Joint numerical ranges and compressions of powers of operators

TL;DR

This paper explores the relationship between joint numerical ranges and the spectra of operator tuples, providing new insights into operator convergence, approximation, and generalizations of classical theorems in operator theory.

Contribution

It introduces a novel connection between joint numerical ranges and spectra, extending previous results and generalizing Bourin's pinching theorem in the context of operator powers.

Findings

Characterization of joint numerical ranges via joint spectrum

Transfer of weak convergence into approximation properties

Generalization of Bourin's pinching theorem

Abstract

We identify subsets of the joint numerical range of an operator tuple in terms of its joint spectrum. This result helps us to transfer weak convergence of operator orbits into certain approximation and interpolation properties for powers in the uniform operator topology. This is a far-reaching generalization of one of the main results in our recent paper posted as arXiv:1607.00040. Moreover, it yields an essential (but partial) generalization of Bourin's "pinching" theorem. It also allows us to revisit several basic results on joint numerical ranges, provide them with new proofs and find a number of new results.