# Unitaries Permuting Two Orthogonal Projections

## TL;DR

This paper investigates conditions under which a unitary operator can interchange two orthogonal projections on a Hilbert space, providing a new proof using supersymmetric methods.

## Contribution

It offers a novel proof of a known characterization for unitaries permuting two orthogonal projections, utilizing supersymmetric techniques.

## Key findings

- Existence condition for unitaries swapping projections
- New proof using supersymmetric machinery
- Extension to infinite-dimensional cases

## Abstract

Let $P$ and $Q$ be two orthogonal projections on a separable Hilbert space, $\calH$. Wang, Du and Dou proved that there exists a unitary, $U$, with $UPU^{-1} =Q, \quad UQU^{-1} = P$ if and only if $\dim(\ker P \cap \ker(1-Q)) = \dim(\ker Q \cap \ker(1-P))$ (both may be infinite). We provide a new proof using the supersymmetric machinery of Avron, Seiler and Simon.