Universal Contractive Projections and A.E. Convergence

Abstract

Let (X, A, μ) be a probability space and for 1 ≤ p ≤ ∞ let L ^p := L ^P(X, A, μ) denote the usual Lebesgue space and ||.||_p the L ^p-norm. The structure of contractive projections of L ^pspaces has been characterized by Ando [1]. He proved that if 1 < p < ∞ and p ≠ 2 then every projection P :L _p → L ^p with ||P||_P ≤ 1 can be generated by a function ϕ ∈ L ^pand by a σ-field B of A-measurable subsets of {ϕ ≠ 0}. Namely P is of the form (1) where on {ϕ ≠ 0} and 0 elsewhere and E(.|B) is the conditional expectation relative to B. If the generating function ϕ is unimodular, i.e. if |ϕ| = 1 then = and the operator ({xi1|1}) is contractive for any L ^pwith 1 ≤ p ≤ ∞. Operators of this type will be called universal contractive projections.

1

This research was completed while the author was a visiting professor at the University of Tennessee, Knoxville, 1991.