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	~~Hi there~~, ~~I am Alyson Pomerleau~~ and ~~I believe it seems quite good when you say~~ it. ~~I am~~ an ~~invoicing officer and I~~'ll be ~~promoted quickly~~. ~~Mississippi~~ is the ~~only location I~~'~~ve been residing~~ in ~~but I will have to transfer~~ in a ~~year~~ or ~~two. Playing badminton~~ is a ~~factor~~ that he is ~~completely addicted~~ to.<br><br>~~Feel free to surf to my website~~ [http://~~Trackchairclips~~.com/~~profile.php?u=EmKinard certified psychics~~]		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 (algebra)\|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]].

			==Properties ==
			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 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

			#''W''*''W'' is called the '''initial projection''' of ''W''.
			#''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

			:<math>W=WW^*W.</math>

			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 [[Francis Joseph Murray (mathematician)\|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.

			== Examples ==

			For example, In the two-dimensional complex Hilbert space '''C'''<sup>2</sup> the matrix

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

			is a partial isometry with initial subspace

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

			and final subspace

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

			== References ==
			*John B. Conway (1999). "A course in operator theory", AMS Bookstore, ISBN 0-8218-2065-6
			*Alan L. T. Paterson (1999). "Groupoids, inverse semigroups, and their operator algebras", Springer, ISBN 0-8176-4051-7
			*Mark V. Lawson (1998). "Inverse semigroups: the theory of partial symmetries". [[World Scientific]] ISBN 981-02-3316-7

			==External links ==
			*[http://www.math.ucla.edu/~pskoufra/OANotes-PartialIsometries.pdf Important properties and proofs]
			*[http://math.stackexchange.com/questions/614331/equivalent-algebraic-definition-of-a-partial-isometry-in-a-c-algebra Alternative proofs]
			{{DEFAULTSORT:Partial Isometry}}
			[[Category:Operator theory]]
			[[Category:C*-algebras]]
			[[Category:Semigroup theory]]

178.176.33.30 at 11:08, 17 July 2012

2012-07-17T11:08:46Z

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Hi there, I am Alyson Pomerleau and I believe it seems quite good when you say it. I am an invoicing officer and I'll be promoted quickly. Mississippi is the only location I've been residing in but I will have to transfer in a year or two. Playing badminton is a factor that he is completely addicted to.<br><br>Feel free to surf to my website [http://Trackchairclips.com/profile.php?u=EmKinard certified psychics]

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178.176.33.30 at 11:08, 17 July 2012