Two Problems in the Theory of Toeplitz Operators on the Bergman Space

by Yousef, Abdelrahman Fawzi

In this thesis we deal with the zero product problem and the commuting problem for Toeplitz operators on the Bergman space over the unit disk of the complex plane. For the zero product problem, we show that the zero product of two Toeplitz operators has only the trivial solution when one of the symbols has certain polar decomposition and the other is a general bounded symbol. As for the commuting problem, we show that if the Fourier series of the bounded function f is of the form f(rei?) = ?k=??N eik? fk(r) where N is a positive integer, and Tf commutes with Tz+g?, where g is a bounded analytic function on the open unit disk, then Tf is a nontrivial linear combination of Tz+g? and the identity operator I. Also, we describe all Toeplitz operators Tf that commutes with Tz+z?, when the symbol f is integrable, with respect to the Lebesgue area measure, on the unit disk.