The set of partial isometries as a quotient Finsler space

Abstract

A known general program, designed to endow the quotient space ${\mathcal{U}}_{\mathcal{A}}/{\mathcal{U}}_{\mathcal{B}}$ of the unitary groups ${\mathcal{U}}_{\mathcal{A}}$, ${\mathcal{U}}_{\mathcal{B}}$ of the C${}^{\ast }$ algebras $\mathcal{B}\subset \mathcal{A}$ with an invariant Finsler metric, is applied to obtain a metric for the space $\mathcal{I}(\mathcal{H})$ of partial isometries of a Hilbert space $\mathcal{H}$. $\mathcal{I}(\mathcal{H})$ is a quotient of the unitary group of $\mathcal{B}(\mathcal{H})\times \mathcal{B}(\mathcal{H})$, where $\mathcal{B}(\mathcal{H})$ is the algebra of bounded linear operators in $\mathcal{H}$. 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.