Commuting families of polygonal type operators on Hilbert space

TL;DR

This work develops a dilation-based functional-calculus framework for commuting families of polygonal-type operators on a Hilbert space. By constructing a concrete unitary dilation for polynomially bounded polygonal-type operators, the authors obtain joint dilation results for $d$-tuples under suitable polygonal-type assumptions, enabling a generalized von Neumann inequality when enough components have polygonal type. They further prove that all commuting, polynomially bounded polygonal-type operators are jointly similar to contractions, and they extend these ideas to a multivariable polygonal functional calculus with bounds tied to a polygonal domain $\Delta^d$. Overall, the results generalize and extend prior work on Ritt operators and polygonal-type operators, providing explicit dilations and a robust multivariate calculus.

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 $Δ\subset {\mathbb D}$, with $\overlineΔ\cap{\mathbb T}\neq\emptyset$, such that the spectrum $σ(T)$ is included in $\overlineΔ$ and the resolvent $R(z,T)$ satisfies an estimate $\Vert R(z,T)\Vert \lesssim \max\{\vert z-ξ\vert^{-1}\, :\, ξ\in \overlineΔ\cap{\mathbb T}\}$ 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 $\Vert φ(T_1,\ldots,T_d)\Vert \leq C\Vertφ\Vert_{\infty,{\mathbb D}^d}$ for polynomials $φ$ 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 $k=1,\ldots,d$.

Figures (1)

• Figure 1: Polygon containing the spectrum of $T$

Theorems & Definitions (22)

• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 2.4
• Theorem 2.5
• Proposition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof