Some Properties For Toeplitz Operators And Dual Toeplitz Operators On The Dirichlet Space

As an active research field in functional analysis, the key problem of operator theory in function spaces is to characterize the properties of operators by using the analysis and geometry properties of the symbols, and hence establish connections between complex analysis and oper-ator theory. There is a great value in theory and practical application to study Toeplitz operator and Hankel operator, since they are wised used in control theory, informatics, probability theory and other fields in mathematics. In this paper, we mainly deal with some properties of Toeplitz operators and dual Toeplitz operators on the Dirichlet space, such as compactness, commutativ-ity and product problem.In chapter 1, we will introduce some basic conceptions and induce the research history and current development status about algebraic properties of (dual) Toeplitz operator.In chapter 2, by the use of the decomposition of Sobolev space and quasihomogeneous decomposition, we characterize the symbols for (semi-)commuting dual Toeplitz operators on the orthogonal complement of the harmonic Dirichlet space.In chapter 3, we use the properties of Riesz’s functions to characterize the compactness of Toeplitz operators on the weighted Dirichlet space.In chapter 4, we give a relation between Toeplitz operators on the Dirichlet space and their analogues defined on the Hardy space of the unit ball. Based on this, we give the sufficient and necessary condition for the problem of when finite sum of products of several Toeplitz operators with pluriharmonic symbols is of finite rank. Also we characterize the symbols for (semi-)commuting Toeplitz operators with pluriharmonic symbols on the Dirichlet space of the unit ball.In chapter 5, we characterize when is f1g1+…+fNgN pluriharmonic for holomorphic functions f1,…,fN and g1,…,gN in the unit ball. Based on this, we completely characterize the pluriharmonic symbols for (semi-)commuting dual Toeplitz operators with pluriharmonic symbols on the orthogonal complement of the pluriharmonic Dirichlet space in Sobolev space of the unit ball.