Algebras of Toeplitz operators on the unit ball

One of the common strategies in the study of Toeplitz operators consists in selecting of various special symbol classes S ⊂ L∞ so that the properties of both the individual Toeplitz operators Ta, with a ∈ S, and of the algebra generated by such Toeplitz operators can be characterized. A motivation to study an algebra generated by Toeplitz operators (rather than just Toeplitz operators themselves) lies in a possibility to apply more tools, in particular those coming from the algebraic toolbox, and furthermore the results obtained are applicable not only for generating Toeplitz operators but also for a whole variety of elements of the algebra in question. To make our approach more transparent we restrict the presentation to the case of the twodimensional unit ball B. We consider various sets S of symbols that are invariant under a certain subgroup of biholomorphisms of B ({1} × T in the talk). Such an invariance permits us to lower the problem dimension and to give a recipe, supplied by various concrete examples, on how the known results for the unit disk D can be applied to the study of various algebras (both commutative and non-commutative) that are generated by Toeplitz operators on the two-dimensional ball B. Although we consider the operators acting on the weighted Bergman space on B with a fixed weight parameter, the Berezin quantization effects (caused by a growing weight parameter of the corresponding weighted Bergman spaces on the unit disk D) have to be taken into account. Talk time: 07/18/2016 2:30PM— 07/18/2016 2:50PM Talk location: Cupples I Room 215 Special Session: Toeplitz operators and related topics. Organized by S. Grudsky and N. Vasilevski.