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.This notion of minimality was used in [8, Section 6], and in the finite-dimensional case it corresponds precisely to the notion of minimal pencil used earlier in [17] (and to the notion of σ-minimal pencil used in [23]). Using matricial polar duality [11], one can show that 2010 Mathematics Subject Classification. 47A20, 47A13, 46L07, 47L25. Key words and phrases. Matrix convex set; matrix range; matrix extreme point; operator system; structure of compact tuples.

Strongly peaking representations and compressions of operator systems

“…The following result provides a necessary and sufficient condition to produce a fully compressed presentation of S, answering [28,Question 4.19]. Later, we will also consider some special cases in the language of matrix convexity.…”

“…In [28], certain special cases of fully compressed d-tuples were classified without using strongly peaking representations, but rather a different concept known as a crucial matrix extreme point. Our classification of fully compressed operator systems (or d-tuples) subsumes these results, so we now address how these two notions are related.…”

“…For T which act on infinite-dimensional spaces, minimality is insufficient. See [1, p.207], [33,Example 3.12], [28,Example 4.7], and [27,Example 3.14], noting that the first three examples are irreducible, and the fourth is compact. For compact tuples T ∈ K(H) d , one may include a nonsingularity assumption to fix the issue [10,Theorem 6.9] or instead require T to be fully compressed [28,Corollary 4.3].…”

“…See [1, p.207], [33,Example 3.12], [28,Example 4.7], and [27,Example 3.14], noting that the first three examples are irreducible, and the fourth is compact. For compact tuples T ∈ K(H) d , one may include a nonsingularity assumption to fix the issue [10,Theorem 6.9] or instead require T to be fully compressed [28,Corollary 4.3]. While uniqueness of fully compressed compact tuples may be shown without reference to nonsingularity, a compact tuple is in fact fully compressed if and only if it is both minimal and nonsingular [28,Theorem 4.4].…”

“…In this pursuit, we rely heavily on peaking properties of an operator system. The concept of a crucial matrix extreme point (see Definition 3.13), a noncommutative generalization of an isolated extreme point of a compact convex set, is used repeatedly in [28]. Arveson's earlier notion of strongly peaking representation, as in Definition 2.2, plays a similar role but is technically distinct.…”