Let (X, μ) be a σ-finite measure space and let τ be an ergodic invertible measure preserving transformation. We study the a.e. convergence of the Cesàro-α ergodic averages associated with τ and the boundedness of the corresponding maximal operator in the setting of Lp,q(wdμ) spaces.

It is well-known that a probability measure $\mu$ on the circle $𝕋$ satisfies $\parallel {\mu }^n{*f-\int f\mathrm{d}m\parallel }_p\to 0$ for every $f\in L_p$, every (some) $p\in [1,\infty )$, if and only if $|\widehat{\mu }\left(n\right)|lt;1$ for every non-zero $n\in \mathbb{Z}$ ($\mu$ is strictly aperiodic). In this paper we study the a.e. convergence of ${\mu }^n*f$ for every $f\in L_p$ whenever $pgt;1$. We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of $\mu$, for the strong sweeping out property (existence of a Borel set $B$ with $lim\; sup{\mu }^n*1_B=1$ a.e. and $lim\; inf{\mu }^n*1_B=0$ a.e.). The results are extended to general compact Abelian groups $G$ with Haar...

Bellow and Calderón proved that the sequence of convolution powers ${\mu }_{n}f\left(x\right)={\sum }_{k\in \mathbb{Z}}{\mu }^n\left(k\right)f\left(T^{k}x\right)$ converges a.e, when $\mu$ is a strictly aperiodic probability measure on $\mathbb{Z}$ such that the expectation is zero, $E\left(\mu \right)=0$, and the second moment is finite, $m_2\left(\mu \right)\<\infty$. In this paper we extend this result to cases where $m_2\left(\mu \right)=\infty$.

We show that some results of Gaposhkin about a.e. convergence of series associated to a unitary operator U acting on L²(X,Σ,μ) (μ is a σ-finite measure) may be extended to the case where U is an invertible power-bounded operator acting on $L^p(X,\Sigma ,\mu )$, p > 1. The proofs make use of the spectral integration initiated by Berkson-Gillespie and, more specifically, of recent results of the author.

In the normed space of bounded operators between a pair of normed spaces, the set of operators which are "bounded below" forms the interior of the set of one-one operators. This note is concerned with the extension of this observation to certain spaces of pairs of operators.

This paper is chiefly a survey of results obtained in recent years on the asymptotic behaviour of semigroups of bounded linear operators on a Banach space. From our general point of view, discrete families of operators $T^n:n=0,1,...$ on a Banach space X (discrete one-parameter semigroups), one-parameter $C_0$-semigroups $T\left(t\right):t\ge 0$ on X (strongly continuous one-parameter semigroups), are particular cases of representations of topological abelian semigroups. Namely, given a topological abelian semigroup S, a family of bounded...

We investigate the characterization of almost periodic C-semigroups, via the Hille-Yosida space Z₀, in case of R(C) being non-dense. Analogous results are obtained for C-cosine functions. We also discuss the almost periodicity of integrated semigroups.