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
Famous quotes containing the words informal and/or presentation:
“We as a nation need to be reeducated about the necessary and sufficient conditions for making human beings human. We need to be reeducated not as parents—but as workers, neighbors, and friends; and as members of the organizations, committees, boards—and, especially, the informal networks that control our social institutions and thereby determine the conditions of life for our families and their children.”
—Urie Bronfenbrenner (b. 1917)
“He uses his folly like a stalking-horse, and under the presentation of that he shoots his wit.”
—William Shakespeare (1564–1616)