In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant 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 the mathematician Erik Ivar Fredholm.

Fredholm determinants have had many applications in mathematical physics, the most celebrated example being Gábor Szegő's limit formula, proved in response to a question raised by Lars Onsager and C. N. Yang on the spontaneous magnetization of the Ising model.

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

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

$|lambda^k(A)|\_1\; le\; |A|\_1^k/k!.$

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

$|\{rm\; det\},\; (I+\; zA)|\; le\; exp\; (|z|cdot\; |A|\_1).$

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

$|\{rm\; det\}(I+A)\; -\{rm\; det\}(I+B)|\; le\; |A-B|\_1\; exp\; (|A|\_1\; +\; |B|\_1\; +1).$

• 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

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

Fredholm determinants of commutators

A function F(t) from (a, b) into G is said to be differentiable if F(t) -I is differentiable as a map into the trace-class operators, i.e. if the limit

$dot\{F\}(t)\; =\; lim\_\{hrightarrow\; 0\}\; \{F(t+h)\; -\; F(t)over\; h\}$

exists in trace-class norm.

If g(t) is a differentiable function with values in trace-class operators, then so too is exp g(t) and

$F^\{-1\}\; dot\{F\}\; =\; \{\{rm\; id\}\; -\; exp\; -\; \{rm\; ad\}\; g(t)over\; \{rm\; ad\}\; g(t)\}\; cdot\; dot\{g\}(t),$

where

$\{rm\; ad\}(X)cdot\; Y\; =\; XY\; -YX.$

Israel Gohberg and Mark Krein proved that if F is a differentiable function into G, then f = det F is a differentiable map into C* with

$f^\{-1\}\; dot\{f\}\; =\; det\; F^\{-1\}\; dot\{F\}.$

This result was used by Joel Pincus, William Helton and Roger Howe to prove that if A and B are bounded operators with trace-class commutator AB -BA, then

$\{rm\; det\},\; e^A\; e^B\; e^\{-A\}\; e^\{-B\}\; =\; exp\; \{rm\; Tr\}\; (AB-BA).$

Szegő limit formula

Let H = L² (S¹) and let P be the orthogonal projection onto the Hardy space H² (S¹).

If f is a smooth function on the circle, let m(f) denote the corresponding multiplication operator on H.

The commutator

Pm(f) - m(f)P

is trace-class.

Let T(f) is the Toeplitz operator on H² (S¹) defined by

$T(f)\; =\; Pm(f)P,$

then the additive commutator

$T(f)\; T(g)\; -\; T(g)\; T(f)$

is trace-class if f and g are smooth.

Berger and Shaw proved that

$\{rm\; tr\}(T(f)\; T(g)\; -\; T(g)\; T(f))\; =\; \{1over\; 2pi\; i\}\; int\_0^\{2pi\}\; f\; dg.$

If f and g are smooth, then

$T(e^\{f+g\})T(e^\{-f\})\; T(e^\{-g\})$

is in G.

Harold Widom used the result of Pincus-Helton-Howe to prove that

$\{rm\; det\}\; ,\; T(e^f)\; T(e^\{-f\})\; =\; exp\; sum\_\{\; n>0\}\; na\_n\; a\_\{-n\},$

where

$f(z)\; =sum\; a\_n\; z^n.$

He used this to give a new proof of Gábor Szegő's celebrated limit formula:

$lim\_\{Nrightarrow\; infty\}\; \{rm\; det\}\; P\_N\; m(e^f)\; P\_N\; =\; exp\; sum\_\{\; n>0\}\; na\_n\; a\_\{-n\},$

where P_N is the projection onto the subspace of H spanned by 1, z, ..., z^N and a₀ = 0.

Szegő's limit formula was proved in 1951 in response to a question raised by the work Lars Onsager and C. N. Yang on the calculation of the spontaneous magnetization for the Ising model. The formula of Widom, which leads quite quickly to Szegő's limit formula, is also equivalent to the duality between bosons and fermions in conformal field theory. A singular version of Szegő's limit formula for functions supported on an arc of the circle was proved by Widom; it has been applied to establish probabilistic results on the eigenvalue distribution of random unitary matrices.

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

$zeta(s)\; =\; frac\{1\}\{det(I-s\; K)\}.$

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.

References

External links

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