Cesàro and Abel ergodic theorems for integrated semigroups

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 sh...

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Autor principal:	Barki Fatih
Formato:	article
Lenguaje:	EN
Publicado:	De Gruyter 2021
Materias:	cesàro means abel averages integrated semigroups uniform abel ergodic uniform cesàro ergodic 47d62 Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/d09c407aa8594ec0a71e736659c3b80e
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spelling	oai:doaj.org-article:d09c407aa8594ec0a71e736659c3b80e2021-12-05T14:10:45ZCesàro and Abel ergodic theorems for integrated semigroups2299-328210.1515/conop-2020-0119https://doaj.org/article/d09c407aa8594ec0a71e736659c3b80e2021-10-01T00:00:00Zhttps://doi.org/10.1515/conop-2020-0119https://doaj.org/toc/2299-3282Let {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.Barki FatihDe Gruyterarticlecesàro meansabel averagesintegrated semigroupsuniform abel ergodicuniform cesàro ergodic47d62MathematicsQA1-939ENConcrete Operators, Vol 8, Iss 1, Pp 135-149 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	cesàro means abel averages integrated semigroups uniform abel ergodic uniform cesàro ergodic 47d62 Mathematics QA1-939
spellingShingle	cesàro means abel averages integrated semigroups uniform abel ergodic uniform cesàro ergodic 47d62 Mathematics QA1-939 Barki Fatih Cesàro and Abel ergodic theorems for integrated semigroups
description	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.
format	article
author	Barki Fatih
author_facet	Barki Fatih
author_sort	Barki Fatih
title	Cesàro and Abel ergodic theorems for integrated semigroups
title_short	Cesàro and Abel ergodic theorems for integrated semigroups
title_full	Cesàro and Abel ergodic theorems for integrated semigroups
title_fullStr	Cesàro and Abel ergodic theorems for integrated semigroups
title_full_unstemmed	Cesàro and Abel ergodic theorems for integrated semigroups
title_sort	cesàro and abel ergodic theorems for integrated semigroups
publisher	De Gruyter
publishDate	2021
url	https://doaj.org/article/d09c407aa8594ec0a71e736659c3b80e
work_keys_str_mv	AT barkifatih cesaroandabelergodictheoremsforintegratedsemigroups
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Cesàro and Abel ergodic theorems for integrated semigroups

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