We show that the existence of a nontrivial bounded uniformly continuous (BUC) complete trajectory for a -semigroup generated by an operator in a Banach space is equivalent to the existence of a solution to the homogenous operator equation . Here generates the shift -group in a closed translation-invariant subspace of , and is the point evaluation at the origin. If, in addition, is operator-invariant and is any solution of , then all functions , , are complete trajectories for in . We connect these results to the study of regular admissibility of Banach function spaces for ; among the new results are perturbation theorems for regular admissibility and complete trajectories. Finally, we show how strong stability of a -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 .
Submitted April 17, 2005. Published June 30, 2005.
Math Subject Classifications: 47D03.
Key Words: Sylvester operator equation; regularly admissible space; complete nontrivial trajectory; -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
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