Operator-Lipschitz functions in Schatten–von Neumann classes
By D. Potapov, Fedor Sukochev. 2009
124 citations recorded by the source corpus.
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 $$ a - b \in {S^{\alpha }} $$.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/835623bc2e8da6f881a6d0cf66b14d4e67902c9b
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Citations in the corpus
- Operator smoothness in Schatten norms for functions of several variables: Lipschitz conditions, differentiability and unbounded derivations (2012, methodology): “Our analysis of the problem of boundedness of such integrals uses a multidimensional version of the Marcinkiewicz theorem for multipliers on UMD-spaces, so it is very different from the approach in [32].”
- Spectral shift function of higher order (2009, background): “Theorem 1.1 (see Section 2) for n ≥ 2 is based on a new powerful estimateformultipleoperatorintegrals(seeTheorem2.1inSection2andTheorem 5.3 and Remark 5.4 in Section 5), which extends the advances of [25] for the first order case to the higher order ones. The spectral shift function of order n, with ”