# Functions of noncommuting operators under perturbation of class $\boldsymbol{S}_p$

## TL;DR

This paper demonstrates that for certain classes of functions and operators, small perturbations in the Schatten class can lead to larger changes in the functions of these operators, highlighting limitations in perturbation stability.

## Contribution

The paper provides counterexamples showing that functions in the Besov class can amplify perturbations beyond the Schatten class for pairs and triples of operators.

## Key findings

- Counterexamples for $p>2$ with pairs of self-adjoint operators
- Extension of results to functions of contractions
- Analogous results for triples of self-adjoint operators for all $p",

## Abstract

In this article we prove that for $p>2$, there exist pairs of self-adjoint operators $(A_1,B_1)$ and $(A_2,B_2)$ and a function $f$ on the real line in the homogeneous Besov class $B_{\infty,1}^1({\Bbb R}^2)$ such that the differences $A_2-A_1$ and $B_2-B_1$ belong to the Schatten--von Neumann class $\boldsymbol{S}_p$ but $f(A_2,B_2)-f(A_1,B_1)\not\in\boldsymbol{S}_p$. A similar result holds for functions of contractions. We also obtain an analog of this result in the case of triples of self-adjoint operators for any $p\ge1$