Generalized Summing Sequences and the Mean Ergodic Theorem

Generalized Summing Sequences and the Mean Ergodic Theorem

Conditions are found on a sequence of probability measures pn on a locally compact abelian group G so that, for any strongly continuous unitary representation of G, J" Utfd/x„ will converge to a [/-invariant function. These conditions are applied in the case where the group is the integers. Introduction. In a recent paper [3] Greenleaf discusses summing sequences in amenable locally compact gro...

Strongly Ergodic Sequences of Integers and the Individual Ergodic Theorem

Let S = {ki,ki, ...} be an increasing sequence of positive integers. We call S strongly ergodic if for every measure preserving transformation T on a probability space (Cl, J, P) and every / £ Li(f2) we have limn-»oo(l/n) J^^j f(TkiuJ) = Pf(w) a.e. where Pf is the appropriate limit guaranteed by the individual ergodic theorem. We give sufficient conditions for a sequence S to be strongly ergodi...

On the Mean Ergodic Theorem for Subsequences

With these assumptions we have T defined for every integer n as a 1-1, onto, bimeasurable transformation. Henceforth we shall assume that every set considered is measurable, i.e. an element of a. We shall say that P is invariant if P(A) =P(TA) for every set A, P is ergodic if P is invariant and if P(U^L_oo TA) = 1 for every set A for which P(A) > 0 , and finally P is strongly mixing if P is inv...

Application of the Mean Ergodic Theorem to Certain Semigroups

We study the asymptotic behaviour of solutions of the Cauchy problem u′ = (∑n j=1(Aj + A −1 j ) − 2nI ) u, u(0) = x as t → ∞, for invertible isometries A1, . . . , An.

A Mean Ergodic Theorem For Asymptotically Quasi-Nonexpansive Affine Mappings in Banach Spaces Satisfying Opial's Condition