Fourier analysis methods in operator ergodic theory on super-reflexive Banach spaces

Earl Berkson

doi:10.3934/era.2010.17.90

Abstract

On reflexive spaces trigonometrically well-bounded operators (abbreviated twbo's'') have an operator-ergodic-theory characterization as the invertible operators $U$ whose rotates transfer'' the discrete Hilbert averages $(C,1)$-boundedly. Twbo's permeate many settings of modern analysis, and this note treats advances in their spectral theory, Fourier analysis, and operator ergodic theory made possible by applying classical analysis techniques pioneered by Hardy-Littlewood and L.C. Young to the R.C. James inequalities for super-reflexive spaces. When the James inequalities are combined with spectral integration methods and Young-Stieltjes integration for the spaces $V_{p}(\mathbb{T}) $ of functions having bounded $p$-variation, it transpires that every twbo on a super-reflexive space $X$ has a norm-continuous $V_{p}(\mathbb{T}) $-functional calculus for a range of values of $p>1$, and we investigate the ways this outcome logically simplifies and simultaneously advances the structure theory of twbo's on $X$. In particular, on a super-reflexive space $X$ (but not on the general reflexive space) Tauberian-type theorems emerge which improve to their $(C,0) $ counterparts the $(C,1) $ averaging and convergence associated with twbo's.

Highlights

Introduction and NotationThe symbol “K” with a set of subscripts will be used to denote a constant which depends only on its subscripts, and which can change in value from one occurrence to another
E(λ) → I as λ → ∞ and E(λ) → 0 as λ → −∞, each limit being with respect to the strong operator topology
Given a spectral family E(·) in the Banach space X concentrated on a compact interval J = [a, b], an associated notion of spectral integration can be developed as follows

Summary

Introduction and Notation

The symbol “K” with a (possibly empty) set of subscripts will be used to denote a constant which depends only on its subscripts, and which can change in value from one occurrence to another. Ergodic Hilbert transform, super-reflexive Banach space, spectral decomposition, p-variation, trigonometrically well-bounded operator. The (C, 1) averages appearing in the uniform boundedness condition of Proposition (1.1) can be replaced by the rotated ergodic Hilbert averages of U : (1.3) This set W is precompact relative to the strong operator topology of B (X). This circle of ideas is facilitated by the development of a suitable convergence theorem for the spectral integrals of Vp (T)-functions (Theorem 4.7).

Background

Functions of Higher Variation

Applications to Operator Ergodic Theory and Abstract Fourier Series