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| In [[mathematics]], a '''Fredholm kernel''' is a certain type of a [[kernel (integral operator)|kernel]] on a [[Banach space]], associated with [[nuclear operator]]s on the Banach space. They are an abstraction of the idea of the [[Fredholm integral equation]] and the [[Fredholm operator]], and are one of the objects of study in [[Fredholm theory]]. Fredholm kernels are named in honour of [[Erik Ivar Fredholm]]. Much of the abstract theory of Fredholm kernels was developed by [[Alexander Grothendieck]] and published in 1955.
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| ==Definition==
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| Let ''B'' be an arbitrary [[Banach space]], and let ''B''<sup>*</sup> be its dual, that is, the space of [[bounded linear functional]]s on ''B''. The [[tensor product]] <math>B^*\otimes B</math> has a [[complete space|completion]] under the norm
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| :<math>\Vert X \Vert_\pi =
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| \inf \sum_{\{i\}} \Vert e^*_i\Vert \Vert e_i \Vert</math>
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| where the [[infimum]] is taken over all finite representations
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| :<math>X=\sum_{\{i\}} e^*_i e_i \in B^*\otimes B</math>
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| The completion, under this norm, is often denoted as
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| :<math>B^* \widehat{\,\otimes\,}_\pi B </math>
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| and is called the '''[[topological tensor product|projective topological tensor product]]'''. The elements of this space are called '''Fredholm kernels'''. | |
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| ==Properties==
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| Every Fredholm kernel has a representation in the form
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| :<math>X=\sum_{\{i\}} \lambda_i e^*_i \otimes e_i</math>
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| with <math>e_i \in B</math> and <math>e^*_i \in B^*</math> such that <math>\Vert e_i \Vert = \Vert e^*_i \Vert = 1</math> and | |
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| :<math>\sum_{\{i\}} \vert \lambda_i \vert < \infty. \, </math>
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| Associated with each such kernel is a linear operator
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| :<math>\mathcal {L}_X : B \to B</math>
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| which has the canonical representation
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| :<math>\mathcal{L}_X f =\sum_{\{i\}} \lambda_i e^*_i(f) \otimes e_i. \, </math>
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| Associated with every Fredholm kernel is a trace, defined as
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| :<math>\mbox{tr} X = \sum_{\{i\}} \lambda_i e^*_i(e_i). \,</math>
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| ==''p''-summable kernels==
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| A Fredholm kernel is said to be ''' ''p''-summable''' if
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| :<math>\sum_{\{i\}} \vert \lambda_i \vert^p < \infty</math>
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| A Fredholm kernel is said to be of '''order q''' if ''q'' is the [[infimum]] of all <math>0<p\le 1</math> for all ''p'' for which it is ''p''-summable.
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| ==Nuclear operators on Banach spaces==
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| An operator <math>\mathcal{L}:B \to B</math> is said to be a [[nuclear operator]] if there exists an
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| <math>X\in B^* \widehat{\,\otimes\,}_\pi B</math> such that <math>\mathcal{L} = \mathcal{L}_X</math>. Such an operator is said to be ''p''-summable and of order ''q'' if ''X'' is. In general, there may be more than one ''X'' associated with such a nuclear operator, and so the trace is not uniquely defined. However, if the order <math>q \le 2/3</math>, then there is a unique trace, as given by a theorem of Grothendieck.
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| ==Grothendieck's theorem==
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| If <math>\mathcal{L}:B\to B</math> is an operator of order <math>q \le 2/3</math> then a trace may be defined, with
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| :<math>\mbox{Tr} \mathcal {L} = \sum_{\{i\}} \rho_i</math>
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| where <math>\rho_i</math> are the [[eigenvalue]]s of <math>\mathcal{L}</math>. Furthermore, the [[Fredholm determinant]]
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| :<math>\det \left( 1-z\mathcal{L}\right)=
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| \prod_i \left(1-\rho_i z \right)</math>
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| is an [[entire function]] of ''z''. The formula
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| :<math>\det \left( 1-z\mathcal{L}\right)=
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| \exp \mbox{Tr} \log\left( 1-z\mathcal{L}\right) </math>
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| holds as well. Finally, if <math>\mathcal{L}</math> is parameterized by some [[complex number|complex]]-valued parameter ''w'', that is, <math>\mathcal{L}=\mathcal{L}_w</math>, and the parameterization is [[holomorphic]] on some domain, then
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| :<math>\det \left( 1-z\mathcal{L}_w\right)</math>
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| is holomorphic on the same domain.
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| ==Examples==
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| An important example is the Banach space of holomorphic functions over a domain <math>D\subset \mathbb{C}^k</math>. In this space, every nuclear operator is of order zero, and is thus of [[trace-class]].
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| ==Nuclear spaces==
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| The idea of a nuclear operator can be adapted to [[Fréchet space]]s. A '''[[nuclear space]]''' is a Fréchet space where every bounded map of the space to an arbitrary Banach space is nuclear.
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| ==References==
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| *{{cite journal |author=Grothendieck A |title=Produits tensoriels topologiques et espaces nucléaires |journal=Mem. Am. Math.Soc. |volume=16 |year=1955}}
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| *{{cite journal |author=Grothendieck A |title=La théorie de Fredholm |journal=Bull. Soc. Math. France |volume=84 |pages=319–84 |year=1956}}
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| * {{springer|id=f/f041440|title=Fredholm kernel|author=B.V. Khvedelidze, G.L. Litvinov}}
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| *{{cite journal |author=Fréchet M |title=On the Behavior of the nth Iterate of a Fredholm Kernel as n Becomes Infinite |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=18 |issue=11 |pages=671–3 |date=November 1932 |pmid=16577494 |pmc=1076308 |doi=10.1073/pnas.18.11.671 }}
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| [[Category:Fredholm theory]]
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| [[Category:Banach spaces]]
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| [[Category:Topology of function spaces]]
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Nice to meet you, my name is Refugia. Hiring is my occupation. The preferred pastime for my children and me is to play baseball but I haven't made a dime with it. Her spouse and her live in Puerto Rico but she will have to move 1 day or another.
Here is my weblog at home std testing (linked internet site)