An Extension of the Admissibility-Type Conditions for the Exponential Dichotomy of C 0-Semigroups期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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An Extension of the Admissibility-Type Conditions for the Exponential Dichotomy of C 0-Semigroups

Authors:	Ciprian Preda Petre Preda

Institution:	(1) Department of Mathematics, University of California, Los Angeles, CA, 90095, U.S.A.;(2) Present address: West University of Timisoara, Bd. V. Parvan, No. 4, 300223 Timişoara, Romania;(3) Department of Mathematics, West University of Timişoara, Bd. V. Parvan, No. 4, 300223 Timişoara, Romania

Abstract:	In the present paper we obtain a sufficient condition for the exponential dichotomy of a strongly continuous, one-parameter semigroup $\{T(t)\}_{t\geq 0}$ , in terms of the admissibility of the pair $(L^{p}({\mathbb{R}}_{+}, X), L^{q}({\mathbb{R}}_{+}, X))$ . It is already known the equivalence between the $(L^{p}({\mathbb{R}}_{+}, X), L^{q}({\mathbb{R}}_{+}, X))$ -admissibility condition $(1 \leq p \leq q \leq \infty$ and $(p, q) \neq (1,\infty))$ and the hyperbolicity of a C ₀-semigroup $\{T(t)\}_{t\geq 0}$ , when we assume a priori that the kernel of the dichotomic projector (denoted here by X ₂) is T(t)-invariant and $T(t)\|_{X_2}$ is an invertible operator. We succeed to prove in this paper that the admissibility of the pair $(L^{p}({\mathbb{R}}_{+}, X), L^{q}({\mathbb{R}}_{+},X))$ still implies the existence of an exponential dichotomy for a C ₀-semigroup ${\bf T} = \{T(t)\}_{t\leq 0}$ even in the general case where the kernel of the dichotomic projector, X ₂, is not assumed to be T(t)-invariant.

Keywords:	" target="_blank"> C ₀-semigroup exponential dichotomy admissibility
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