Unitaries Permuting Two Orthogonal Projections

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.