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

Let $(T_1,\ldots,T_d)$ be a commuting $d$-tuple of Ritt$_E$ operators on some UMD Banach space $X$.

We show that $(T_1,\ldots,T_d)$ admits a bounded $H^\infty$-functional calculus if and only if $T_k$ is an $R$-Ritt$_E$ operator for every $k=1,\ldots,d$, and the $d$-tuple $(T_1,\ldots,T_d)$ admits an isometric dilation $(U_1,\ldots,U_d)$ on some UMD Banach space $Y$ such that $(U_1,\ldots,U_d)$ is polynomially bounded.

In the case where $X$ further possesses property $(\alpha)$, we establish other characterizations of the $H^\infty$-functional calculus property for $(T_1,\ldots,T_d)$ in terms of isometric dilations.