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Weak real integral characterizations for exponential stability of semigroups in reflexive spaces. (English) Zbl 1393.47019
Summary: Let \((t_n)\) be a sequence of nonnegative real numbers tending to \(\infty\), such that \(1\leq t_{n+1}-t_n\leq\alpha\) for all natural numbers \(n\) and some positive \(\alpha\). We prove that a strongly continuous semigroup \(\{T(t)\}_{t\geq 0}\), acting on a Hilbert space \(H\), is uniformly exponentially stable if \[ \sum\limits_{n=0}^\infty\varphi\left(|\langle T(t_n)x, y\rangle|\right)<\infty, \] for all unit vectors \(x\), \(y\) in \(H\). We obtain the same conclusion under the assumption that the inequality \[ \sum\limits_{n=0}^\infty\varphi\left(|\langle T(t_n)x,x^\ast\rangle|\right)<\infty, \] is fulfilled for all unit vectors \(x\in X\) and \(x^\ast\in X^\ast\), \(X\) being a reflexive Banach space. These results are stated for functions \(\phi\) belonging to a special class of functions, such as defined in the second section of this paper. We conclude our paper with a Rolewicz’s type result in the continuous case on Hilbert spaces.

MSC:
47D03 Groups and semigroups of linear operators
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