On a trace formula for functions of noncommuting operators

A. B. Aleksandrov; V. V. Peller; D. S. Potapov

paper

On a trace formula for functions of noncommuting operators

arXiv:1901.09495

Abstract

The main result of the paper is that the Lifshits--Krein trace formula cannot be generalized to the case of functions of noncommuting self-adjoint operators. To prove this, we show that for pairs $(A_{1}, B_{1})$ and $(A_{2}, B_{2})$ of bounded self-adjoint operators with trace class differences $A_{2} - A_{1}$ and $B_{2} - B_{1}$ , it is impossible to estimate the modulus of the trace of the difference $f (A_{2}, B_{2}) - f (A_{1}, B_{1})$ in terms of the norm of $f$ in the Lipschitz class.

8 pages