# On a trace formula for functions of noncommuting operators

## TL;DR

This paper demonstrates that the Lifshits--Krein trace formula cannot be extended to functions of noncommuting self-adjoint operators, highlighting fundamental limitations in trace estimates for such operator functions.

## Contribution

It proves the impossibility of generalizing the Lifshits--Krein trace formula to noncommuting operators by showing trace estimate limitations.

## Key findings

- Lifshits--Krein trace formula cannot be extended to noncommuting operators
- Trace differences cannot be bounded by Lipschitz norm of functions
- Fundamental limitations in trace estimates for noncommuting operator functions

## 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.