A and B are bounded operators with trace-class commutator AB − BA, then 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 is trace-class. Let T(f) be the Toeplitz operator on H2 (S1) 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. Informal presentation for the case of integral operators 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(x,x). 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 T 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 (1937, Phys. Rev. 52:1107) to help provide mathematical description of the wavefunction for a composite nucleus composed of antisymmetrized