Products of Toeplitz operators on the Bergman space
by
Zeljko Cuckovic
University of Toledo, USA
Coauthors: Patrick Ahern (University of Wisconsin), Sheldon Axler (San Francisco State University), N. V. Rao (University of Toledo)

This talk concerns two problems involving products of Toeplitz operators on the Bergman space:

1. When is a product of two Toeplitz operators equal to a Toeplitz operator? This seemingly innocent and very natural question is still waiting for an answer. The corresponding Hardy space problem was solved in 1964 by Brown and Halmos. They proved that a product of two Hardy Toeplitz operators is again Toeplitz only in the trivial cases. We consider Toeplitz operators with harmonic symbols. The first step is to find an equivalent function theoretic formulation of our problem that involves the Berezin transform. Then using function theoretic tools, we give several necessary conditions for the product to occur in a nontrivial way. Roughly speaking, we show that the symbols of Toeplitz operators are somewhat smooth, but not too smooth. Finally we consider a special case of the zero product question and give a condition for nontriviality of the product in terms of cyclic vectors of the backward shift. This is a joint work with Patrick Ahern. 2. Characterize symbols of commuting Toeplitz operators. In 1991, S. Axler and I completely characterized commuting Toeplitz operators with harmonic symbols. Recently, together with S. Axler and N. V. Rao, I showed that there is a sort of rigidity of the symbol if one of them is a radial or analytic function. I will present these results and pose more questions. (T)

Date received: December 13, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cado-72.