Abel ergodic theorems for $\alpha$-times integrated semigroups
DOI:
https://doi.org/10.33044/revuma.2661Abstract
Let $\{T(t)\}_{t\geq 0}$ be an $\alpha$-times integrated semigroup of bounded linear operators on the Banach space $\mathcal{X}$ and let $A$ be their generator. In this paper, we study the uniform convergence of the Abel averages $\mathcal{A} (\lambda) = \lambda^{\alpha +1} \int_{0}^{\infty} e^{-\lambda t} T(t)\,dt$ as $\lambda \to 0^+$, with $\alpha \geq 0$. More precisely, we show that the following conditions are equivalent: (i) $T(t)$ is uniformly Abel ergodic; (ii) $\mathcal{X}= \mathcal{R}(A)\oplus \mathcal{N}(A)$, with $\mathcal{R}(A)$ closed; (iii) $\|\lambda^2 R(\lambda,A)\| \longrightarrow 0$ as $\lambda\to 0^+$, and $\mathcal{R}(A^k)$ is closed for some integer $k$; (iv) $A$ is $a$-Drazin invertible and $\mathcal{R}(A^k)$ is closed for some $k\geq 1$; where $\mathcal{N}(A)$, $\mathcal{R}(A)$ and $R(\lambda,A)$ are the kernel, the range, and the resolvent function of $A$, respectively. Additionally, we show that if $T(t)$ satisfies $\lim_{t\to \infty}\|T(t)\|/ t^{\alpha+1}=0$, then $T(t)$ is uniformly Abel ergodic if and only if $\frac{1}{t^{\alpha+1}}\int_{0}^{t}T(s)\,ds$ converges uniformly as $t \to +\infty$. Finally, we examine simultaneously this theory with the uniform power convergence of the Abel averages $\mathcal{A} (\lambda)$ for some $\lambda>0$.
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