Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 71, pp. 1-14.

Title: On Sylvester operator equations, complete trajectories,
       regular admissibility, and stability of $C_0$-semigroups
       
Author: Eero Immonen (Tampere Univ. of Technology, Finland)

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 $\Pi S|_\mathcal{M} = A\Pi$.
  
Submitted April 17, 2005. Published June 30, 2005.
Math Subject Classifications: 47D03.
Key Words: Sylvester operator equation; regularly admissible space; 
 complete nontrivial trajectory; $C_0$-semigroup; exponential stability;
 strong stability; exponential dichotomy.