Two Questions on Products of Toeplitz Operators on the Bergman Space

The zero product problem and the commuting problem for Toeplitz operators on the Bergman space over the unit disk are some of the most interesting unsolved problems. For bounded harmonic symbols these are solved but for general bounded symbols it is still far from being complete. This paper shows that the zero product problem holds for a special case where one of the symbols has certain polar decomposition and the other is a general bounded symbol. We also prove that the commutant of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_z+\bar{z}$$\end{document} is sum of powers of itself.