Commuting families of polygonal type operators on Hilbert space

TL;DR

This paper investigates polygonal type operators on Hilbert space, establishing functional calculus properties and similarity results for commuting operator tuples under polygonal spectrum and resolvent conditions.

Contribution

It introduces new functional calculus results and similarity criteria for commuting polygonal type operators, extending the class of Ritt operators.

Findings

Establishes a generalized von Neumann inequality for certain operator tuples.

Proves the existence of similarity transforms to contractions for polynomially bounded polygonal type operators.

Extends functional calculus theory to multivariable settings with polygonal spectral conditions.

Abstract

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 $k=1,\ldots,d$.