Relative position of three subspaces in a Hilbert space

TL;DR

This paper investigates the geometric relationships among three subspaces in infinite-dimensional Hilbert spaces, extending finite-dimensional results and exploring conditions for specific decompositions.

Contribution

It extends Brenner's finite-dimensional classification of three subspaces to certain infinite-dimensional cases and provides conditions for pentagon decompositions.

Findings

Extended Brenner's classification to some infinite-dimensional subspaces

Identified conditions for systems to have pentagon decompositions

Provided partial results on subspace arrangements in Hilbert spaces

Abstract

We study the relative position of three subspaces in a separable infinite-dimensional Hilbert space. In the finite-dimensional case, Brenner described the general position of three subspaces completely. We extend it to a certain class of three subspaces in an infinite-dimensional Hilbert space. We also give a partial result which gives a condition on a system to have a (dense) decomposition containing a pentagon.