Cesàro and Abel ergodic theorems for integrated semigroups

• Published in: Concrete operators (Warsaw, Poland) Vol. 8; no. 1; pp. 135 - 149
• Main Author: Barki, Fatih
• Format: Journal Article
• Language: English
• Published: De Gruyter 11-10-2021
• Subjects: 47D62; Abel averages; Cesàro means; Integrated semigroups; Uniform Abel ergodic; Uniform Cesàro ergodic
• ISSN: 2299-3282; 2299-3282
• DOI: 10.1515/conop-2020-0119

Let {
)}
be an integrated semigroup of bounded linear operators on the Banach space 𝒳 into itself and let
be their generator. In this paper, we consider some necessary and sufficient conditions for the Cesàro mean and the Abel average of
) converge uniformly on ℬ(𝒳). More precisely, we show that the Abel average of
) converges uniformly if and only if 𝒳 = ℛ(
) ⊕ 𝒩(
), if and only if ℛ(
) is closed for some integer
and ∥
,
) ∥ → 0 as
→ 0
, where ℛ(
), 𝒩(
) and
,
), be the range, the kernel, the resolvent function of
, respectively. Furthermore, we prove that if
)/
→ 0 as
→ 1, then the Cesàro mean of
) converges uniformly if and only if the Abel average of
) is also converges uniformly.