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 **C**2 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 *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 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

*W***W*is called the**initial projection**of*W*.*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":

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... because U is a

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### 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.”

—Angela Carter (1940–1992)