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.

在本文中，我们研究了为作用于一般Banach空间X上的通勤Ritt算子d-元组\（（T_1，\ ldots，T_d）\）定义的平方函数的作用。首先，我们证明如果d元组接受\（H ^ \ infty \）联合函数演算，则它可以验证各种平方函数估计。然后我们研究当每个\（T_k \）是R -Ritt运算符时的逆。在最后一个假设下，当X是一个K凸空间时，我们证明平方函数估计在某些Bochner空间\（L_p（\ Omega; X）\上产生\（（T_1，\ ldots，T_d）\）的扩张。 ）进入带有\（C（\ mathbb {T} ^ d）\）有界微积分的d元同构。最后，对于Ritt算子的d元组，将其\（H ^ \ infty \）联合函数演算与将其扩张为多项式有界同构的d元组进行比较。