Partial isometry

 ( Operator Theory ) 

In functional analysis, a partial isometry is a linear map between 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.

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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 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.

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Characterization in finite dimensions In finite-dimensional , a matrix $A$ is a partial isometry if and only if $A^*\; A$ is the projection onto its support. Contrast this with the more demanding definition of isometry: a matrix $V$ is an isometry if and only if $V^*\; V=I$. In other words, an isometry is an injective partial isometry.

Any finite-dimensional partial isometry can be represented, in some choice of basis, as a matrix of the form , that is, as a matrix whose first $\backslash operatorname\{rank\}(A)$ columns form an isometry, while all the other columns are identically 0.

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

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Operator Algebras For , one introduces the initial and final subspaces:

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

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

$(W^*W)^2\; =\; W^*W\; \backslash iff\; WW^*W\; =\; W\; \backslash iff\; W^*WW^*\; =\; W^*\; \backslash iff\; (WW^*)^2\; =\; WW^*$

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:

$P\; =\; W^*W,\backslash ,Q\; =\; WW^*$

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.

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Special Classes

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

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

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Embeddings Any isometric embedding is one with full initial subspace:

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

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

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

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

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Examples

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Nilpotents On the two-dimensional complex number Hilbert space the matrix

is a partial isometry with initial subspace

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

and final subspace

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

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Generic finite-dimensional examples Other possible examples in finite dimensions are This is clearly not an isometry, because the columns are not orthonormal. However, its support is the span of $\backslash mathbf\; e\_1\backslash equiv\; (1,0,0)$ and $\backslash frac\{1\}\{\backslash sqrt2\}(\backslash mathbf\; e\_2+\backslash mathbf\; e\_3)\backslash equiv\; (0,1/\backslash sqrt2,1/\backslash sqrt2)$, and restricting the action of $A$ on this space, it becomes an isometry (and in particular, a unitary). One can similarly verify that $A^*\; A\; =\; \backslash Pi\_\{\backslash operatorname\{supp\}(A)\}$, that is, that $A^*\; A$ is the projection onto its support.

Partial isometries do not necessarily correspond to squared matrices. Consider for example, This matrix has support the span of $\backslash mathbf\; e\_1\backslash equiv\; (1,0,0)$ and $\backslash mathbf\; e\_2+\backslash mathbf\; e\_3\backslash equiv\; (0,1,1)$, and acts as an isometry (and in particular, as the identity) on this space.

Yet another example, in which this time $A$ acts like a non-trivial isometry on its support, isOne can readily verify that $A\backslash mathbf\; e\_1=\backslash mathbf\; e\_2$, and $A\; \backslash left(\backslash frac\{\backslash mathbf\; e\_2\; +\; \backslash mathbf\; e\_3\}\{\backslash sqrt2\}\backslash right)\; =\; \backslash mathbf\; e\_1$, showing the isometric behavior of $A$ between its support $\backslash operatorname\{span\}(\backslash \{\backslash mathbf\; e\_1,\; \backslash mathbf\; e\_2+\backslash mathbf\; e\_3\backslash \})$ and its range $\backslash operatorname\{span\}(\backslash \{\backslash mathbf\; e\_1,\backslash mathbf\; e\_2\backslash \})$.

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Leftshift and Rightshift On the square summable sequences, the operators

$R:\; \backslash ell^2(\backslash mathbb\{N\})\; \backslash to\; \backslash ell^2(\backslash mathbb\{N\}):\; (x\_1,x\_2,\backslash ldots)\; \backslash mapsto\; (0,x\_1,x\_2,\backslash ldots)$

$L:\; \backslash ell^2(\backslash mathbb\{N\})\; \backslash to\; \backslash ell^2(\backslash mathbb\{N\}):\; (x\_1,x\_2,\backslash ldots)\; \backslash mapsto\; (x\_2,x\_3,\backslash ldots)$

which are related by

$R^*\; =\; L$

are partial isometries with initial subspace

$LR(x\_1,x\_2,\backslash ldots)=(x\_1,x\_2,\backslash ldots)$

and final subspace:

$RL(x\_1,x\_2,\backslash ldots)=(0,x\_2,\backslash ldots)$.

• John B. Conway (1999). "A course in operator theory", AMS Bookstore,
• Alan L. T. Paterson (1999). " Groupoids, inverse semigroups, and their operator algebras", Springer,
• Mark V. Lawson (1998). " Inverse semigroups: the theory of partial symmetries". World Scientific

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External links

• Important properties and proofs
• Alternative proofs

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