Partial isometry

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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, i.e. it is an isometry. (Conversely a surjective isometry is a unitary operator.) Projections are of course another example of partial isometry.

Partial isometries appear in the polar decomposition.


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.


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

 \begin{bmatrix}0 & 1 \\ 0 & 0 \end{bmatrix}

is a partial isometry with initial subspace

 \{0\} \oplus \mathbb{C}  \subseteq \mathbb{C} \oplus \mathbb{C}

and final subspace

  \mathbb{C} \oplus \{0\}.


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