Fredholm Determinant - Fredholm Determinants of Commutators

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 g(t) is a differentiable function with values in trace-class operators, then so too is exp g(t) 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 A and B are bounded operators with trace-class commutator AB -BA, then