# $H^\infty$-functional calculus for commuting families of Ritt operators and sectorial operators

## TL;DR

This paper develops a comprehensive theory for $H^al$-functional calculus for commuting families of Ritt and sectorial operators on Banach spaces, providing characterizations, dilation properties, and multivariable inequalities.

## Contribution

It introduces and characterizes $H^al$-functional calculus for commuting Ritt and sectorial operators, extending known results and establishing new dilation and inequality properties.

## Key findings

- Characterization of $H^al$-calculus for Ritt operators on Banach lattices and spaces with property $(al)$.
- Dilation-based criteria for $H^al$-calculus on $L^p$ spaces.
- Multivariable von Neumann inequality for commuting contractions.

## Abstract

We introduce and investigate $H^\infty$-functional calculus for commuting finite families of Ritt operators on Banach space $X$. We show that if either $X$ is a Banach lattice or $X$ or $X^*$ has property $(\alpha)$, then a commuting $d$-tuple $(T_1,\ldots, T_d)$ of Ritt operators on $X$ has an $H^\infty$ joint functional calculus if and only if each $T_k$ admits an $H^\infty$ functional calculus. Next for $p\in(1,\infty)$, we characterize commuting $d$-tuple of Ritt operators on $L^p(\Omega)$ which admit an $H^\infty$ joint functional calculus, by a joint dilation property. We also obtain a similar characterisation for operators acting on a UMD Banach space with property $(\alpha)$. Then we study commuting $d$-tuples $(T_1,\ldots, T_d)$ of Ritt operators on Hilbert space. In particular we show that if $\Vert T_k\Vert\leq 1$ for every $k=1,\ldots,d$, then $(T_1,\ldots, T_d)$ satisfies a multivariable analogue of von Neumann's inequality. Further we show analogues of most of the above results for commuting finite families of sectorial operators.