Commuting families of polygonal type operators on Hilbert space

Let $T\text{\hspace{0.17em}⁣}:H\to H$ be a bounded operator on Hilbert space. We say that $T$ has a polygonal type if there exists an open convex polygon $\mathrm{\Delta }\subset \mathbb{D}$, with $\bar{\mathrm{\Delta }}\cap \mathbb{T}\mathrm{\ne }\mathrm{\varnothing }$, such that the spectrum $\sigma (T)$ is included in $\bar{\mathrm{\Delta }}$ and the resolvent $R(z,T)$ satisfies an estimate $\mathrm{\parallel }R(z,T)\mathrm{\parallel }\lesssim \max \{\mathrm{\mid }z-\xi {\mathrm{\mid }}^{-1}:\xi \in \bar{\mathrm{\Delta }}\cap \mathbb{T}\}$ for $z\in {\bar{\mathbb{D}}}^c$. The class of polygonal type operators (which goes back to De Laubenfels and Franks-McIntosh) contains the class of Ritt operators. Let $T_1,\dots ,T_d$ be commuting operators on $H$, with $d\ge 3$. We prove functional calculus properties of the $d$-tuple $(T_1,\dots ,T_d)$ under various assumptions involving poygonal type. The main ones are the following. (1) If the $T_k$ are contractions for all $k=1,\dots ,d$ and if $T_1,\dots ,T_{d-2}$ have a polygonal type, then $(T_1,\dots ,T_d)$ satisfies a generalized von Neumann inequality $\mathrm{\parallel }\phi (T_1,\dots ,T_d)\mathrm{\parallel }\le C\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty },{\mathbb{D}}^d}$ for polynomials $\phi$ in $d$ variables; (2) If $T_k$ is polynomially bounded with a polygonal type for all $k=1,\dots ,d$, then there exists an invertible operator $S\text{\hspace{0.17em}⁣}:H\to H$ such that $\mathrm{\parallel }{S}^{-1}{T}_{k}S\mathrm{\parallel }\le 1$ for all $k=1,\dots ,d$.