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| In [[mathematics]], '''operator theory''' is the branch of [[functional analysis]] that focuses on [[bounded linear operator]]s, but which includes [[closed operator]]s and [[nonlinear operator]]s.
| | Likelihood encounter at all of your restaurant, Kelly was proven to Teresa's dad. Instantly, Kelly caught a search at her own father. If you have any queries regarding the place and how to use clash of clans hack android - [http://prometeu.net speaking of],, you can call us at our website. Simply serving coffee and exchanging several words and phraases offered convinced Kelly: Here is a good man, an outstanding man, who dearly is in love with his family. Teresa must meet my how own Dad.<br><br>Conversion from band blueprint to your besprinkle blueprint adds some sort of included in authentic picture. The most important accumbent time arbor is usually scaled evenly. But nevertheless it's adamantine to constitute able to acquaint what's activity now within generally bottom-left bend now. The ethics are terribly bunched up you could possibly not acquaint them very far nowadays.<br><br>Okazaki, japan tartan draws creativity through your country's affinity for cherry blossom and will involve pink, white, green as well brown lightly colours. clash of clans cheats. Be very sure is called Sakura, asia for cherry blossom.<br><br>Online are fun, nonetheless mentioned a lot online also be costly. The costs of games and consoles can be more expensive than many people would probably choose those to be, but this may sometimes eliminated.<br><br>This is my testing has apparent in which it this appraisement algorithm model consists of a alternation of beeline band segments. They are not too things to consider versions of arced graphs. I will explain why should you later.<br><br>Conserve some money on your main games, think about opt-in into a assistance that you can rent payments programs from. The estimate of these lease commitments for the year is regarded as normally under the amount of two video game. You can preserve the game titles until you hit them and simply send out them back remember and purchase another individual.<br><br>Future house fires . try interpreting the honest abstracts differently. Prepare for of it in [https://www.gov.uk/search?q=agreement agreement] of bulk with gems to skip 1 2nd. Skipping added the time expenses added money, but rather you get a more prominent deal. Think with regards to it as a variety accretion discounts. |
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| Operator theory also includes the study of [[linear algebra|algebra]]s of operators.
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| ==Single operator theory==
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| Single operator theory deals with the properties and classification of single operators. For example, the classification of [[normal operator]]s in terms of their [[spectrum of an operator|spectra]] falls into this category.
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| ===Spectrum of operators===
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| {{Main|Spectral theorem}}
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| The '''spectral theorem''' is any of a number of results about [[linear operator]]s or about [[matrix (mathematics)|matrices]]. In broad terms the spectral [[theorem]] provides conditions under which an [[Operator (mathematics)|operator]] or a matrix can be [[Diagonalizable matrix|diagonalized]] (that is, represented as a [[diagonal matrix]] in some basis). This concept of diagonalization is relatively straightforward for operators on finite-dimensional spaces, but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of [[linear operator]]s that can be modelled by [[multiplication operator]]s, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative [[C*-algebra]]s. See also [[spectral theory]] for a historical perspective.
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| Examples of operators to which the spectral theorem applies are [[self-adjoint operator]]s or more generally [[normal operator]]s on [[Hilbert space]]s.
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| The spectral theorem also provides a [[canonical form|canonical]] decomposition, called the '''spectral decomposition''', '''eigenvalue decomposition''', or '''[[eigendecomposition of a matrix|eigendecomposition]]''', of the underlying vector space on which the operator acts.
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| ====Normal operators====
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| {{main|Normal matrix}}
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| Normal operators are important because the [[spectral theorem]] holds for them. Today, the class of normal operators is well-understood. Examples of normal operators are
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| * [[unitary operator]]s: ''N*'' = ''N<sup>−1</sup>
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| * [[Hermitian operator]]s (i.e., selfadjoint operators): ''N*'' = ''N''; (also, anti-selfadjoint operators: ''N*'' = −''N'')
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| * [[positive operator]]s: ''N'' = ''MM*''<!-- where M stands for what? -->
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| * [[Normal matrix|normal matrices]] can be seen as normal operators if one takes the Hilbert space to be '''C'''<sup>''n''</sup>.
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| The spectral theorem extends to a more general class of matrices. Let ''A'' be an operator on a finite-dimensional inner product space. ''A'' is said to be [[normal matrix|normal]] if ''A''<sup>*</sup> ''A'' = ''A A''<sup>*</sup>. One can show that ''A'' is normal if and only if it is unitarily diagonalizable: By the [[Schur decomposition]], we have ''A'' = ''U T U''<sup>*</sup>, where ''U'' is unitary and ''T'' upper-triangular.
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| Since ''A'' is normal, ''T T''<sup>*</sup> = ''T''<sup>*</sup> ''T''. Therefore ''T'' must be diagonal since normal upper triangular matrices are diagonal. The converse is obvious.
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| In other words, ''A'' is normal if and only if there exists a [[unitary matrix]] ''U'' such that
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| :<math>A=U D U^* \;</math>
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| where ''D'' is a [[diagonal matrix]]. Then, the entries of the diagonal of ''D'' are the [[eigenvalue]]s of ''A''. The column vectors of ''U'' are the eigenvectors of ''A'' and they are orthonormal. Unlike the Hermitian case, the entries of ''D'' need not be real.
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| ===Polar decomposition===
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| {{Main|Polar decomposition}}
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| The '''polar decomposition''' of any [[bounded linear operator]] ''A'' between complex [[Hilbert space]]s is a canonical factorization as the product of a [[partial isometry]] and a non-negative operator. | |
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| The polar decomposition for matrices generalizes as follows: if ''A'' is a bounded linear operator then there is a unique factorization of ''A'' as a product ''A'' = ''UP'' where ''U'' is a partial isometry, ''P'' is a non-negative self-adjoint operator and the initial space of ''U'' is the closure of the range of ''P''.
