Fredholm determinant

In mathematics, a Fredholm determinant is a complex analytic function which generalizes the characteristic polynomial of a matrix. It is defined for bounded operators on a Hilbert space which differ from the identity operator by a trace-class operator. The function is named after mathematician Erik Ivar Fredholm.

Definition

Let H be a Hilbert space and G the set of bounded invertible operators on H of the form I + T, where T is a trace-class operator. G is a group because

$(I+T)^{-1}-I=T(I+T)^{-1}.$

It has a natural metric given by d(X, Y) = ||X - Y||₁, where || · ||₁ is the trace-class norm.

If H is a Hilbert space, then so too is the kth exterior power λ^k H with inner product

$(v_{1}\wedge v_{2}\wedge \cdots \wedge v_{k},w_{1}\wedge w_{2}\wedge \cdots \wedge w_{k})={\rm {det}}\,(v_{i},w_{j}).$

In particular

$e_{i_{1}}\wedge e_{i_{2}}\wedge \cdots \wedge e_{i_{k}},\qquad (i_{1}<i_{2}<\cdots <i_{k})$

gives an orthonormal basis of λ^k H if (e_i) is an orthonormal basis of H. If A is a bounded operator on H, then A functorially defines a bounded operator λ^k(A) on λ^k H by

$\lambda ^{k}(A)v_{1}\wedge v_{2}\wedge \cdots \wedge v_{k}=Av_{1}\wedge Av_{2}\wedge \cdots \wedge Av_{k}.$

If A is trace-class, then λ^k(A) is also trace-class with

$\|\lambda ^{k}(A)\|_{1}\leq \|A\|_{1}^{k}/k!.$

This shows that the definition of the Fredholm determinant given by

${\rm {det}}\,(I+A)=\sum _{k=0}^{\infty }{\rm {Tr}}\lambda ^{k}(A)$

makes sense.

Properties

• If A is a trace-class operator

${\rm {det}}\,(I+zA)=\sum _{k=0}^{\infty }z^{k}{\rm {Tr}}\lambda ^{k}(A)$

defines an entire function such that

$|{\rm {det}}\,(I+zA)|\leq \exp(|z|\cdot \|A\|_{1}).$

• The function det(I + A) is continuous on trace-class operators, with

$|{\rm {det}}(I+A)-{\rm {det}}(I+B)|\leq \|A-B\|_{1}\exp(\|A\|_{1}+\|B\|_{1}+1).$

• If A and B are trace-class then

${\rm {det}}(I+A)\cdot {\rm {det}}(I+B)={\rm {det}}(I+A)(I+B).$

• The function det defines a homomorphism of G into the multiplicative group C* of non-zero complex numbers.

• If T is in G and X is invertible,

${\rm {det}}\,XTX^{-1}={\rm {det}}\,T.$

• If A is trace-class, then

${\rm {det}}\,e^{A}=\exp \,{\rm {Tr}}(A).$

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

$\det(I-\lambda K)=\exp \left[-\sum _{n}{\frac {\lambda ^{n}}{n}}\operatorname {Tr} K^{n}\right]$

where K is an integral operator, the Fredholm operator. The trace of the operator is given by

$\operatorname {Tr} K=\int K(x,x)\,dx$

and

$\operatorname {Tr} K^{2}=\iint K(x,y)K(y,x)\,dxdy$

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

$\zeta (s)={\frac {1}{\det(I-sK)}}$

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.

External links

The Front for the Math arXiv has several papers utilizing Fredholm determinants.

References

Simon, Barry (2005), Trace Ideals and Their Applications, Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, ISBN 0821835815