In mathematics, a Fredholm kernel is a certain type of a kernel on a Banach space, associated with nuclear operators on the Banach space.
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.
Let B be an arbitrary Banach space, and let B* be its dual, that is, the space of bounded linear functionals on B.
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.
Every Fredholm kernel has a representation in the form with
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
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.
Furthermore, the Fredholm determinant is an entire function of z.
is parameterized by some complex-valued parameter w, that is,
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.
The idea of a nuclear operator can be adapted to Fréchet spaces.