Editing Partial isometry

In mathematical [[functional analysis]], a '''partial isometry''' is a [[linear map]] between [[Hilbert space]]s such that it is an [[isometry]] on the [[orthogonal complement]] of its [[kernel (algebra)|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 definition ==

The concept of partial isometry can be defined in other equivalent ways.  If ''U'' is an isometric map defined on a closed subset  ''H''<sub>1</sub> 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''<sub>1</sub>. 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.

== Characterization in finite dimensions ==
In [[dimension (vector space)|finite-dimensional]] [[vector space]]s, a [[matrix (mathematics)|matrix]] <math>A</math> is a partial isometry [[if and only if]] <math> A^* A</math> is the projection onto its support. Contrast this with the more demanding definition of [[isometry]]: a matrix <math>V</math> is an isometry if and only if <math>V^* V=I</math>. In other words, an isometry is an [[injective]] partial isometry.

Any finite-dimensional partial isometry can be represented, in some choice of [[basis (linear algebra)|basis]], as a matrix of the form <math>A=\begin{pmatrix}V & 0\end{pmatrix}</math>, that is, as a matrix whose first <math>\operatorname{rank}(A)</math> columns form an isometry, while all the other columns are identically 0. 

Note that for any isometry <math>V</math>, the Hermitian conjugate <math>V^*</math> is a partial isometry, although not every partial isometry has this form, as shown explicitly in the given examples.

== Operator Algebras ==

For [[operator algebra]]s, one introduces the initial and final subspaces:
:<math>\mathcal{I}W:=\mathcal{R}W^*W,\,\mathcal{F}W:=\mathcal{R}WW^*</math>

== C*-Algebras ==

For [[C*-algebra]]s, one has the chain of equivalences due to the C*-property:
:<math>(W^*W)^2 = W^*W \iff WW^*W = W \iff W^*WW^* = W^* \iff (WW^*)^2 = WW^*</math>
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]]:
:<math>P = W^*W,\,Q = WW^*</math>
It plays an important role in [[K-theory]] for C*-algebras and in the [[Francis Joseph Murray|Murray]]-[[John von Neumann|von Neumann]] theory of projections in a [[von Neumann algebra]].

== Special Classes ==

=== Projections ===

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

:<math>P:\mathcal{H}\rightarrow\mathcal{H}:\quad\mathcal{I}P=\mathcal{F}P</math>

=== Embeddings ===

Any isometric embedding is one with full initial subspace:

:<math>J:\mathcal{H}\hookrightarrow\mathcal{K}:\quad\mathcal{I}J=\mathcal{H}</math>

=== Unitaries ===

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

:<math>U:\mathcal{H}\leftrightarrow\mathcal{K}:\quad\mathcal{I}U=\mathcal{H},\,\mathcal{F}U=\mathcal{K}</math>

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

== Examples ==

=== Nilpotents ===

On the two-dimensional [[complex number|complex]] Hilbert space the matrix

:<math>\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}</math>

is a partial isometry with initial subspace

: <math>\{0\} \oplus \mathbb{C}</math>

and final subspace

: <math>\mathbb{C} \oplus \{0\}.</math>

=== Generic finite-dimensional examples ===
Other possible examples in finite dimensions are
<math display="block">A\equiv \begin{pmatrix}1&0&0\\0&\frac1{\sqrt2}&\frac1{\sqrt2}\\0&0&0\end{pmatrix}.</math>
This is clearly not an isometry, because the columns are not [[orthonormal]]. However, its support is the span of <math>\mathbf e_1\equiv (1,0,0)</math> and <math>\frac{1}{\sqrt2}(\mathbf e_2+\mathbf e_3)\equiv (0,1/\sqrt2,1/\sqrt2)</math>, and restricting the action of <math>A</math> on this space, it becomes an isometry (and in particular, a unitary). One can similarly verify that <math>A^* A = \Pi_{\operatorname{supp}(A)}</math>, that is, that <math>A^* A</math> is the projection onto its support.

