In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a finite dimensional linear operator. 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 so (I+T)−1−I is trace class if T is. It has a natural metric given by , where 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 (ei) 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. defines an entire function such that The function det(I + A) is continuous on trace-class operators, with One can improve this inequality slightly to the following, as noted in Chapter 5 of Simon: If A and B are trace-class then The function det defines a homomorphism of G into the multiplicative group C* of nonzero complex numbers (since elements of G are invertible). If T is in G and X is invertible, If A is trace-class, then 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 exists in trace-class norm. If is a differentiable function with values in trace-class operators, then so too is 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