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

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 $X^{\ast }$ has property $(\alpha )$, then a commuting $d$-tuple $(T_1,\dots ,T_d)$ 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 $(T_1,\dots ,T_d)$ of Ritt operators on Hilbert space. In particular we show that if $\Vert T_k\Vert \le 1$ for every $k=1,\dots ,d$, then $(T_1,\dots ,T_d)$ 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.

Accepted for publication in "Operators and Matrices". In this revised version, the appendix is slightly improved