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Exponential stability of linear systems in Banach spaces. (English) Zbl 0694.47027
Let $$e^{tA}$$ be a $$C_ 0$$-semigroup in a Banach space X, let $$w_ s=\sup \{Re \lambda:\quad \lambda \in \sigma (A)\}<0$$ and let $$\sup \{\| (\lambda -A)^{-1}\|:\quad Re \lambda \geq \sigma \}<\infty$$ where $$w_ s<\sigma <0$$. It is shown that
(a) for every $$f\in X^*$$ and $$x\in X$$ we have $$\int^{\infty}_{0}e^{-\sigma t}| f(e^{tA}x)| dt<\infty,$$
(b) there exists $$M>0$$ such that $$\| e^{tA}x\| \leq Ne^{\sigma t}\| Ax\|$$ for every $$x\in D(A)$$, and
(c) there exists a Banach space $$\hat X\supset X$$ for which $$\| e^{tA}x\|_{\hat X}\leq e^{\sigma t}\| x\|_{\hat X}$$ for all $$x\in X.$$
This paper continues the work begun by the author in Ann. Diff. Eqs. 1, No.1, 43-56 (1985; Zbl 0593.34048).
Reviewer: R.Cross

MSC:
 47D03 Groups and semigroups of linear operators 47A50 Equations and inequalities involving linear operators, with vector unknowns 34G10 Linear differential equations in abstract spaces