JournalsrmiVol. 30, No. 4pp. 1149–1190

HH^{\infty} functional calculus and square function estimates for Ritt operators

  • Christian Le Merdy

    Université de Franche-Comté, Besançon, France
$H^{\infty}$ functional calculus and square function estimates for Ritt operators cover
Download PDF

Abstract

A Ritt operator T ⁣:XXT\colon X\to X on a Banach space is a power bounded operator satisfying an estimate nTnTn1Cn\| T^{n}-T^{n-1}\| \leq C. When X=Lp(Ω)X=L^p(\Omega) for some 1p1\leq p \leq \infty, we study the validity of square functions estimates (kkTk(x)Tk1(x)2)1/2Lp xLp\| (\sum_k k|T^{k}(x) - T^{k-1}(x)|^2)^{1/2}\|_{L^p}\lesssim\ \|x\|_{L^p} for such operators. We show that TT and TT^* both satisfy such estimates if and only if TT admits a bounded functional calculus with respect to a Stolz domain. This is a single operator analogue of the famous Cowling–Doust–McIntosh–Yagi characterization of bounded HH^\infty-calculus on LpL^p-spaces by the boundedness of certain Littlewood–Paley–Stein square functions. We also prove a similar result for Hilbert spaces. 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 LpL^p-spaces, where square functions are quite explicit, and we give applications, examples, and illustrations on such spaces, as well as on classical LpL^p.

Cite this article

Christian Le Merdy, HH^{\infty} functional calculus and square function estimates for Ritt operators. Rev. Mat. Iberoam. 30 (2014), no. 4, pp. 1149–1190

DOI 10.4171/RMI/811