Partial isometries do not necessarily correspond to squared matrices. Consider for example,
<math display="block">A\equiv \begin{pmatrix}1&0&0\\0&\frac12&\frac12\\ 0 & 0 & 0 \\ 0& \frac12 & \frac12\end{pmatrix}.</math>This matrix has support the span of <math>\mathbf e_1\equiv (1,0,0)</math> and <math>\mathbf e_2+\mathbf e_3\equiv (0,1,1)</math>, and acts as an isometry (and in particular, as the identity) on this space.

Yet another example, in which this time <math>A</math> acts like a non-trivial isometry on its support, is<math display="block">A = \begin{pmatrix}0 & \frac1{\sqrt2} & \frac1{\sqrt2} \\ 1&0&0\\0&0&0\end{pmatrix}.</math>One can readily verify that <math>A\mathbf e_1=\mathbf e_2</math>, and <math>A \left(\frac{\mathbf e_2 + \mathbf e_3}{\sqrt2}\right) = \mathbf e_1</math>, showing the isometric behavior of <math>A</math> between its support <math>\operatorname{span}(\{\mathbf e_1, \mathbf e_2+\mathbf e_3\})</math> and its range <math>\operatorname{span}(\{\mathbf e_1,\mathbf e_2\})</math>.

=== Leftshift and Rightshift ===

On the square summable sequences, the operators

:<math>R: \ell^2(\mathbb{N}) \to \ell^2(\mathbb{N}): (x_1,x_2,\ldots) \mapsto (0,x_1,x_2,\ldots)</math>

:<math>L: \ell^2(\mathbb{N}) \to \ell^2(\mathbb{N}): (x_1,x_2,\ldots) \mapsto (x_2,x_3,\ldots)</math>

which are related by

:<math>R^* = L</math>

are partial isometries with initial subspace

:<math>LR(x_1,x_2,\ldots)=(x_1,x_2,\ldots)</math>

and final subspace:

:<math>RL(x_1,x_2,\ldots)=(0,x_2,\ldots)</math>.

== References ==
*John B. Conway (1999). "A course in operator theory", AMS Bookstore, {{ISBN|0-8218-2065-6}}
*{{cite journal|last1=Carey|first1=R. W. | last2= Pincus | first2= J. D. |title= An Invariant for Certain Operator Algebras| journal=[[Proceedings of the National Academy of Sciences]] | volume=71|number=5|pages=1952–1956|date=May 1974|doi=10.1073/pnas.71.5.1952 |pmid=16592156 |pmc=388361 |bibcode=1974PNAS...71.1952C |doi-access=free }}
*Alan L. T. Paterson (1999). "[https://books.google.com/books?id=aUPhBwAAQBAJ&q=%22Partial+isometry%22 Groupoids, inverse semigroups, and their operator algebras]", Springer, {{ISBN|0-8176-4051-7}}
*Mark V. Lawson (1998). "[https://books.google.com/books?id=2805q4tFiCkC&q=%22partial+isometry%22 Inverse semigroups: the theory of partial symmetries]". [[World Scientific]] {{ISBN|981-02-3316-7}}
*{{cite arXiv |eprint=1903.11648|author1=Stephan Ramon Garcia |author2=Matthew Okubo Patterson |last3=Ross |first3=William T. |title=Partially isometric matrices: A brief and selective survey |year=2019 |class=math.FA }}

==External links ==
*[https://web.archive.org/web/20141027161712/http://www.math.tamu.edu/~pskoufra/OANotes-PartialIsometries.pdf Important properties and proofs]
*[https://math.stackexchange.com/q/614331 Alternative proofs]

{{DEFAULTSORT:Partial Isometry}}
[[Category:Operator theory]]
[[Category:C*-algebras]]
[[Category:Semigroup theory]]