The compactness of a class of radial operators on weighted Bergman spaces

Abstract

In this paper, we study some connection between the compactness of radial operators and the boundary behavior of the corresponding Berezin transform on weighted Bergman spaces. More precisely, we prove that, under some mild condition, the vanishing of the Berezin transform on the unit circle is equivalent to the compactness of a class of radial operators on weighted Bergman spaces. Moreover, we also study the radial essential commutant of the Toeplitz operator $T_z$.