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

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.