Spectral conditions for stability of one-parameter semigroups - Batty, C (Elsevier, 1996)
Let {S(t): t≥0} be a C0-semigroup on a Banach space Y with generator B and {T(t): t≥0} be a bounded C0-semigroup on a Banach space X with generator A. Suppose that σ(B) ∩ iR is countable, Pσ(A*) ∩ iR is empty and that there is a bounded linear operator C: Y → X with dense range which intertwines the two semigroups. Then ∥T(t)x∥X → 0 as t → ∞, for each x in X. This generalises results of W. Arendt and the author, Yu. I. Lyubich and Vũ Quôc Phóng, and Falun Huang. © 1996 Academic Press, Inc.