Fredholm determinant

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.

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 d(X, Y) = ||X - Y||₁, where || · ||₁ is the trace-class norm.

If H is a Hilbert space with inner product ${\displaystyle (\cdot ,\cdot )}$, then so too is the kth exterior power ${\displaystyle \Lambda ^{k}H}$ with inner product

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