Geometric spectral theory for compact operators

TL;DR

This paper develops a spectral theory for tuples of compact operators, characterizing their commutativity through the structure of their joint spectrum and polynomial factorizability.

Contribution

It introduces a joint spectrum concept for compact operators and links commutativity to the geometric structure of this spectrum and polynomial factorization.

Findings

Operators commute iff joint spectrum consists of countably many hyperplanes.

Normal matrices commute iff their associated polynomial is completely reducible.

Provides a spectral criterion for operator commutativity.

Abstract

We introduce a notion of joint spectrum for a tuple of compact operators on a separable Hilbert space and show that in many situations these operators commute if and only if the joint spectrum consists of countably many, locally finite, complex hyperplanes. In particular, we show that normal matrices (of the same size) $A_1,\cdots,A_n$ commute if and only if the polynomial $\det(z_1A_1+\cdots+z_nA_n+I)$ is completely reducible, that is, it can be factored into a product of linear polynomials.