# Unitary operators with decomposable corners

## TL;DR

This paper investigates pairs of unitary operators and subspaces where the restriction of the unitary to the subspace admits a singular value decomposition, providing abstract characterizations and exploring geometric relations.

## Contribution

It introduces a new framework for analyzing unitary operators with decomposable corners and offers abstract characterizations and geometric insights.

## Key findings

- Characterization of pairs with decomposable corners
- Relations between projections and operator decompositions
- Concrete examples illustrating the theory

## Abstract

We study pairs $(U,L_0)$, where $U$ is a unitary operator in $H$ and $L_0\subset H$ is a closed subspace, such that $$ P_{L_0}U|_{L_0}:L_0\to L_0 $$ has a singular value decomposition. Abstract characterizations of this condition are given, as well as relations to the geometry of projections and pairs of projections. Several concrete examples are examined.