# Partial isometry

In functional analysis a partial isometry is a linear map between Hilbert spaces such that it is an isometry on the orthogonal complement of its kernel.

The orthogonal complement of its kernel is called the initial subspace and its range is called the final subspace.

Partial isometries appear in the polar decomposition.

## Properties

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 subspace of W respectively.

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

$W=WW^*W.$

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.

## Classes

### Projections

Any isometric projection is one with full final subspace:

$P:\mathcal{H}\twoheadrightarrow\mathcal{K}:\quad\mathcal{F}P=\mathcal{K}$

(Especially, the orthogonal projections when restricted to their range.)

### Embeddings

Any isometric embedding is one with full initial subspace:

$J:\mathcal{H}\hookrightarrow\mathcal{K}:\quad\mathcal{I}J=\mathcal{H}$

(Precisely, these are the isometries.)

### Isomorphisms

Any unitary operator is one with full initial and final subspace:

$U:\mathcal{H}\leftrightarrow\mathcal{K}:\quad\mathcal{I}U=\mathcal{H},\,\mathcal{F}U=\mathcal{K}$

(Precisely, these are simultaneously projections and embeddings.)

### Others

Apart from these there are far more partial isometries.

## Examples

### Nilpotents

On the two-dimensional complex Hilbert space the matrix

$\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}$

is a partial isometry with initial subspace

$\{0\} \oplus \mathbb{C}$

and final subspace

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

### Leftshift and Rightshift

On the square summable sequences the operators

$R:\ell^2(\mathbb{N})\to\ell^2(\mathbb{N}):(x_1,x_2,\ldots)\mapsto(0,x_1,x_2,\ldots)$
$L:\ell^2(\mathbb{N})\to\ell^2(\mathbb{N}):(x_1,x_2,\ldots)\mapsto(x_2,x_3,\ldots)$

which are related by

$R^*=L$

are partial isometries with initial subspace

$LR(x_1,x_2,\ldots)=(x_1,x_2,\ldots)$

and final subspace:

$RL(x_1,x_2,\ldots)=(0,x_2,\ldots)$.