turkmath.org

The compactness of Toeplitz operators defined on spaces of holomorphic functions has been the subject of extensive research. One of the most important studies on this subject was conducted by Axler-Zheng. The Axler-Zheng theorem characterizes the compactness of finite sums of finite products of Toeplitz operators acting on the Bergman space on the unit disk in terms of the Berezin transform of the operators. There are two significant approaches in the generalizations of the Axler-Zheng theorem for Bergman spaces defined on more general domains: one approach utilizes functional analysis techniques, while the other employs methods from several complex variables. In this talk, I will present the generalizations of the Axler-Zheng theorem, focusing on the techniques used to establish them.