  We give a relation between the exponential stability of $ C_{0}- $semigroup $
\textbf{T}=\left\lbrace T(t) \right\rbrace_{t\geq 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^{*}) $, 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\geq 0 $, and also the left invertibility of the $ C_{0}- $semigroup
T.
