Toeplitz Operators with Essentially Radial Symbols

For Topelitz operators with radial symbols on the disk, there are important results that characterize boundedness, compactness, and its relation to the Berezin transform. The notion of essentially radial symbol is a natural extension, in the context of multiply-connected domains, of the notion of radial symbol on the disk. In this paper we analyze the relationship between the boundary behavior of the Berezin transform and the compactness of Tφ when φ ∈ L2 Ω is essentially radial and Ω is multiply-connected domains.