Operator-Lipschitz functions in Schatten–von Neumann classes

Denis Potapov School of Mathematics & Statistics, University of New South Wales Fedor Sukochev School of Mathematics & Statistics, University of New South Wales

Analysis of PDEs Spectral Theory and Operator Algebra mathscidoc: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
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@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.
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