Compact Hankel Operators on Generalized Bergman Spaces of the Polydisc

Abstract

Let \({\vartheta}\) be a measure on the polydisc \({\mathbb{D}^n}\) which is the product of n regular Borel probability measures so that \({\vartheta([r,1)^n\times\mathbb{T}^n) >0 }\) for all 0 < r < 1. The Bergman space \({A^2_{\vartheta}}\) consists of all holomorphic functions that are square integrable with respect to \({\vartheta}\). In one dimension, it is well known that if f is continuous on the closed disc \({\overline{\mathbb{D}}}\), then the Hankel operator H f is compact on \({A^2_\vartheta}\). In this paper we show that for n ≥ 2 and f a continuous function on \({{\overline{\mathbb{D}}}^n}\), H f is compact on \({A^2_\vartheta}\) if and only if there is a decomposition f = h + g, where h belongs to \({A^2_\vartheta}\) and \({\lim_{z\to\partial\mathbb{D}^n}g(z)=0}\).