Commutator Estimates for Interpolation Scales with Holomorphic Structure

Abstract

The main result of this paper is some quantitative estimates for nonlinear commutators under the complex interpolation methods and more general interpolation scales with holomorphic structures. We also investigate the spectral behaviour of bounded linear operators under this kind of interpolation methods. Mathematics Subject Classification (2000)Primary 46B70 · 46M35; Secondary 47A10 · 47B47 IntroductionIn 1983, Rochberg and Weiss [15] studied the behaviour of the commutators for bounded linear operators and some derivation operators, which are usually unbounded and nonlinear, under the complex interpolation methods. The similar results for the real interpolation methods were obtained by Jawerth et al. [11] in 1985. Recently, Cwikel et al. constructed a general interpolation method with holomorphic structure in [8]. This new setting includes the classical real and complex methods, and even the so called ± methods given by Peetre and Gustavsson as special cases. The main idea behind this approach is to give a unified treatment on several different interpolation methods currently in use and the corresponding commutator estimates.

Nigel Kalton and the interpolation theory of commutators

“…See Proposition 1.5 on p. 166 of [20] for a detailed discussion of a non-commutative version of this example. Cf.…”