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.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 withIn 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.An important example is the Banach space of holomorphic functions over a domainIn 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.

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