Weak real integral characterizations for exponential stability of semigroups in reflexive spaces

Abstract

Let (t n ) be a sequence of nonnegative real numbers tending to ∞, such that 1≤t n+1−t n ≤α for all natural numbers n and some positive α. We prove that a strongly continuous semigroup {T(t)} t≥0, acting on a Hilbert space H, is uniformly exponentially stable if $$\sum_{n=0}^\infty\varphi\bigl(\bigl|\bigl\langle T(t_n)x, y\bigr\rangle\bigr|\bigr)<\infty, $$ for all unit vectors x, y in H. We obtain the same conclusion under the assumption that the inequality $$\sum_{n=0}^\infty\varphi\bigl(\bigl|\bigl\langle T(t_n)x, x^\ast\bigr\rangle\bigr|\bigr)<\infty, $$ is fulfilled for all unit vectors x∈X and x ∗∈X ∗, X being a reflexive Banach space. These results are stated for functions φ belonging to a special class of functions, such as defined in the second section of this paper. We conclude our paper with a Rolewicz’s type result in the continuous case on Hilbert spaces.