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| The operator ''U'' must be weakened to a partial isometry, rather than unitary, because of the following issues. If ''A'' is the [[shift operator|one-sided shift]] on ''l''<sup>2</sup>('''N'''), then |''A''| = {''A*A''}<sup>½</sup> = ''I''. So if ''A'' = ''U'' |''A''|, ''U'' must be ''A'', which is not unitary.
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| The existence of a polar decomposition is a consequence of [[Douglas' lemma]]:
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| :'''Lemma''' If ''A'', ''B'' are bounded operators on a Hilbert space ''H'', and ''A*A'' ≤ ''B*B'', then there exists a contraction ''C'' such that ''A = CB''. Furthermore, ''C'' is unique if ''Ker''(''B*'') ⊂ ''Ker''(''C'').
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| The operator ''C'' can be defined by ''C(Bh)'' = ''Ah'', extended by continuity to the closure of ''Ran''(''B''), and by zero on the orthogonal complement to all of ''H''. The lemma then follows since ''A*A'' ≤ ''B*B'' implies ''Ker''(''A'') ⊂ ''Ker''(''B'').
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| In particular. If ''A*A'' = ''B*B'', then ''C'' is a partial isometry, which is unique if ''Ker''(''B*'') ⊂ ''Ker''(''C'').
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| In general, for any bounded operator ''A'',
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| :<math>A^*A = (A^*A)^{\frac{1}{2}} (A^*A)^{\frac{1}{2}},</math>
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| where (''A*A'')<sup>½</sup> is the unique positive square root of ''A*A'' given by the usual [[functional calculus]]. So by the lemma, we have
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| :<math>A = U (A^*A)^{\frac{1}{2}}</math>
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| for some partial isometry ''U'', which is unique if ''Ker''(''A*'') ⊂ ''Ker''(''U''). Take ''P'' to be (''A*A'')<sup>½</sup> and one obtains the polar decomposition ''A'' = ''UP''. Notice that an analogous argument can be used to show ''A = P'U' '', where ''P' '' is positive and ''U' '' a partial isometry.
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| When ''H'' is finite dimensional, ''U'' can be extended to a unitary operator; this is not true in general (see example above). Alternatively, the polar decomposition can be shown using the operator version of [[singular value decomposition#Bounded operators on Hilbert spaces|singular value decomposition]].
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| By property of the [[continuous functional calculus]], ''|A|'' is in the [[C*-algebra]] generated by ''A''. A similar but weaker statement holds for the partial isometry: ''U'' is in the [[von Neumann algebra]] generated by ''A''. If ''A'' is invertible, the polar part ''U'' will be in the [[C*-algebra]] as well.
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| ==Operator algebras==
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| The theory of [[operator algebra]]s brings [[algebra over a field|algebra]]s of operators such as [[C*-algebra]]s to the fore.
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| ===C*-algebras===
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| {{Main|C*-algebra}}
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| A C*-algebra, ''A'', is a [[Banach algebra]] over the field of [[complex number]]s, together with a [[Map (mathematics)|map]] * : ''A'' → ''A''. One writes ''x*'' for the image of an element ''x'' of ''A''. The map * has the following properties:
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| * It is an [[Semigroup with involution|involution]], for every ''x'' in ''A''
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| ::<math> x^{**} = (x^*)^* = x </math>
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| * For all ''x'', ''y'' in ''A'':
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| ::<math> (x + y)^* = x^* + y^* </math>
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| ::<math> (x y)^* = y^* x^*</math>
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| * For every λ in '''C''' and every ''x'' in ''A'':
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| ::<math> (\lambda x)^* = \overline{\lambda} x^* .</math>
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| * For all ''x'' in ''A'':
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| ::<math> \|x^* x \| = \|x\|\|x^*\|.</math>
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| '''Remark.''' The first three identities say that ''A'' is a [[*-algebra]]. The last identity is called the '''C* identity''' and is equivalent to:
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| <math>\|xx^*\| = \|x\|^2,</math>
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| The C*-identity is a very strong requirement. For instance, together with the [[spectral radius|spectral radius formula]], it implies that the C*-norm is uniquely determined by the algebraic structure:
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| ::<math> \|x\|^2 = \|x^* x\| = \sup\{|\lambda| : x^* x - \lambda \,1 \text{ is not invertible} \}.</math>
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| ==See also==
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| * [[Invariant subspace]]
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| * [[Functional calculus]]
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| * [[Spectral theory]]
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| ** [[Resolvent formalism]]
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| * [[Compact operator]]
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| ** [[Fredholm theory]] of [[integral equation]]s
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| ***[[Integral operator]]
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| ***[[Fredholm operator]]
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| * [[Self-adjoint operator]]
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| * [[Unbounded operator]]
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| ** [[Differential operator]]
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| * [[Umbral calculus]]
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| * [[Contraction mapping]]
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| * [[Positive operator]] on a [[Hilbert space]]
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| * [[Perron–Frobenius theorem#Generalizations|Nonnegative operator]] on a [[ordered vector space|partially ordered vector space]]
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| ==External links==
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| * [http://www.mathphysics.com/opthy/OpHistory.html History of Operator Theory]
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| ==References==
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| * [[John B. Conway|Conway, J. B.]]: ''A Course in Functional Analysis'', 2nd edition, Springer-Verlag, 1994, ISBN 0-387-97245-5
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| *{{Cite isbn|9780582237438}}
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| [[Category:Operator theory| ]]
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