Ergodic Theory and Dynamical Systems

Table of Contents - 01 September 1990 - Volume 10, Issue 03  

Research Article

Minimal convergence on Lp spaces

I. Assania1

a1 Department of Mathematics, University of Toronto, Ontario, Canada, M5S 1A1

Abstract

Let (X, F, μ) be a probability measure space, p and β real numbers such that 1≤p<+∞ and 0<β<p. For any linear positive operator T satisfying T1, T*1 = 1 we prove the norm and pointwise convergence of the sequence

S0143385700005666_eqnU1

We get then the pointwise and norm convergence in Lp, 0 < β ≥ 1 < p < 2, of the sequence
S0143385700005666_eqnU1

sgn Sif for any positive linear operator on Lp(Ω, A, μ) (μ-σ-finite) verifying xs2225(1 − α)I + αSxs2225p ≤ 1 for a real number 0 < α < 1. In the particular case α = 1, (S is a contraction), β = p−l, this result gives the pointwise and norm convergence of the sequences S0143385700005666_inline1 introduced by Beauzamy and Enflo in 1985 to the asymptotic center of the sequence S0143385700005666_inline2.

(Received May 26 1988)

(Revised August 14 1989)