Partial isometry

Summary

A partial isometry is a bounded operator on a Hilbert space that is isometric on the orthogonal complement of its kernel. It maps an initial subspace isometrically onto a final subspace and kills everything orthogonal to the initial subspace. Partial isometries appear in the polar decomposition $T = V\abs{T}$ of an operator and in the comparison of projections in von Neumann algebras.