Partial Isometry

In functional analysis a partial isometry is a linear map W between Hilbert spaces H and K such that the restriction of W to the orthogonal complement of its kernel is an isometry. We call the orthogonal complement of the kernel of W the initial subspace of W, and the range of W is called the final subspace of W.

Any unitary operator on H is a partial isometry with initial and final subspaces being all of H.

For example, In the two-dimensional complex Hilbert space C2 the matrix

is a partial isometry with initial subspace

and 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 are also characterized by the condition that W W* or W* W is a projection. In that case, both W W* and W* W are projections (of course, since orthogonal projections are self-adjoint, each orthogonal projection is a partial isometry). This allows us to define partial isometry in any C*-algebra as follows:

If A is a C*-algebra, an element W in A is a partial isometry if and only if W W* or W* W is a projection (self-adjoint idempotent) in A. In that case W W* and W* W are both projections, and

1. W*W is called the initial projection of W.
2. W W* is called the final projection of W.

When A is an operator algebra, the ranges of these projections are the initial and final subspaces of W respectively.

It is not hard to show that partial isometries are characterised by the equation

A pair of projections one of which is the initial projection of a partial isometry and the other a final projection of the same isometry are said to be equivalent. This is indeed an equivalence relation and 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.

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