A generalization of Pincus' formula and Toeplitz operator determinants (Q1401617)

The author gives a generalization of Pincus' formula. Let \(H\) be a separable Hilbert space and \({\mathcal C}_1(H)\) the set of all trace class operators on \(H\). The main result contains the following Theorem. Let \(A,B \in {\mathcal L} (H),\) and assume that \(AB-BA\in {\mathcal C}_1(H)\). Then \(\det\;e^A e^B e^{-A-B} =\exp( \frac 12 \operatorname{trace} (AB-BA)).\) Applications to Toeplitz and Toeplitz-Hankel determinants are also considered.