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

Abstract

We give a relation between the exponential stability of $C_0-$semigroup $\text{T}={\left\{T(t)\right\}}_{t\ge 0}$ and the solutions of Lyapunov inequality \( \left\langle QAx,x\right\langle +\left\langle Qx,Ax\right\langle \leq -||x||^{2}, \) in $B^{+}(X,X^{\ast })$, with $X$ is a Banach space. The solutions of this inequality characterizes, the boundedness of the resolvent $R(\lambda ,A)$ inside and outside of the left half-plane $\Re \lambda \ge 0$, and also the left invertibility of the $C_0-$semigroup T.