Berezin's operator calculus and higher order Schwarz-Pick lemma

Author： Li Bo

ISSN： 1747-6933

Source： Complex Variables and Elliptic Equations, Vol.57, Iss.5, 2012-05, pp. : 523-538

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Abstract

We provide a new and simple proof to the result in our study of Berezin's operator calculus [B. Li, The Berezin transform and mth order Bergman metric, Trans. Amer. Math. Soc. (to appear)] that the mth order Bergman metric (B _m∂_v)(z) is a constant multiple of {(B ₁∂_v)(z)}^m on the unit ball, where (B ₁∂_v)(z) is the classical Bergman metric. Based on the reproducing-kernel theory, an approximation approach is developed to treat (B _m∂_v)(z) on the unit ball and ⁿ in a uniform way. Secondly, we discuss the interplay between our analysis in Berezin's operator calculus and the higher order Schwarz-Pick lemma in Dai et al. [S. Dai, H. Chen, and Y. Pan, The Schwarz-Pick lemma of high order in several variables, Michigan Math. J. (to appear)]. As a consequence, the mth order Carathéodory-Reiffen metric (C _m∂_v)(z) is shown to be a constant multiple of {(C ₁∂_v)(z)}^m also on the unit ball, where (C ₁∂_v)(z) is the classical infinitesimal Carathéodory-Reiffen metric.