Systems of two subspases in a Hilbert space

TL;DR

This paper classifies systems of two subspaces in an infinite-dimensional Hilbert space up to isomorphism, linking their structure to properties of bounded operators and providing explicit examples of non-isomorphic systems.

Contribution

It establishes a classification framework for two subspace systems based on operator ranges and nullity, and constructs explicit non-isomorphic examples.

Findings

Isomorphism classes determined by unitarily equivalent operator ranges

Nullity of bounded operators influences subspace system classification

Constructed explicit non-isomorphic subspace system examples

Abstract

We study two subspace systems in a separable infinite-dimensional Hilbert space up to (bounded) isomorphism. One of the main result of this paper is the following: Isomorphism classes of two subspace systems given by graphs of bounded operators are determined by unitarily equivalent classes of the operator ranges and the nullity of the original bounded operators giving graphs. We construct several non-isomorphic examples of two subspace systems in an infinite-dimensional Hilbert space.