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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

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