Square Functions for Commuting Families of Ritt Operators

In this paper, we investigate the role of square functions defined for a d-tuple of commuting Ritt operators (T-1, ..., T-d) acting on a general Banach space X. Firstly, we prove that if the d-tuple admits a H-infinity joint functional calculus, then it verifies various square function estimates. Then we study the converse when every T-k is a R-Ritt operator. Under this last hypothesis, and when X is a K -convex space, we show that square function estimates yield dilation of (T-1, ..., T-d) on some Bochner space L-p (Omega; X) into a d-tuple of isomorphisms with a C(T-d) bounded calculus. Finally, we compare for a d-tuple of Ritt operators its H-infinity joint functional calculus with its dilation into a d-tuple of polynomially bounded isomorphisms.