论$$C_0$$-半群的均匀指数稳定性与解析量的有界性之间的等价性

Pub Date : 2024-07-24 DOI:10.1007/s00233-024-10455-5

Abdelhadi El Harfi

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引用次数: 0

摘要

我们考虑一个巴拿赫空间上的$C_0$-半群，它的解析量在右半平面上是均匀有界的。在本文中，我们提供了一个关于解析量的条件，这个条件对于这种半群的均匀指数稳定性是充分和必要的。因此，我们给出了 Gearhart 定理的另一个证明（Trans.Amer.Math.236, 385-394 (1978) ）。这种方法依赖于一个复杂的反转公式和调和分布。

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On the equivalence between the uniform exponential stability of a $$C_0$$ -semigroup and the boundedness of the resolvent

We consider a $C_0$-semigroup on a Banach space such that the resolvent is uniformly bounded on the right half-plane. In this paper we provide a condition on the resolvent which is sufficient and necessary for the uniform exponential stability of such a semigroup. As a consequence, we give an alternative proof of Gearhart’s theorem (Trans. Amer. Math. Soc. 236, 385–394 (1978)). The approach lies on a complex inversion formula and tempered distributions.

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