# 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.

### Other articles related to "isometry, partial isometry":

Self-adjoint Operator - Extensions of Symmetric Operators
... It associates a partially defined isometry to any symmetric densely defined operator ... isometric operator with closed domain is called a partial isometry ... Given a partial isometry V, the deficiency indices of V are defined as the dimension of the orthogonal complements of the domain and range Theorem ...
Polar Decomposition - Bounded Operators On Hilbert Space
... factorization as the product of a partial isometry and a non-negative operator ... then there is a unique factorization of A as a product A = UP where U is a partial isometry, P is a non-negative self-adjoint operator and the initial ... The operator U must be weakened to a partial isometry, rather than unitary, because of the following issues ...
Quasinormal Operator - Definition and Some Properties - Properties
... because U is a partial isometry whose initial space is closure of range P ... This is because that in the finite dimensional case, the partial isometry U in the polar decomposition A = UP can be taken to be unitary ... This then gives In general, a partial isometry may not be extendable to a unitary operator and therefore a quasinormal operator need not be normal ...

### Famous quotes containing the word partial:

The one-eyed man will be King in the country of the blind only if he arrives there in full possession of his partial faculties—that is, providing he is perfectly aware of the precise nature of sight and does not confuse it with second sight ... nor with madness.
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