Acta Mathematica

Operator-Lipschitz functions in Schatten–von Neumann classes

Denis Potapov and Fedor Sukochev

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Abstract

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 Sα, 1 < α < ∞. Alternatively, for every 1 < α < ∞, there is a constant cα > 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 Sα, and a and b are self-adjoint linear operators such that $ a - b \in {S^{\alpha }} $.

Note

2000 Math. Subject Classification: 47A56, 47B10, 47B47.

Article information

Source
Acta Math., Volume 207, Number 2 (2011), 375-389.

Dates
Received: 20 May 2009
Revised: 15 October 2009
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892583

Digital Object Identifier
doi:10.1007/s11511-012-0072-8

Mathematical Reviews number (MathSciNet)
MR2892613

Zentralblatt MATH identifier
1242.47013

Keywords
Operator-Lipschitz functions Schatten–von Neumann ideals

Rights
2011 © Institut Mittag-Leffler

Citation

Potapov, Denis; Sukochev, Fedor. Operator-Lipschitz functions in Schatten–von Neumann classes. Acta Math. 207 (2011), no. 2, 375--389. doi:10.1007/s11511-012-0072-8. https://projecteuclid.org/euclid.acta/1485892583


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