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

Abstract

A Ritt operator $T:X\to X$ on a Banach space is a power bounded operator satisfying an estimate $n\Vert T^n-T^{n-1}\Vert \le C$. When $X=L^p(\Omega )$ for some $1\le p\le \infty$, we study the validity of square functions estimates $\Vert ({\sum }_{k}k|T^k(x)-T^{k-1}(x){|}^2{)}^{1/2}{\Vert }_{L^p}\lesssim \Vert x{\Vert }_{L^p}$ for such operators. We show that $T$ and $T^{\ast }$ 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 analogue of the famous Cowling–Doust–McIntosh–Yagi characterization of bounded $H^{\infty }$-calculus on $L^p$-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 $L^p$-spaces, where square functions are quite explicit, and we give applications, examples, and illustrations on such spaces, as well as on classical $L^p$.