Cayley--Hamilton tuples: an interplay between algebraic varieties and joint spectra

TL;DR

This paper introduces Cayley--Hamilton tuples, linking algebraic varieties and joint spectra of commuting operator tuples, with applications to characterizing distinguished varieties and polynomial zero sets.

Contribution

It defines Cayley--Hamilton tuples, explores their spectral properties, and provides new characterizations of algebraic varieties and polynomial zero sets in operator theory.

Findings

Cayley--Hamilton tuples are annihilated by polynomials with joint spectrum matching algebraic varieties.

The Taylor and Waelbroeck joint spectra coincide for Cayley--Hamilton tuples.

Supports of annihilating ideals match the joint spectrum of the tuple.

Abstract

We introduce the notion of Cayley--Hamilton tuples: these are commuting operator tuples that are annihilated by a non-zero polynomial and such that its Taylor joint spectrum coincides with the algebraic variety determined by its annihilating ideal. Commuting matrix tuples are Cayley--Hamilton tuples. We provide two families of Cayley--Hamilton tuples in the infinite dimensional setting with additional details. What arises as a by-product is a concrete characterization of distinguished varieties in the polydisk in terms of Taylor joint spectrum of commuting isometries. These varieties have been of interest in various fields of mathematics over the last two decades. The Taylor and Waelbroeck joint spectrum of a Cayley--Hamilton tuple are shown to be the same. It is also shown that the support of the annihilating ideal of a Cayley--Hamilton tuple is the same as its joint spectrum. As an application, we deduce an algebraic characterization of bi-variate polynomials whose zero set intersected with the closed bidisk is the joint spectrum of a commuting isometric pair.