Similarity between two projections

TL;DR

This paper explores the relationships between two orthogonal projections and unitary operators that interchange them, comparing different construction methods and providing a new proof using a supersymmetric approach.

Contribution

It offers a new proof for characterizing unitaries related to two projections, contrasting with existing methods based on the two projections theorem.

Findings

All such unitaries are characterized using the supersymmetric approach.

Comparison of different construction methods for these unitaries.

Provides a new proof based on supersymmetry rather than the two projections theorem.

Abstract

Given two orthogonal projections P and Q, we are interested in all unitary operators U such that UP=QU and UQ=PU. Such unitaries U have previously been constructed by Wang, Du, and Dou and also by one of the authors. One purpose of this note is to compare these constructions. Very recently, Dou, Shi, Cui, and Du described all unitaries U with the required property. Their proof is via the two projections theorem by Halmos. We here give a proof based on the supersymmetric approach by Avron, Seiler, and one of the authors.