Fredholm determinant
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In mathematics, a Fredholm determinant is a complex analytic function which generalizes the characteristic polynomial of a matrix. It is defined for those operators which have continuous kernels, i.e., kernels in the sense of mathematical analysis. The function is named in honour of the mathematician Erik Ivar Fredholm; it is one of the parts of Fredholm theory.
[edit] 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) = \exp \left[ -\sum_n \frac{\lambda^n}{n} \operatorname{Tr } K^n \right]](../../../math/d/0/e/d0e3e5e209deb7cc7eb613a218a9871b.png)
where K is an integral operator, the Fredholm operator. The trace of the operator is given by
and
and so on. The trace is well-defined for the Fredholm kernels, since these are trace-class or nuclear operators, which follows from the fact that the Fredholm operator is a compact operator.
The corresponding zeta function is
The zeta function can be thought of as the determinant of the resolvent.
The zeta function plays an important role in studying dynamical systems. Note that this is the same general type of zeta function as the Riemann zeta function; however, in this case, the corresponding kernel is not known. The hypothesis stating the existence of such a kernel is known as the Hilbert-Pólya conjecture.
[edit] References
The Front for the Math arXiv has several papers utilizing Fredholm determinants.
