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.
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.
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
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 spaces
An operator is said to be a nuclear operator if there exists an such that . 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 , then there is a unique trace, as given by a theorem of Grothendieck.
If is an operator of order then a trace may be defined, with
is an entire function of z. The formula
is holomorphic on the same domain.
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.
- Grothendieck A (1955). "Produits tensoriels topologiques et espaces nucléaires". Mem. Am. Math.Soc. 16.
- Grothendieck A (1956). "La théorie de Fredholm". Bull. Soc. Math. France 84: 319–84.
- B.V. Khvedelidze, G.L. Litvinov (2001), "Fredholm kernel", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- 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.