Connecting $H^{\mathrm{\infty }}$-functional calculus and isometric dilations for commuting families of Ritt${}_E$ operators

Let $(T_1,\dots ,T_d)$ be a commuting $d$-tuple of Ritt${}_E$ operators on some UMD Banach space $X$. We show that $(T_1,\dots ,T_d)$ admits a bounded $H^{\mathrm{\infty }}$-functional calculus if and only if $T_k$ is an $R$-Ritt${}_E$ operator for every $k=1,\dots ,d$, and the $d$-tuple $(T_1,\dots ,T_d)$ admits an isometric dilation $(U_1,\dots ,U_d)$ on some UMD Banach space $Y$ such that $(U_1,\dots ,U_d)$ is polynomially bounded. In the case where $X$ further possesses property $(\alpha )$, we establish other characterizations of the $H^{\mathrm{\infty }}$-functional calculus property for $(T_1,\dots ,T_d)$ in terms of isometric dilations.