On Sylvester Operator Equations, Complete Trajectories, Regular Admissibility, and Stability of C0-semigroups

We show that the existence of a nontrivial bounded uniformly continuous (BUC) complete trajectory for a C0-semigroup TA(t) generated by an operator A in a Banach space X is equivalent to the existence of a solution Π = δ0 to the homogenous operator equation ΠS|M = AΠ. Here S|M generates the shift C0-group TS(t)|M in a closed translation-invariant subspaceM of BUC(R, X), and δ0 is the point evaluation at the origin. If, in addition,M is operator-invariant and 0 6= Π ∈ L(M, X) is any solution of ΠS|M = AΠ, then all functions t→ ΠTS(t)|Mf , f ∈ M, are complete trajectories for TA(t) inM. We connect these results to the study of regular admissibility of Banach function spaces for TA(t); among the new results are perturbation theorems for regular admissibility and complete trajectories. Finally, we show how strong stability of a C0-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 ΠS|M = AΠ.