H^\infty functional calculus and square function estimates for Ritt operators

TL;DR

This paper investigates the conditions under which Ritt operators on Banach spaces, especially Lp spaces, satisfy square function estimates linked to their bounded functional calculus, extending results to noncommutative Lp-spaces.

Contribution

It establishes a characterization of Ritt operators with bounded H^-calculus via square function estimates, generalizing classical results to noncommutative and broader Banach spaces.

Findings

Ritt operators on Lp spaces satisfy specific square function estimates if and only if they have a bounded functional calculus.

The results extend to Hilbert spaces and general Banach spaces using Rademacher averages.

Applications and examples are provided for noncommutative Lp-spaces and classical Lp-spaces.

Abstract

A Ritt operator T : X --> X on Banach space is a power bounded operator such that the sequence of all n(T^{n} -T^{n-1}) is bounded. When X=Lp for some 1<p<\infty, we study the validity of square functions estimates Norm{(\sum_k k |T^{k}(x) - T^{k-1}(x)|^2)^{1/2}}_{Lp} \leq K Norm{x}_{Lp} for such operators. We show that T and its adjoint T^* both satisfy such estimates if and only if T admits a bounded functional calculus with respect to a Stolz domain. This is a single operator analog of the famous Cowling-Doust-McIntosh-Yagi characterization of bounded H^\infty-calculus on $Lp$-spaces by the boundedness of certain Littlewood-Paley-Stein square functions. We also prove a similar result on Hilbert space. Then we extend the above to more general Banach spaces, where square functions have to be defined in terms of certain Rademacher averages. We focus on noncommutative Lp-spaces, where square functions are quite explicit, and we give applications, examples and illustrations on those spaces, as well as on classical Lp.