On the asymptotic behaviour of a semigroup of linear operators

Let T = {T(t)}t ≥ 0 be a C0-semigroup on a Banach space X. In this paper, we study the relations between the abscissa ωLp(T) of weak p-integrability of T (1 ≤ p < ∞), the abscissa ωpR(A) of p-boundedness of the resolvent of the generator A of T (1 ≤ p ≤ ∞), and the growth bounds ωβ(T), β ≥ 0, of T. Our main results are as follows. 1.(i) Let T be a C0-semigroup on a B-convex Banach space such that the resolvent of its generator is uniformly bounded in the right half plane. Then ω1 − ϵ(T) < 0 for some ϵ > 0.2.(ii) Let T be a C0-semigroup on Lp such that the resolvent of the generator is uniformly bounded in the right half plane. Then ωβ(T) < 0 for all β>¦1p − 1p′¦, 1p + 1p′ = 1.3.(iii) Let 1 ≤ p ≤ 2 and let T be a weakly Lp-stable C0-semigroup on a Banach space X. Then for all β>1p we have ωβ(T) ≤ 0.Further, we give sufficient conditions in terms of ωqR(A) for the existence of Lp-solutions and W1,p-solutions (1 ≤ p ≤ ∞) of the abstract Cauchy problem for a general class of operators A on X.