Some Problems For Dual Toeplitz Operators In Harmonic Function Spaces

Operator theory in function spaces plays an important role in operator theory. Toeplitz operators on Bergman space become a popular branch in the field of operator theory because of its close relationship with other branches of mathematics, such as Banach algera, Complex analysis and the extensive applications in Physics, Quantum mechanics, Control Theory and so on. Furthermore, the research of Toeplitz operators on Bergman space induces many im-portant problems about complex analysis and differential equations. These improve the theory development of mathematics greatly and make the Toeplitz operators on Bergman space receive widespread attention.Recently, the study of Toeplitz operators was not only restricted on classical Hardy s-pace and Bergman space, but has been extended to more complicated function spaces, such as Fock space, harmonic Bergman space and Segal-Bargmann space. Meanwhile, the study of dual Toeplitz operators receives more attention because of the close relationship between Toeplitz operators and dual Toeplitz operators. This paper deals with the finite rank product and (semi)commutators of quasihomogeneous Toeplitz operators on the pluriharmonic Bergman s-pace, and the commutativity and semi-commutativity of dual Toeplitz operators on the harmonic Bergman space and harmonic Dirichlet space.Chapter1includes the review of relevant results on Toeplitz operators and dual Toeplitz operators as well as the introduction of research history and current development status about algebraic properties of (dual)Toeplitz operator.In Chapter2, the algebraic properties of Toeplitz operators on pluriharmonic Bergman space has been discussed by using Mellin transform. First of all, a necessary and sufficient condition for the product of two Toeplitz operators with radial symbols to be a Toeplitz operator is given, meanwhile the zero-product problem for several Toeplitz operators with radial symbols is discussed. Then the study on the finite rank product and finite rank (semi)commutators of Toeplitz operators with qusihomogeneous symbols has been conducted. Some examples are given here as the application of the theory.In Chapter3, the commutativity and semi-commutativity of dual Toeplitz operators with harmonic symbols are characterized on the orthogonal complement space of harmonic Bergman space on the open unit disk D. At first, the result can be obtained by using function theory that (anti)analytic dual Toeplitz operators (semi)commute on (Lh2(D))⊥. Then, by the application of complex analysis and SφSΨ=SΨSφ,SφSΨ=SφΨ on (Lh2(D))⊥, the symbol functions φ and Ψ have been further characterized. The results are analogous to the well-known Brown-Halmos’s results about (semi)commutativity on the classical Hardy space.In Chapter4, the commuting dual Toeplitz operators on harmonic Dirichlet space with symbols in W1,∞(D) has been studied.In Chapter5, conclusions, innovations along with the prospects of this thesis have been stated.