Informal Presentation
The section below provides an informal definition for the Fredholm determinant. A proper definition requires a presentation showing that each of the manipulations are well-defined, convergent, and so on, for the given situation for which the Fredholm determinant is contemplated. Since the kernel K may be defined on a large variety of Hilbert spaces and Banach spaces, this is a non-trivial exercise.
The Fredholm determinant may be defined as
![\det(I-\lambda K) = \left[
\sum_{n=0}^\infty (-\lambda)^n \operatorname{Tr } K^n \right]= \exp{(\sum_{n=0}^\infty(-1)^{n+1}\frac{\operatorname{Tr} A^n}{n}z^n})](http://upload.wikimedia.org/math/0/6/5/065b8569fb98929c2f92c830a65d0cc7.png)
where K is an integral operator. The trace of the operator is given by
and
and in generall : . The trace is well-defined for these kernels, since these are trace-class or nuclear operators.
Read more about this topic: Fredholm Determinant
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