Anti-commutators and the related trace formulas (Q1072088)

The main results of this paper are that if A and B are bounded self- adjoint operators with trace class anti-commutator \(AB+BA\), then \(tr[A^ kB^{\ell},A^ mB^ n]=0\), where k, \(\ell\), m, n are non-negative integers with \(kn+\ell m\) even. The author even proved that if the bounded self-adjoint operator A and B with trace class anti-commutator also satisfy \(A\geq 0\) and \(B\geq 0\), then AB is in the trace class.