Algebraic Properties Of Toeplitz Operators On The Function Spaces

Operator theory in function spaces becomes a hot issue of research in recent decades. As the carrier of research is function spaces, the operator defined on com-mon spaces must be derived from certain function, and hence we need to investigate the inner relation between the algebraic properties of the operator and the analytical properties of the corresponding function. This paper mainly discusses some algebraic properties of Toeplitz operators in function spaces, including the commutativity of two Toeplitz operators, zero-product problem and when the product of two Toeplitz opera-tors is a Toeplitz operator.First, we discuss some algebraic properties of Toeplitz operators on the Bergman space of the unit ball. We give necessary and sufficient conditions for the product of two Toeplitz operators whose symbols are radial functions to be a Toeplitz operator, and the corresponding commuting problem of Toeplitz operators with quasihomogeneous symbols is studied. Also, we provide a decomposition of square integrable functions on the unit ball, then we use it to study the zero-product problem and the commutativity of two Toeplitz operators.Next, we discuss some algebraic properties of Toeplitz operators on the Bergman space of the unit polydisk. We determine when the product of two Toeplitz operators with separately quasihomogeneous symbols is a Toeplitz operator. Then using similar polar decomposition, we also study zero-product problem and the commutativity of two Toeplitz operators.Finally, we discuss whether the product of two Toeplitz operators with quasihomo-geneous symbols on the harmonic Bergman space of the unit disk is a Toeplitz operator, then get a completely different conclusion than Bergman space.