The set of partial isometries as a quotient Finsler space

A known general program, designed to endow the quotient space ${\cal U}{\cal A} / {\cal U}{\cal B}$oftheunitarygroups${\cal U}{\cal A}$,${\cal U}{\cal B}$oftheC$^*$algebras${\cal B}\subset{\cal A}$ with an invariant Finsler metric, is applied to obtain a metric for the space ${\cal I}({\cal H})$ofpartialisometriesofaHilbertspace${\cal H}$\mathrm{.}${\cal I}({\cal H})$isaquotientoftheunitarygroupof${\cal B}({\cal H})\times{\cal B}({\cal H})$,where${\cal B}({\cal H})$isthealgebraofboundedlinearoperatorsin${\cal H}$. 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.