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
It has a natural metric given by d(X, Y) = ||X - Y||1, where || ยท ||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 (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 (A) is also trace-class with
This shows that the definition of the Fredholm determinant given by
makes sense.
Read more about this topic: Fredholm Determinant
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