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

We introduce and investigate $H^{\mathrm{\infty }}$-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 $H^{\mathrm{\infty }}$ joint functional calculus if and only if each $T_k$ admits an $H^{\mathrm{\infty }}$ functional calculus. Next for $p\in (1,\mathrm{\infty })$, we characterize commuting $d$-tuple of Ritt operators on $L^p(\mathrm{\Omega })$ which admit an $H^{\mathrm{\infty }}$ 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 $(T_1,\ldots, T_d)$ofRittoperatorsonHilbertspace\mathrm{.}Inparticularweshowthatif$\Vert T_k\Vert\leq 1$forevery$k=1,\ldots,d$,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.