Analytic Toeplitz Operators, Distinguished Varieties and Boundary Behavior of Symbols

Based on our paper (Dan et al. in J Funct Anal 273:559–597, 2017), we continue the study on totally Abelian analytic Toeplitz operators on the Hardy space. However, rather than to focus on the geometry of the symbol curves associated with these operators as done before, this paper is to consider the Riemann surfaces in ℂ^2 defined by these symbols. In this way, we reveal the connection between the totally Abelian property of these operators and the distinguished varieties arising from the Riemann surfaces, thus widely extend the classes of function symbols considered in the previous paper. Furthermore, an alternative method is used to provide a sharp characterization for analytic Toeplitz operators being not totally Abelian.