Cesàro and Abel ergodic theorems for integrated semigroups

IF 0.3 Q4 MATHEMATICS
F. Barki
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引用次数: 0

Abstract

Abstract Let {S(t)}t≥ 0 be an integrated semigroup of bounded linear operators on the Banach space 𝒳 into itself and let A be their generator. In this paper, we consider some necessary and sufficient conditions for the Cesàro mean and the Abel average of S(t) converge uniformly on ℬ(𝒳). More precisely, we show that the Abel average of S(t) converges uniformly if and only if 𝒳 = ℛ(A) ⊕ 𝒩(A), if and only if ℛ(Ak) is closed for some integer k and ∥ λ2R(λ, A) ∥ → 0 as λ→ 0+, where ℛ(A), 𝒩(A) and R(λ, A), be the range, the kernel, the resolvent function of A, respectively. Furthermore, we prove that if S(t)/t2 → 0 as t → 1, then the Cesàro mean of S(t) converges uniformly if and only if the Abel average of S(t) is also converges uniformly.
Cesàro和Abel遍历定理
摘要:设{S(t)}t≥0是Banach空间上有界线性算子的积分半群,它们归为自身,设A是它们的生成子。本文研究了S(t)的Cesàro均值和Abel均值在∑(f)上一致收敛的几个充分必要条件。更精确地说,我们证明了S(t)的Abel平均是一致收敛的,当且仅当∫f =∑(A)⊕(A),当且仅当∑(Ak)对于某个整数k和∥λ 2r (λ, A)∥→0为λ→0+是闭的,其中∑(A),∑(A)和R(λ, A)分别是A的范围,核,解函数。进一步证明了当t→1时S(t)/t2→0,则S(t)的Cesàro均值当且仅当S(t)的Abel平均值也均匀收敛。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Concrete Operators
Concrete Operators MATHEMATICS-
CiteScore
1.00
自引率
16.70%
发文量
10
审稿时长
22 weeks
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