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. Kreǐn. 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, where f is a Lipschitz function with, {double pipe}·{double pipe} α is the norm is S α, and a and b are self-adjoint linear operators such that a-bεS α. © 2011 Institut Mittag-Leffler.