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