On Sylvester operator equations, complete trajectories, regular admissibility, and stability of $C_0$-semigroups

Immonen, Eero

Electronic Journal of Differential Equations, Tome 2005 (2005).

Résumé

Summary: 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 $\Pi S|_\mathcal{M} = A\Pi$.

Détail

Classification : 47D03
Keywords: Sylvester operator equation, regularly admissible space, complete nontrivial trajectory, $C_0$-semigroup, exponential stability, strong stability, exponential dichotomy