## 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.

Original language | English |
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Pages (from-to) | 264-283 |

Number of pages | 20 |

Journal | Linear Algebra and Its Applications |

Volume | 564 |

DOIs | |

State | Published - 1 Mar 2019 |

Externally published | Yes |

## Keywords

- Matrix convex set
- Matrix extreme point
- Matrix range
- Operator system
- Structure of compact tuples

## ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics