- Fredholm kernel
In
mathematics , a Fredholm kernel is a certain type of a kernel on aBanach space , associated withnuclear operator s on the Banach space. They are an abstraction of the idea of theFredholm integral equation and theFredholm operator , and are one of the objects of study inFredholm theory . Fredholm kernels are named in honour ofErik Ivar Fredholm . Much of the abstract theory of Fredholm kernels was developed byAlexander Grothendieck and published in1955 .Definition
Let "B" be an arbitrary
Banach space , and let "B"* be its dual, that is, the space ofbounded linear functional s on "B". Thetensor product B^*otimes B has a completion under the norm:Vert X Vert_pi = inf sum_{{i Vert e^*_iVert Vert e_i Vert
where the
infimum is taken over all finite representations:X=sum_{{i e^*_i e_i in B^*otimes B
The completion, under this norm, is often denoted as
:B^* widehat{,otimes,}_pi B
and is called the projective topological tensor product. The elements of this space are called Fredholm kernels.
Properties
Every Fredholm kernel has a representation in the form
:X=sum_{{i lambda_i e^*_i otimes e_i
with e_i in B and e^*_i in B^* such that Vert e_i Vert = Vert e^*_i Vert = 1 and
:sum_{{i vert lambda_i vert < infty
Associated with each such kernel is a linear operator
:mathcal {L}_X : B o B
which has the canonical representation
:mathcal{L}_X f =sum_{{i lambda_i e^*_i(f) otimes e_i
Associated with every Fredholm kernel is a trace, defined as
:mbox{tr} X = sum_{{i lambda_i e^*_i(e_i)
"p"-summable kernels
A Fredholm kernel is said to be "p"-summable if
:sum_{{i vert lambda_i vert^p < infty
A Fredholm kernel is said to be of order q if "q" is the
infimum of all 0for all "p" for which it is "p"-summable. Nuclear operators on Banach spaces
An operator mathcal{L}:B o B is said to be a
nuclear operator if there exists an Xin B^* widehat{,otimes,}_pi B such that mathcal{L} = mathcal{L}_X. 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 q le 2/3, then there is a unique trace, as given by a theorem of Grothendieck.Grothendieck's theorem
If mathcal{L}:B o B is an operator of order q le 2/3 then a trace may be defined, with
:mbox{Tr} mathcal {L} = sum_{{i ho_i
where ho_i are the
eigenvalue s of mathcal{L}. Furthermore, theFredholm determinant :det left( 1-zmathcal{L} ight)=prod_i left(1- ho_i z ight)
is an
entire function of "z". The formula:det left( 1-zmathcal{L} ight)= exp mbox{Tr} logleft( 1-zmathcal{L} ight)
holds as well. Finally, if mathcal{L} is parameterized by some complex-valued parameter "w", that is, mathcal{L}=mathcal{L}_w, and the parameterization is
holomorphic on some domain, then:det left( 1-zmathcal{L}_w ight)
is holomorphic on the same domain.
Examples
An important example is the Banach space of holomorphic functions over a domain Dsubset mathbb{C}^k. In this space, every nuclear operator is of order zero, and is thus of
trace-class .Nuclear spaces
The idea of a nuclear operator can be adapted to
Fréchet space s. Anuclear space is a Fréchet space where every bounded map of the space to an arbitrary Banach space is nuclear.References
* A. Grothendieck, Produits tensoriels topologiques et espace nucleaires, (1955) "Mem. Am. Math.Soc." 16.
* A. Grothendieck, La theorie de Fredholm, (1956) "Bull. Soc. Math. France", 84:319-384.
*
* Maurice Fréchet, [http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1076308 On the Behavior of the nth Iterate of a Fredholm Kernel as n Becomes Infinite] (1932) "Proc Natl Acad Sci U S A". 18(11): 671–673.
Wikimedia Foundation. 2010.