Year: 2012
Type: article
Abstract: We study in this paper properties of functions of perturbed normal operators and develop earlier results obtained in Aleksandrov, Peller, Potapov and Sukochev [‘Functions of normal operators under perturbations’, Adv. Math. 226 (2011) 5216–5251.]. We study operator Lipschitz and commutator Lipschitz functions on closed subsets of the plane. For such functions we introduce the notions of the operator modulus of continuity and of various commutator moduli of continuity. Our estimates lead to estimates of the norms of quasicommutators f(N1)R−Rf(N2) in terms of ‖N1R−RN2‖, where N1 and N2 are normal operator and R is a bounded linear operator. In particular, we show that if 0 < α < 1 and f is a Hölder function of order α, then, for normal operators N1 and N2, ‖ f ( N 1 ) R − R f ( N 2 ) ‖ ⩽ const ( 1 − α ) − 2 ‖ f ‖ Λ α ‖ N 1 R − R N 2 ‖ α ‖ R ‖ 1 − α . In the last section, we obtain lower estimates for constants in operator Hölder estimates.
DOI: 10.1112/plms/pds012
Authors: A. B. Aleksandrov, V. V. Peller
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2031742908