous left shift semigroup in $\mathcal{M}$.
\begin{thm} \label{ss}
Let $X \neq \set{0}$ and let $T_A(t)$ be a $C_0$-semigroup in $X$
generated by $A$. Then $T_A(t)$ is strongly stable if and only if
there exists a nontrivial closed translation-invariant subspace
$\mathcal{M} \subset C_0(\mathbb{R}_+,X)$ such that the operator equation
$\Pi S|_\mathcal{M} = A\Pi$ has a surjective solution $\Pi \in
\mathcal{L}(\mathcal{M},X)$\footnote{In analogy to Section \ref{soln}, by
a surjective solution of $\Pi S|_\mathcal{M} = A\Pi$ we mean a
bounded linear surjective operator $\Pi$ such that
$\Pi(\mathcal{D}(S|_\mathcal{M}))\subset \mathcal{D}(A)$ and $\Pi
S|_\mathcal{M}f = A\Pi f$ for each $f \in \mathcal{D}(S|_\mathcal{M})$.}.
\end{thm}
\begin{proof}
Let $T_A(t)$ be strongly stable and let $\mathcal{M} =
\overline{\mathop{\rm span}}\setm{T_A(\cdot)x}{x \in X}$ where closure is
taken in the $\sup$-norm. Then $0 \neq \mathcal{M} \subset
C_0(\mathbb{R}_+,X)$. Let $\Pi = \delta_0$, the point evaluation operator
in $\mathcal{M}$ centered at the origin. Then $\delta_0 \in
\mathcal{L}(\mathcal{M},X)$ and $\delta_0$ is surjective; for any $x \in
X$ we have $x = \delta_0 T_A(t)x$. Moreover, for every trajectory
$f_x(t) = T_A(t)x$ we have $\delta_0 T_S(t)|_\mathcal{M} f_x =
f_x(t) = T_A(t)x = T_A(t)\delta_0 f_x$. Extension by continuity
and linearity shows that $\delta_0 T_S(t)|_\mathcal{M} =
T_A(t)\delta_0$ throughout $\mathcal{M}$ for each $t \geq 0$. Let
$f \in \mathcal{D}(S|_\mathcal{M})$. Then
\begin{equation}
\begin{aligned}
\frac{T_A(h)\delta_0 f - \delta_0 f}{h}
& = \frac{T_A(h)\delta_0 f - \delta_0 T_S(h)|_\mathcal{M} f}{h}
+ \frac{\delta_0 T_S(h)|_\mathcal{M}f-\delta_0 f}{h} \\
& = \frac{\delta_0 T_S(h)|_\mathcal{M}f-\delta_0 f}{h} \quad
\forall h > 0 \label{conv2}
\end{aligned}
\end{equation}
which by the boundedness of $\delta_0$ shows that $\delta_0 f \in \mathcal{D}(A)$
and that $A\delta_0 f = \delta_0 S|_\mathcal{M} f$ for each
$f \in \mathcal{D}(S|_\mathcal{M})$. Consequently $\delta_0$ is a surjective solution
of $\Pi S|_\mathcal{M} = A\Pi$.
Conversely, assume that there exists a nontrivial closed translation-invariant
subspace $\mathcal{M} \subset C_0(\mathbb{R}_+,X)$ such that the operator equation
$\Pi S|_\mathcal{M} = A\Pi$ has a surjective solution
$\Pi \in \mathcal{L}(\mathcal{M},X)$. Then since $\Pi(\mathcal{D}(S|_\mathcal{M}))
\subset \mathcal{D}(A)$, we have for every $t \geq 0$ and
$f \in \mathcal{D}(S|_\mathcal{M})$ that
\begin{align*}
\Pi T_S(t)|_\mathcal{M}f - T_A(t)\Pi f
& = \big|_{\tau = 0}^t T_A(t-\tau)\Pi T_S(\tau)|_\mathcal{M}f d\tau\\
& = \int_0^t \frac{d}{d\tau} T_A(t-\tau)\Pi T_S(\tau)|_\mathcal{M}f d\tau \\
& = \int_0^t T_A(t-\tau)[\Pi S|_\mathcal{M} - A\Pi]T_S(\tau)|_\mathcal{M}f d\tau
= 0
\end{align*}
and by continuity $\Pi T_S(t)|_\mathcal{M}f - T_A(t)\Pi f = 0$ for each
$f \in \mathcal{M}$ and $t \geq 0$. Let $x \in X$ be arbitrary.
Then by the surjectivity of $\Pi$ there exists $f \in \mathcal{M}$ such that
$x = \Pi f$. Moreover,
\begin{equation}
\lim_{t \to \infty} T_A(t)x = \lim_{t \to \infty} T_A(t)\Pi f
= \lim_{t \to \infty} \Pi T_S(t)|_\mathcal{M} f = 0
\end{equation}
since $T_S(t)|_\mathcal{M}$ is strongly stable and $\Pi \in \mathcal{L}(\mathcal{M},X)$.
Consequently $T_A(t)$ is strongly stable.
\end{proof}
In a very similar way we obtain the following corollary.
\begin{cor} \label{vs2}
Let $X \neq \set{0}$ and let $T_A(t)$ be a $C_0$-semigroup in $X$ generated by
$A$. Then $T_A(t)$ is strongly stable if and only if $T_A(t)$ is uniformly
bounded and there exists a nontrivial closed translation-invariant subspace
$\mathcal{M} \subset C_0(\mathbb{R}_+,X)$ such that the operator equation
$\Pi S|_\mathcal{M} = A\Pi$ has a solution $\Pi \in \mathcal{L}(\mathcal{M},X)$
such that $\mathop{\rm ran}(\Pi)$ is dense in $X$.
\end{cor}
\begin{rem} \rm
Theorem \ref{ss} and Corollary \ref{vs2} are related to, but
independent of, a result of Batty \cite{batty}. He showed that if
$T_S(t)$ is a $C_0$-semigroup in some Banach space $Y$ with
generator $S$, if $T_A(t)$ is a uniformly bounded $C_0$-semigroup
in $X$ with generator $A$, if $\sigma(S)\cap i\mathbb{R}$ is countable and
$\sigma_P(A^*)\cap i\mathbb{R} = \emptyset$, and if $\Pi T_S(t) =
T_A(t)\Pi$ for some $\Pi \in \mathcal{L}(Y,X)$ with a dense range, then
$T_A(t)$ is strongly stable. In the above, we had to assume that
$T_S(t)$ is the translation semigroup in some $\mathcal{M} \subset
C_0(\mathbb{R}_+,X)$. However, we also obtained complete characterizations
for strong stability.
\end{rem}
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\end{document}