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.
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Definition
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
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.
Properties
Every Fredholm kernel has a representation in the form
with
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 kernels
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
Nuclear operators on Banach spaces
An operator L : B→B is said to be a nuclear operator if there exists an X ∈
Grothendieck's theorem
If
where
is an entire function of z. The formula
holds as well. Finally, if
is holomorphic on the same domain.
Examples
An important example is the Banach space of holomorphic functions over a domain
Nuclear spaces
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.