Dilation of Ritt operators on L^{p}-spaces

Abstract

For any Ritt operator T:L^{p}(\Omega) --> L^{p}(\Omega), for any positive real number \alpha, and for any x in L^{p}, we consider the square functions |x |_{T,\alpha} = \Bigl| \Bigl(\sum_{k=1}^{\infty} k^{2\alpha -1}\bigl |T^{k-1}(I-T)^\alpha x \bigr|^2 \Bigr)^{1/2}_{L^{p}}. We show that if T is actually an R-Ritt operator, then these square functions are pairwise equivalent. Then we show that T and its adjoint T* acting on L^{p'} both satisfy uniform estimates |x|_{T,1} \lesssim |x|_{L^{p}} and |y|_{T*,1} \lesssim |y|_{L^{p'}} for x in L^{p} and y in L^{p'} if and only if T is R-Ritt and admits a dilation in the following sense: there exist a measure space \widetilde{\Omega}, an isomorphism U of L^{p}(\widetilde{\Omega}) such that the sequence of all U^{n} for n varying in Z is bounded, as well as two bounded maps J : L^{p}(\Omega) --> L^{p}(\widetilde{\Omega}) and Q : L^{p}(\widetilde{\Omega}) --> L^p(\Omega) such that T^{n}=QU^{n}J for any nonnegative integer n. We also investigate functional calculus properties of Ritt operators and analogs of the above results on noncommutative L^{p}-spaces.