# 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 *H*_{1} 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 *H*_{1}. 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: