Functions of perturbed operators

Abstract

We prove that if 0 < α < 1 and f is in the Hölder class Λ α ( R ) , then for arbitrary selfadjoint operators A and B with bounded A − B , the operator f ( A ) − f ( B ) is bounded and ‖ f ( A ) − f ( B ) ‖ ⩽ const ‖ A − B ‖ α . We prove a similar result for functions f of the Zygmund class Λ 1 ( R ) : ‖ f ( A + K ) − 2 f ( A ) + f ( A − K ) ‖ ⩽ const ‖ K ‖ , where A and K are selfadjoint operators. Similar results also hold for all Hölder–Zygmund classes Λ α ( R ) , α > 0 . We also study properties of the operators f ( A ) − f ( B ) for f ∈ Λ α ( R ) and selfadjoint operators A and B such that A − B belongs to the Schatten–von Neumann class S p . We consider the same problem for higher order differences. Similar results also hold for unitary operators and for contractions. To cite this article: A. Aleksandrov, V. Peller, C. R. Acad. Sci. Paris, Ser. I 347 (2009).