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 "partial isometry, isometry":

Polar Decomposition - Bounded Operators On Hilbert Space
... complex Hilbert spaces is a canonical factorization as the product of a partial isometry and a non-negative operator ... bounded linear 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 space of U is the closure of the range ... 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 ...
Self-adjoint Operator - Extensions of Symmetric Operators
... It associates a partially defined isometry to any symmetric densely defined operator ... A partially defined 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 ...

Famous quotes containing the word partial:

    You must not be partial in judging: hear out the small and the great alike; you shall not be intimidated by anyone, for the judgment is God’s.
    Bible: Hebrew, Deuteronomy 1:17.