Spectral properties of commutators on harmonic Bergman spaces of the unit disk

Abstract

In this paper, we determine the asymptotic behavior of the singular values of the commutator [Formula: see text] acting on [Formula: see text], where [Formula: see text] is the operator of multiplication by a subharmonic function [Formula: see text] on [Formula: see text] and harmonic outside the origin, with [Formula: see text], and [Formula: see text] is the orthogonal projection onto the space [Formula: see text] of harmonic functions on [Formula: see text] that are square-integrable with respect to the weighted measure [Formula: see text]. We prove that if [Formula: see text], where [Formula: see text] is a holomorphic function on [Formula: see text] and [Formula: see text] is the Lelong number of [Formula: see text] at [Formula: see text], then [Formula: see text] In particular, the operator [Formula: see text] belongs to the Von Newman-Schatten class [Formula: see text] for any [Formula: see text].