The Algebraic Properties Of Monomial-type Toeplitz Operators And The Complex Symmetry Of Some Common Operators

Toeplitz operator and weighted composition operator are two important objects in the operator theory of holomorphic function spaces.The main goal of the research is to relate the operator theoretic properties to the function theoretic properties of the symbols.This papers concerns the complex symmetry of both operators and some algebraic properties of Toeplitz operator,which includes the commutativity,productiveness and the finite rank problem of the commutator and semicommutator.There exist relatively complete characterizations for these algebraic properties of Toeplitz operators over the Hardy space.However,it turns to be difficult on the Bergman space and some new and interesting phenomena happen in higher dimension.The research of complex symmet-ric operators is a young area in recent years and we also make a investigation on the complex symmetry of Toeplitz operator and weighted composition operator.This paper consists of six chapters which are arranged as follows.In the first chap-ter,we introduce some definitions and notations,then we take a quick look at the basic background and some related results in literature.The second chapter investigates the finite rank commutator and semicommutator of two Toeplitz operators with monomial-type symbols both on the holomorphic Bergman space and the pluriharmonic Bergman space of the unit polydisk,where some properties of the separately quasihomogeneous Toeplitz operators is provided.The third chapter studies the finite rank commutator and semicommutator of monomial-type Toeplitz operators on the Bergman space over certain weakly pseudoconvex domains.The fourth chapter study the complex symme-try of Toeplitz operators on Bergman space and pluriharmonic Bergman space over the unit polydisk and unit ball.The fifth chapter studies the complex symmetry of weight-ed composition operators on the Hardy space,in which we provide a class of complex symmetric weighted composition operators to includes both the unitary and Hermitian weighted composition operators,as well as a class of normal weighted composition op-erators identified by Bourdon and Narayan;A characterization of algebraic weighted composition operator with degree no more than two is provided to illustrate that the weight function of a complex symmetric weighted composition operator is not neces-sarily linear fractional.In the last chapter,we provide some open problems for further research.