Compressions of compact tuples

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.

arXiv comment17 pages. To appear in Linear Algebra and its Applications

References in corpus (10)

• The noncommutative Choquet boundary
• The Choquet boundary of an operator system
• The matricial relaxation of a linear matrix inequality
• Dilations, inclusions of matrix convex sets, and completely positive maps
• Minimal and maximal matrix convex sets
• Spectrahedral Containment and Operator Systems with Finite-Dimensional Realization
• Extreme points of matrix convex sets, free spectrahedra and dilation theory
• Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets
• Arveson extreme points span free spectrahedra
• Matrix Convex Sets Without Absolute Extreme Points

Cited by in corpus (6)

• Strongly peaking representations and compressions of operator systems
• Noncommutative partial convexity via $\Gamma$-convexity
• Empirical properties of optima in free semidefinite programs
• Matrix convex sets over the Euclidean ball and polar duals of real free spectrahedra
• Extreme points of matrix convex sets and their spanning properties
• On the matrix range of random matrices