Commuting families of polygonal type operators on Hilbert space

Let \(T:H\rightarrow H\) be a bounded operator on Hilbert space H. We say that T has a polygonal type if there exists an open convex polygon \(\Delta \subset {\mathbb {D}}\), with \(\overline{\Delta }\cap {\mathbb {T}}\ne \emptyset \), 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 \(d\ge 3\). 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 operator \(T_k\) is a contraction 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 \(\Vert \phi (T_1,\ldots ,T_d)\Vert \le C\Vert \phi \Vert _{\infty ,{\mathbb {D}}^d}\) 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 \(S:H\rightarrow H\) such that \(\Vert S^{-1}T_kS\Vert \le 1\) for all \(k=1,\ldots ,d\).