Two questions on products of Toeplitz operators on the Bergman space
The authors study the so-called zero product and commuting problems for Toeplitz operators acting on the Bergman space over the unit disk \(\mathbb{D}\). The main results of the paper are as follows. Let \(f , g \, \in L^{\infty}(\mathbb{D})\) and \(g(re^{i\theta})=\sum_{k=-\infty}^{N}e^{ik\theta}g_k(r)\); if \(T_fT_g=0\), then \(f=0\). Let \(f(re^{i\theta})=\sum_{k=-\infty}^{N}e^{ik\theta}f_k(r)\) and let \(T_f\) commute with \(T_{z+\overline{z}}\), then \(T_f=Q(T_{z+\overline{z}})\), where \(Q\) is a polynomial of degree at most 3. The authors show as well that \(T^2_{z+\overline{z}}\) and \(T^3_{z+\overline{z}}\) are Toeplitz operators and calculate their symbols.
scientific article; zbMATH DE number 5659751