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 so −1-I is trace class if T is. It has a natural metric given by d = ||X - Y||1, where || · ||1 is the trace-class norm. If H is a Hilbert space with inner product, then so too is the kth exterior power with inner product In particular gives an orthonormal basis of if is an orthonormal basis of H. If A is a bounded operator on H, then A functorially defines a bounded operator on by If A is trace-class, then is also trace-class with This shows that the definition of the Fredholm determinant given by makes sense.
Properties
If A is a trace-class operator.
The function det is continuous on trace-class operators, with
A function F from into G is said to be differentiable if F -I is differentiable as a map into the trace-class operators, i.e. if the limit exists in trace-class norm. If g is a differentiable function with values in trace-class operators, then so too is exp g and where 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 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
Szegő limit formula
Let H = L2 and let P be the orthogonal projection onto the Hardy spaceH2. If f is a smooth function on the circle, let m denote the corresponding multiplication operator on H. The commutator is trace-class. Let T be the Toeplitz operator on H2 defined by then the additive commutator is trace-class if f and g are smooth. Berger and Shaw proved that If f and g are smooth, then is in G. Harold Widom used the result of Pincus-Helton-Howe to prove that where He used this to give a new proof of Gábor Szegő's celebrated limit formula: where PN is the projection onto the subspace of H spanned by 1, z,..., zN 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.
The section below provides an informal definition for the Fredholm determinant of I-T when the trace-class operator T is an integral operator given by a kernel K. 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 for a large variety of Hilbert spaces and Banach spaces, this is a non-trivial exercise. The Fredholm determinant may be defined as where K is an integral operator. The trace of the operator T and its alternating powers is given in terms of the kernel K by and and in general The trace is well-defined for these kernels, since these are trace-class or nuclear operators.
Applications
The Fredholm determinant was used by physicist John A. Wheeler to help provide mathematical description of the wavefunction for a composite nucleus composed of antisymmetrized combination of partial wavefunctions by the method of Resonating Group Structure. This method corresponds to the various possible ways of distributing the energy of neutrons and protons into fundamental boson and fermion nucleon cluster groups or building blocks such as the alpha-particle, helium-3, deuterium, triton, di-neutron, etc. When applied to the method of Resonating Group Structure for beta and alpha stable isotopes, use of the Fredholm determinant: determines the energy values of the composite system, and determines scattering and disintegration cross sections. The method of Resonating Group Structure of Wheeler provides the theoretical bases for all subsequent Nucleon Cluster Models and associated cluster energy dynamics for all light and heavy mass isotopes.
