Electron. J. Diff. Eqns., Vol. 2005(2005), No. 71, pp. 1-14.

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

Eero Immonen

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.

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Eero Immonen
Institute of Mathematics
Tampere University of Technology
PL 553, 33101 Tampere, Finland
email: Eero.Immonen@tut.fi

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