Functions of perturbed normal operators

Abstract

In Peller (1985, 1990) [10,11], Aleksandrov and Peller (2009, 2010, 2010) [1–3] sharp estimates for f ( A ) − f ( B ) were obtained for self-adjoint operators A and B and for various classes of functions f on the real line R . In this Note we extend those results to the case of functions of normal operators. We show that if f belongs to the Hölder class Λ α ( R 2 ) , 0 < α < 1 , of functions of two variables, and N 1 and N 2 are normal operators, then ‖ f ( N 1 ) − f ( N 2 ) ‖ ⩽ const ‖ f ‖ Λ α ‖ N 1 − N 2 ‖ α . We obtain a more general result for functions in the space Λ ω ( R 2 ) = { f : | f ( ζ 1 ) − f ( ζ 2 ) | ⩽ const ω ( | ζ 1 − ζ 2 | ) } for an arbitrary modulus of continuity ω. We prove that if f belongs to the Besov class B ∞ 1 1 ( R 2 ) , then it is operator Lipschitz, i.e., ‖ f ( N 1 ) − f ( N 2 ) ‖ ⩽ const ‖ f ‖ B ∞ 1 1 ‖ N 1 − N 2 ‖ . We also study properties of f ( N 1 ) − f ( N 2 ) in the case when f ∈ Λ α ( R 2 ) and N 1 − N 2 belongs to the Schatten–von Neumann class S p .