On systems of subspaces of a Hilbert space such that every pair of subspaces satisfies one of the angle or commutativity condition

TL;DR

This paper investigates finite systems of Hilbert space subspaces where pairs satisfy angle or commutativity conditions, revealing structural properties and classifications of such configurations.

Contribution

It introduces a novel framework for analyzing systems of subspaces with specified angle and commutativity relations, expanding understanding of their geometric and algebraic structure.

Findings

Characterization of subspace systems with fixed angles

Classification of systems based on commutativity conditions

Structural insights into orthogonal and commuting projections

Abstract

We study finite systems of subspaces of a complex Hilbert space such that each pair of subspaces satisfies a certain condition as described in the following. For each subspace excepting the first one an angle between this subspace and the first one is fixed. The set of all subspaces excluding the first one is divided into disjoint subsets. Each such subset consists of one or two subspaces. Subspaces from different subsets are orthogonal and orthogonal projections onto subspaces from the same subset commute.