Operator-Lipschitz functions in Schatten–von Neumann classes

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 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 }} $.