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arXiv · 2009.02270

Square functions for commuting families of Ritt operators

Olivier Arrigoni
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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 $L_p(Ω;X)$ 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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Olivier Arrigoni. 2020-09-04. Square functions for commuting families of Ritt operators. https://arxiv.org/abs/2009.02270

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