On Sylvester operator equations, complete trajectories, regular admissibility, and stability of C0-semigroups
Texas State University-San Marcos, Department of Mathematics
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 homogeneous operator equation ΠS|M = AΠ. Here S|M generates the shift C0-group TS(t)|M in a closed translation-invariant subspace M of BUC (ℝ, X), and δ0 is the point evaluation at the origin. If, in addition, M is operator-invariant and 0 ≠ Π ∈ L(M, X) is any solution of ΠS|M = AΠ, then all functions t → ΠTs(t)|Mƒ, ƒ ∈ M, are complete trajectories for TA(t) in M. 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 non-trivial bounded complete trajectories for the sun-dual semigroup, and by the surjective solvability of an operator equation ΠS|M = AΠ.
Sylvester operator equation, Regularly admissible space, Complete nontrivial trajectory, C0-semigroup, Exponential stability, Strong stability, Exponential dichotomy
Immonen, E. (2005). On Sylvester operator equations, complete trajectories, regular admissibility, and stability of C0-semigroups. <i>Electronic Journal of Differential Equations, 2005</i>(71), pp. 1-14.