Exponential stability of c0-semigroup via Lyapunov inequality in Banach space

Abstract

We give a relation between the exponential stability of C 0 −semigroup T = {T (t)} t≥0 and the solutions of Lyapunov inequality QAx, x + Qx, Ax ≤ −||x|| 2 , in B + (X, X * ), with X is a Banach space. The solutions of this inequality characterizes, the boundedness of the resolvent R(λ, A) inside and outside of the left half-plane ℜλ ≥ 0, and also the left invertibility of the C 0 −semigroup T.