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| In [[mathematics]], a '''nuclear operator''' is a [[compact operator]] for which a [[trace (linear algebra)|trace]] may be defined, such that the trace is finite and independent of the choice of basis (at least on well behaved spaces; there are some spaces on which nuclear operators do not have a trace).
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| Nuclear operators are essentially the same as '''[[trace class|trace class operators]]''', though most authors reserve the term "trace class operator" for the special case of
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| nuclear operators on [[Hilbert space]]s. The general definition for [[Banach space]]s was given by [[Grothendieck]]. This article concentrates on the general case of nuclear operators on Banach spaces; for the important special case of nuclear (=trace class) operators on Hilbert space see the article on [[trace class|trace class operator]]s.
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| ==Compact operator==
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| An operator <math>\mathcal{L}</math> on a [[Hilbert space]] <math>\mathcal{H}</math>
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| :<math>\mathcal{L}:\mathcal{H} \to \mathcal{H}</math>
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| is said to be a [[compact operator]] if it can be written in the form{{Citation needed|date=September 2011}}
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| :<math>\mathcal{L} = \sum_{n=1}^N \rho_n \langle f_n, \cdot \rangle g_n</math>
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| where <math>1 \le N \le \infty</math> and <math>f_1,\ldots,f_N</math> and <math>g_1,\ldots,g_N</math> are (not necessarily complete) orthonormal sets. Here, <math>\rho_1,\ldots,\rho_N</math> are a set of real numbers, the [[singular value]]s of the operator, obeying <math>\rho_n \to 0</math> if <math>N = \infty</math>. The bracket <math>\langle\cdot,\cdot\rangle</math> is the scalar product on the Hilbert space; the sum on the right hand side must converge in norm.
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| ==Nuclear operator==
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| An operator that is compact as defined above is said to be '''nuclear''' or '''trace-class''' if
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| :<math>\sum_{n=1}^\infty |\rho_n| < \infty</math>
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| ==Properties==
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| A nuclear operator on a Hilbert space has the important property that its [[trace class|trace]] may be defined so that it is finite and is independent of the basis. Given any orthonormal basis <math>\{\psi_n\}</math> for the Hilbert space, one may define the trace as
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| :<math>\mbox{Tr} \mathcal {L} = \sum_n \langle \psi_n , \mathcal{L} \psi_n \rangle</math>
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| since the sum converges absolutely and is independent of the basis{{Citation needed|date=September 2011}}. Furthermore, this trace is identical to the sum over the eigenvalues of <math>\mathcal{L}</math> (counted with multiplicity).
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| ==On Banach spaces==
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| :''See main article [[Fredholm kernel]].''
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| The definition of trace-class operator was extended to [[Banach space]]s by [[Alexander Grothendieck]] in 1955.
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| Let ''A'' and ''B'' be Banach spaces, and ''A''' be the [[continuous dual space|dual]] of ''A'', that is, the set of all [[continuous (topology)|continuous]] or (equivalently) [[bounded linear functional]]s on ''A'' with the usual norm. Then an operator
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| :<math>\mathcal{L}:A \to B</math>
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| is said to be '''nuclear of order ''q'' ''' if there exist sequences of vectors <math>\{g_n\} \in B</math> with <math>\Vert g_n \Vert \le 1</math>, functionals <math>\{f^*_n\} \in A'</math> with <math>\Vert f^*_n \Vert \le 1</math> and [[complex number]]s <math>\{\rho_n\}</math> with
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| :<math>\inf \left\{ p\ge 1 : \sum_n |\rho_n|^p < \infty \right\} = q,</math>
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| such that the operator may be written as
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| :<math>\mathcal{L} = \sum_n \rho_n f^*_n(\cdot) g_n</math>
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| with the sum converging in the operator norm.
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| With additional steps, a trace may be defined for such operators when ''A'' = ''B''.
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| Operators that are nuclear of order 1 are called '''nuclear operators''': these are the ones for which the series ∑''ρ<sub>n</sub>'' is absolutely convergent. Nuclear operators of order 2 are called [[Hilbert-Schmidt operator]]s.
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| More generally, an operator from a [[locally convex topological vector space]] ''A'' to a Banach space ''B'' is called '''nuclear''' if it satisfies the condition above with all ''f<sub>n</sub><sup>*</sup>'' bounded by 1 on some fixed neighborhood of 0 and all ''g<sub>n</sub>'' bounded by 1 on some fixed neighborhood of 0.
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| ==References==
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| * A. Grothendieck (1955), Produits tensoriels topologiques et espace nucléaires,''Mem. Am. Math.Soc.'' '''16'''. {{MR|0075539}}
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| * A. Grothendieck (1956), La theorie de Fredholm, ''Bull. Soc. Math. France'', '''84''':319-384. {{MR|0088665}}
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| * A. Hinrichs and A. Pietsch (2010), ''p''-nuclear operators in the sense of Grothendieck, ''Mathematische Nachrichen'' '''283''': 232-261. {{doi|10.1002/mana.200910128}} {{MR|2604120}}
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| * {{springer|id=Nuclear_operator|author=G. L. Litvinov|title=Nuclear operator }}
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| {{Functional Analysis}}
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| [[Category:Operator theory]]
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