# Schmidt decomposable products of projections

## TL;DR

This paper characterizes when the product of two orthogonal projections on a Hilbert space admits a singular value decomposition, linking it to orthonormal bases, diagonalizability of their difference, and geometric properties of the Grassmann manifold.

## Contribution

It provides a new characterization of operators as products of projections with SVD, connecting algebraic, geometric, and operator-theoretic perspectives.

## Key findings

- Operators $T=PQ$ have SVD iff certain orthonormal bases exist.
- $A=P-Q$ is diagonalizable if and only if $T$ has SVD.
- Connections to Toeplitz, Hankel, Wiener-Hopf operators and Grassmannian geometry.

## Abstract

We characterize operators $T=PQ$ ($P,Q$ orthogonal projections in a Hilbert space $H$) which have a singular value decomposition. A spatial characterizations is given: this condition occurs if and only if there exist orthonormal bases $\{\psi_n\}$ of $R(P)$ and $\{\xi_n\}$ of $R(Q)$ such that $\langle\xi_n,\psi_m\rangle=0$ if $n\ne m$. Also it is shown that this is equivalent to $A=P-Q$ being diagonalizable. Several examples are studied, relating Toeplitz, Hankel and Wiener-Hopf operators to this condition. We also examine the relationship with the differential geometry of the Grassmann manifold of underlying the Hilbert space: if $T=PQ$ has a singular value decomposition, then the generic parts of $P$ and $Q$ are joined by a minimal geodesic with diagonalizable exponent.