We study the convergence properties of Dirichlet series for a bounded linear operator T in a Banach space X. For an increasing sequence $\mu ={\mu }_n$ of positive numbers and a sequence $f=f_n$ of functions analytic in neighborhoods of the spectrum σ(T), the Dirichlet series for $f_n\left(T\right)$ is defined by D[f,μ;z](T) = ∑n=0∞ e-μnz fn(T), z∈ ℂ. Moreover, we introduce a family of summation methods called Dirichlet methods and study the ergodic properties of Dirichlet averages for T in the uniform operator topology.

The purpose of this paper is to show that if σ is the maximal spectral type of Chacon’s transformation, then for any d ≠ d’ we have ${\sigma }^{*d}\perp {\sigma }^{*d^{\text{'}}}$. First, we establish the disjointness of convolutions of the maximal spectral type for the class of dynamical systems that satisfy a certain algebraic condition. Then we show that Chacon’s automorphism belongs to this class.

We create a general framework for describing domains of functions of power-bounded operators given by power series with log-convex coefficients. This sheds new light on recent results of Assani, Derriennic, Lin and others. In particular, we resolve an open problem regarding the "one-sided ergodic Hilbert transform" formulated in a 2001 paper by Derriennic and Lin.

We study conditions under which Dominated Ergodic Theorems hold in rearrangement invariant spaces. Consequences for Orlicz and Lorentz spaces are given. In particular, our results generalize the classical theorems for the spaces $L_p$ and the classes $Llo{g}^{n}L$.

We give sufficient conditions for the strong asymptotic stability of the distributions of dynamical systems with multiplicative perturbations. We apply our results to iterated function systems.

We define the space of trajectories of a doubly stochastic operator on L¹(X,μ) as a shift space $(X^{\mathbb{N}},\nu ,\sigma )$, where ν is a probability measure defined as in the Ionescu-Tulcea theorem and σ is the shift transformation. We study connections between the entropy of a doubly stochastic operator and the entropy of the shift on the space of trajectories of this operator.

Two types of weighted ergodic averages are studied. It is shown that if F = {Fₙ} is an admissible superadditive process relative to a measure preserving transformation, then a Wiener-Wintner type result holds for F. Using this result new good classes of weights generated by such processes are obtained. We also introduce another class of weights via the group of unitary functions, and study the convergence of the corresponding weighted averages. The limits of such weighted averages are also identified....

The purpose of this note is to prove various versions of the ergodic decomposition theorem for probability measures on standard Borel spaces which are quasi-invariant under a Borel action of a locally compact second countable group or a discrete nonsingular equivalence relation. In the process we obtain a simultaneous ergodic decomposition of all quasi-invariant probability measures with a prescribed Radon-Nikodym derivative, analogous to classical results about decomposition of invariant probability...

Let $\left\{T\right(t):t>0\}$ be a strongly continuous semigroup of linear contractions in $L_p$, $1<p<\infty$, of a $\sigma$-finite measure space. In this paper we prove that if there corresponds to each $t>0$ a positive linear contraction $P\left(t\right)$ in $L_p$ such that $\left|T\right(t\left)f\right|\le P\left(t\right)\left|f\right|$ for all $f\in L_p$, then there exists a strongly continuous semigroup $\left\{S\right(t):t>0\}$ of positive linear contractions in $L_p$ such that $\left|T\right(t\left)f\right|\le S\left(t\right)\left|f\right|$ for all $t>0$ and $f\in L_p$. Using this and Akcoglu’s dominated ergodic theorem for positive linear contractions in $L_p$, we also prove multiparameter pointwise ergodic and local ergodic theorems...

We provide sufficient conditions for sums of two unbounded operators on a Banach space to be (pre-)generators of contraction semigroups. Necessary conditions and applications to positive emigroups on Banach lattices are also presented.