The set of partial isometries as a quotient Finsler space

TL;DR

This paper applies a quotient Finsler metric framework to the space of partial isometries on a Hilbert space, deriving minimal geodesics and solutions to a linear approximation problem.

Contribution

It introduces a new Finsler metric on the space of partial isometries as a quotient of unitary groups, enabling analysis of geodesics and approximation solutions.

Findings

Derived a Finsler metric for partial isometries

Computed minimal geodesics in the quotient space

Solved a linear best approximation problem

Abstract

A known general program, designed to endow the quotient space ${\cal U}_{\cal A} / {\cal U}_{\cal B}$ of the unitary groups ${\cal U}_{\cal A}$, ${\cal U}_{\cal B}$ of the C$^*$ algebras ${\cal B}\subset{\cal A}$ with an invariant Finsler metric, is applied to obtain a metric for the space ${\cal I}({\cal H})$ of partial isometries of a Hilbert space ${\cal H}$. ${\cal I}({\cal H})$ is a quotient of the unitary group of ${\cal B}({\cal H})\times{\cal B}({\cal H})$, where ${\cal B}({\cal H})$ is the algebra of bounded linear operators in ${\cal H}$. Under this program, the solution of a linear best approximation problem leads to the computation of minimal geodesics in the quotient space. We find solutions of this best approximation problem, and study properties of the minimal geodesics obtained.