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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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