Fredholm determinant

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 "d"("X", "Y") = ||"X" - "Y"||1, where || · ||1 is the trace-class norm.
If "H" is a Hilbert space with inner product formula_2, then so too is the "k"th exterior power formula_3 with inner product
In particular
gives an orthonormal basis of formula_3 if ("e""i") is an orthonormal basis of "H". 
If "A" is a bounded operator on "H", then "A" functorially defines a bounded operator formula_7
on formula_3 by
If "A" is trace-class, then formula_7 is also trace-class with
This shows that the definition of the Fredholm determinant given by
makes sense.