James Olsen

Subsequence ergodic theorems for ^{} contractions

In this paper certain subsequence ergodic theorems which have previously been known in the case of measure preserving point transformations, or Dunford Schwartz operators, are extended to operators which are positive contractions on L P for p fixed

Weighted ergodic theorems for mean ergodic ₁-contractions

It is shown that any bounded weight sequence which is good for all probability preserving transformations (a universally good weight) is also a good weight for any L1-contraction with mean ergodic (ME) modulus, and for any positive contraction of Lp with 1 < p < ∞. We extend the return times theorem by proving that if S is a Dunford-Schwartz operator (not necessarily positive) on a Lebesgue space, then for any g bounded measurable {Sng(ω)} is a universally good weight for a.e. ω. We prove that if a bounded sequence has ”Fourier coefficents”, then its weighted averages for any L1-contraction with mean ergodic modulus converge in L1-norm. In order to produce weights, good for weighted ergodic theorems for L1-contractions with quasi-ME modulus (i.e., so that the modulus has a positive fixed point supported on its conservative part), we show that the modulus of the tensor product of L1contractions is the product of their moduli, and that the tensor product of positive quasi-ME L1-contractions is quasi-ME.

Subsequence pointwise ergodic theorems for operators inL p

In this paper certain subsequence ergodic theorems which have previously been known in the case of measure preserving point transformations are extended to Dunford-Schwartz operators, positive isometries, and power bounded Lamperti operators.

Dominated estimates of convex combinations of commuting isometries

The principal result of this paper is that the convex combination of two positive, invertible, commuting isometries ofLp(X,F, μ) 1<p<+∞, one of which is periodic, admits a dominated estimate with constantp/p−1. In establishing this, the following analogue of Linderholm’s theorem is obtained: Let σ and ε be two commuting non-singular point transformations of a Lebesgue Space with τ periodic. Then given ε>O, there exists a periodic non-singular point transformation σ′ such that σ′ commutes with τ and μ(x:σ′x≠σx}<ε. Byan approximation argument, the principal result is applied to the convex combination of two isometries ofLp (0, 1) induced by point transformations of the form τx=xk,k>0 to show that such convex combinations admit a dominated estimate with constantp/p−1.

Multi–Parameter Moving Averages

Publisher Summary This chapter discusses multiparameter moving averages. It presents a dynamical system, (X,Σ,m,T), where (X,Σ,m) is a probability space and T is a measure preserving point transformation from X onto itself. When T is induced by nonsingular point transformations, τx is written as Tx. The chapter presents a definition where τ is a nonsingular point transformation if τ:X → X, and if for each A ∈ Σ τ−1 (A) and τ (A) are in Σ and m(τ−1A) > 0 if and only if m(A)>0.

A multiple sequence ergodic theorem

Let (X, cF, JUL) be a cr-finite measure space, { T \ , . . . , Tk} a set of linear operators of Lp(X, ̂ , JUL), some p, 1 < p < oo. If 1 n , l n k l exists a.e. for all / G Lp, we say that the multiple sequence ergodic theorem holds for {7\ , . . . , T J . If / > 0 implies Tf>0 , we say that T is positive. If there exists an operator S such that |T/(x) |<S |/| (x) a.e., we say that T is dominated by S. In this paper we prove that if 7 \ , . . . , Tk are dominated by positive contractions of Lp(X, &, JUL), p fixed, K p < » , then the multiple sequence ergodic theorem holds for {T\, . . . , Tk}.