# MathSciDoc: An Archive for Mathematician ∫

#### Analysis of PDEsSpectral Theory and Operator Algebramathscidoc:1701.03007

Acta Mathematica, 207, (2), 375-389, 2009.5
This paper resolves a number of problems in the perturbation theory of linear operators, linked with the 45-year-old conjecure of M. G. Kreĭn. In particular, we prove that every Lipschitz function is operator-Lipschitz in the Schatten–von Neumann ideals$S$^{$α$}, 1 <$α$< ∞. Alternatively, for every 1 <$α$< ∞, there is a constant$c$_{$α$}> 0 such that $${\left\| {f(a) - f(b)} \right\|_{\alpha }} \leqslant {c_{\alpha }}{\left\| f \right\|_{{{\text{Lip}}\,{1}}}}{\left\| {a - b} \right\|_{\alpha }},$$ where$f$is a Lipschitz function with $${\left\| f \right\|_{{{\text{Lip}}\,{1}}}}: = \mathop{{\sup }}\limits_{{_{{\lambda \ne \mu }}^{{\lambda, \mu \in \mathbb{R}}}}} \left| {\frac{{f\left( \lambda \right) - f\left( \mu \right)}}{{\lambda - \mu }}} \right| < \infty,$$ $${\left\| \cdot \right\|_{\alpha }}$$ is the norm is$S$^{$α$}, and$a$and$b$are self-adjoint linear operators such that $$a - b \in {S^{\alpha }}$$ .
Operator-Lipschitz functions; Schatten–von Neumann ideals
@inproceedings{denis2009operator-lipschitz,
title={Operator-Lipschitz functions in Schatten–von Neumann classes},
author={Denis Potapov, and Fedor Sukochev},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203357666603748},
booktitle={Acta Mathematica},
volume={207},
number={2},
pages={375-389},
year={2009},
}

Denis Potapov, and Fedor Sukochev. Operator-Lipschitz functions in Schatten–von Neumann classes. 2009. Vol. 207. In Acta Mathematica. pp.375-389. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203357666603748.