\documentclass[reqno]{amsart} \usepackage{amssymb} \usepackage{hyperref} \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} \newcommand*{\set} [1]{\{#1\}} \newcommand*{\norm}[1]{\lVert#1\rVert} \newcommand*{\setm}[2]{\{#1\mid#2\}} % Set with middle | \section{Introduction} Consider the abstract Cauchy problem $$\label{acp} \dot{x}(t) = Ax(t), \quad t \geq 0, \quad x(0)=x_0 \in X$$ 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 $$\label{iacp} \dot{x}(t) = Ax(t)+f(t), \quad t \in \mathbb{R}$$ 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 $$x(t) =T_A(t-s)x(s) + \int_s^t T_A(t-\tau)f(\tau)d\tau \quad \forall t \geq s, \quad t,s \in \mathbb{R}$$ 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 $$\label{araj} \norm{\Delta_A x} \leq a \norm{x} + b \norm{Ax} \quad \forall x \in \mathcal{D}(A)$$ If the Banach space $X$ is not reflexive, then the adjoint semigroup $T_A^*(t)$ is not necessarily strongly continuous. However, the subspace $$X^\odot = \setm{\phi \in X^*}{T_A^*(t)\phi \text{ is strongly continuous}}$$ 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 $$\widetilde{f}(\lambda) = \begin{cases} \int_0^\infty e^{-\lambda t}f(t)dt, & \Re(\lambda) > 0 \\ -\int_{-\infty}^0 e^{-\lambda t}f(t)dt, & \Re(\lambda) < 0 \end{cases}$$ 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 $$\label{homog} \Pi S|_\mathcal{M} = A\Pi + \Delta$$ 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 $$\label{miehtaa} \Pi T_S(t)|_\mathcal{M} f = T_A(t) \Pi f + \int_0^t T_A(t-s)\Delta T_S(s)|_\mathcal{M}fds$$ 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 $$\Pi T_S(t)|_\mathcal{M} f = T_A(t) \Pi f + \int_0^t T_A(t-\tau)\Delta T_S(\tau)|_\mathcal{M}f d\tau \quad \forall f \in \mathcal{M} \quad \forall t \geq 0$$ 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 $$\label{mieh} \Pi T_S(t)|_\mathcal{M} f = T_A(t) \Pi f + \int_0^t T_A(t-s)f(s)ds, \quad t \geq 0$$ 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 $$x(\tau+s) = \delta_0 T_S(\tau)|_\mathcal{M}T_S(s)|_\mathcal{M}x = T_A(\tau)x(s) = T_A(\tau)\delta_0 T_S(s)|_\mathcal{M}x$$ 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 $$\sup_{i\omega \in \sigma(S|_\mathcal{M})} a \norm{R(i\omega,A)}+b\norm{AR(i\omega,A)} < 1$$ \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 $$M = \sup \setm{\norm{\Pi}}{\Pi S|_\mathcal{M} = A\Pi + \Delta, \norm{\Delta} = 1}$$ 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 $$\norm{\tau_{A,S|_\mathcal{M}}^{-1}} = \sup_{\norm{\Delta} = 1} \norm{\tau_{A,S|_\mathcal{M}}^{-1} \Delta} = \sup_{\norm{\Delta} = 1} \setm{\norm{\Pi}}{\Pi S|_\mathcal{M} = A\Pi + \Delta} = M$$ 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{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} 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, $$\lim_{t \to \infty} T_A(t)x = \lim_{t \to \infty} T_A(t)\Pi f = \lim_{t \to \infty} \Pi T_S(t)|_\mathcal{M} f = 0$$ 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} \begin{thebibliography}{99} \bibitem{arendtbatty} Arendt,~W., Batty,~C.; \emph{Tauberian Theorems and Stability of One-Parameter Semigroups}. Transactions of the American Mathematical Society 306, 1988. pp. 837-852. \bibitem{abhn} Arendt,~W., Batty,~C., Hieber,~M., Neubrander,~F.; \emph{Vector-valued Laplace transforms and Cauchy problems}. 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