Fredholm Determinant - Properties

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 non-zero complex numbers.
  • If T is in G and X is invertible,
  • If A is trace-class, then

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