# 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.

## General

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 (and projections) can be defined in the more abstract setting of a semigroup with involution; the definition coincides with the one herein.

## Operator Algebras

For operator algebras one introduces the initial and final subspaces:

${\displaystyle {\mathcal {I}}W:={\mathcal {R}}W^{*}W,\,{\mathcal {F}}W:={\mathcal {R}}WW^{*}}$

## C*-Algebras

For C*-algebras one has the chain of equivalences due to the C*-property:

${\displaystyle (W^{*}W)^{2}=W^{*}W\iff WW^{*}W=W\iff W^{*}WW^{*}=W^{*}\iff (WW^{*})^{2}=WW^{*}}$

So one defines partial isometries by either of the above and declares the initial resp. final projection to be W*W resp. WW*.

A pair of projections are partitioned by the equivalence relation:

${\displaystyle P=W^{*}W,\,Q=WW^{*}}$

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.

## Special Classes

### Projections

Any orthogonal projection is one with common initial and final subspace:

${\displaystyle P:{\mathcal {H}}\rightarrow {\mathcal {H}}:\quad {\mathcal {I}}P={\mathcal {F}}P}$

### Embeddings

Any isometric embedding is one with full initial subspace:

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

### Unitaries

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

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

(Apart from these there are far more partial isometries.)

## Examples

### Nilpotents

On the two-dimensional complex Hilbert space the matrix

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

is a partial isometry with initial subspace

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

and final subspace

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

### Leftshift and Rightshift

On the square summable sequences the operators

${\displaystyle R:\ell ^{2}(\mathbb {N} )\to \ell ^{2}(\mathbb {N} ):(x_{1},x_{2},\ldots )\mapsto (0,x_{1},x_{2},\ldots )}$
${\displaystyle L:\ell ^{2}(\mathbb {N} )\to \ell ^{2}(\mathbb {N} ):(x_{1},x_{2},\ldots )\mapsto (x_{2},x_{3},\ldots )}$

which are related by

${\displaystyle R^{*}=L}$

are partial isometries with initial subspace

${\displaystyle LR(x_{1},x_{2},\ldots )=(x_{1},x_{2},\ldots )}$

and final subspace:

${\displaystyle RL(x_{1},x_{2},\ldots )=(0,x_{2},\ldots )}$.