Partial isometry

In functional analysis a partial isometry is a linear map between Hilbert spaces such that it is an isometry on the orthogonal complement of its kernel.

The orthogonal complement of its kernel is called the initial subspace and its range is called the final subspace.

The concept of partial isometry can be defined in other equivalent ways. If U is an isometric map defined on a closed subset H1 of a Hilbert space H then we can define an extension W of U to all of H by the condition that W be zero on the orthogonal complement of H1. Thus a partial isometry is also sometimes defined as a closed partially defined isometric map.

Partial isometries (and projections) can be defined in the more abstract setting of a semigroup with involution; the definition coincides with the one herein.

For C*-algebras one has the chain of equivalences due to the C*-property:

So one defines partial isometries by either of the above and declares the initial resp. final projection to be W*W resp. WW*.

It plays an important role in K-theory for C*-algebras and in the Murray-von Neumann theory of projections in a von Neumann algebra.

Any orthogonal projection is one with common initial and final subspace: