$n$-Subspaces in linear and unitary spaces

TL;DR

This paper explores the relationship between brick n-tuples of subspaces in linear spaces and irreducible n-tuples in Hilbert spaces, demonstrating conditions under which certain systems can be unitarized and characterizing these cases.

Contribution

It establishes a connection between linear and unitary subspace systems and provides criteria for unitarization of brick triples and quadruples of subspaces.

Findings

Brick systems of one-dimensional subspaces can be unitarized.

Coxeter functors preserve unitarizability of certain subspace systems.

Explicit characterizations of characters admitting unitarization for triples and quadruples.

Abstract

We study a relation between brick $n$-tuples of subspaces of a finite dimensional linear space, and irreducible $n$-tuples of subspaces of a finite dimensional Hilbert (unitary) space such that a linear combination, with positive coefficients, of orthogonal projections onto these subspaces equals the identity operator. We prove that brick systems of one-dimensional subspaces and the systems obtained from them by applying the Coxeter functors (in particular, all brick triples and quadruples of subspaces) can be unitarized. For each brick triple and quadruple of subspaces, we describe sets of characters that admit a unitarization.