Abstract
In this paper, we investigate the role of square functions defined for a d-tuple of commuting Ritt operators \((T_1,\ldots ,T_d)\) acting on a general Banach space X. Firstly, we prove that if the d-tuple admits a \(H^\infty \) 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 \((T_1,\ldots ,T_d)\) on some Bochner space \(L_p(\Omega ;X)\) into a d-tuple of isomorphisms with a \(C(\mathbb {T}^d)\) bounded calculus. Finally, we compare for a d-tuple of Ritt operators its \(H^\infty \) joint functional calculus with its dilation into a d-tuple of polynomially bounded isomorphisms.
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