Square functions for commuting families of Ritt operators

TL;DR

This paper explores the relationship between square function estimates and functional calculus for commuting Ritt operators on Banach spaces, establishing conditions for dilation and bounded calculus.

Contribution

It demonstrates that $H^$ joint functional calculus implies square function estimates and shows how these estimates lead to dilations into isomorphisms on Bochner spaces.

Findings

$H^$ calculus implies square function estimates for Ritt tuples.

Square function estimates lead to dilations into isomorphisms on $L_p$ spaces.

Comparison between $H^$ calculus and polynomially bounded dilations for Ritt tuples.

Abstract

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 $H^\infty$ 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 $(T_1,...,T_d)$ 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.