Compressions of compact tuples

TL;DR

This paper investigates the matrix range of compact operator tuples, characterizing minimal, nonsingular, and fully compressed tuples, and establishing their unique determination by matrix ranges.

Contribution

It refines previous characterizations of nonsingular tuples and provides a new proof that fully compressed tuples are uniquely determined by their matrix ranges.

Findings

A compact tuple is fully compressed iff it is multiplicity-free with a trivial Shilov ideal.

Fully compressed tuples are uniquely determined by their matrix ranges up to unitary equivalence.

A new proof shows this uniqueness without relying on nonsingularity.

Abstract

We study the matrix range of a tuple of compact operators on a Hilbert space and examine the notions of minimal, nonsingular, and fully compressed tuples. In this pursuit, we refine previous results by characterizing nonsingular compact tuples in terms of matrix extreme points of the matrix range. Further, we find that a compact tuple $A$ is fully compressed if and only if it is multiplicity-free and the Shilov ideal is trivial, which occurs if and only if $A$ is minimal and nonsingular. Fully compressed compact tuples are therefore uniquely determined up to unitary equivalence by their matrix ranges. We also produce a proof of this fact which does not depend on the concept of nonsingularity.