Fredholm kernel

In mathematics, a Fredholm kernel is a certain type of a kernel on a Banach space, associated with nuclear operators 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.

DefinitionEdit

Let B be an arbitrary Banach space, and let B* be its dual, that is, the space of bounded linear functionals on B. The tensor product   has a completion under the norm

 

where the infimum is taken over all finite representations

 

The completion, under this norm, is often denoted as

 

and is called the projective topological tensor product. The elements of this space are called Fredholm kernels.

PropertiesEdit

Every Fredholm kernel has a representation in the form

 

with   and   such that   and

 

Associated with each such kernel is a linear operator

 

which has the canonical representation

 

Associated with every Fredholm kernel is a trace, defined as

 

p-summable kernelsEdit

A Fredholm kernel is said to be p-summable if

 

A Fredholm kernel is said to be of order q if q is the infimum of all   for all p for which it is p-summable.

Nuclear operators on Banach spacesEdit

An operator L : BB is said to be a nuclear operator if there exists an X  such that L = LX. 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 ≤ 2/3, then there is a unique trace, as given by a theorem of Grothendieck.

Grothendieck's theoremEdit

If   is an operator of order   then a trace may be defined, with

 

where   are the eigenvalues of  . Furthermore, the Fredholm determinant

 

is an entire function of z. The formula

 

holds as well. Finally, if   is parameterized by some complex-valued parameter w, that is,  , and the parameterization is holomorphic on some domain, then

 

is holomorphic on the same domain.

ExamplesEdit

An important example is the Banach space of holomorphic functions over a domain  . In this space, every nuclear operator is of order zero, and is thus of trace-class.

Nuclear spacesEdit

The idea of a nuclear operator can be adapted to Fréchet spaces. A nuclear space is a Fréchet space where every bounded map of the space to an arbitrary Banach space is nuclear.

ReferencesEdit

  • Grothendieck A (1955). "Produits tensoriels topologiques et espaces nucléaires". Mem. Amer. Math. Soc. 16.
  • Grothendieck A (1956). "La théorie de Fredholm". Bull. Soc. Math. France. 84: 319–84.
  • B.V. Khvedelidze, G.L. Litvinov (2001) [1994], "Fredholm kernel", Encyclopedia of Mathematics, EMS Press
  • Fréchet M (November 1932). "On the Behavior of the nth Iterate of a Fredholm Kernel as n Becomes Infinite". Proc. Natl. Acad. Sci. U.S.A. 18 (11): 671–3. doi:10.1073/pnas.18.11.671. PMC 1076308. PMID 16577494.