Dual Toeplitz Operators And Dual Truncated Toeplitz Operators

Operator theory in function spaces is an important part of functional analysis,which has close relationships with some different fields,such as quantum mechanics,probability theory,informatics and control theory,etc.In the last few decades,the study of Toeplitz operators in function spaces has developed quite fast,which obtains many impressive results.This thesis mainly focuses on two issues:The first part is to explore the hyponormality of Toeplitz operators on the harmonic Bergman space and dual Toeplitz operators on its orthogonal complement.For the second problem,we mainly study when two dual truncated Toeplitz operators are essentially semi-commuting or essentially commuting on the orthogonal complement of model space.We are going to organize this thesis as the following six parts:In Chapter 1,we introduce some research background and development of the topics in this thesis and state the main results.Besides,some necessary definitions and notations will be given in this chapter.In Chapter 2,we will introduce Toeplitz operators on the harmonic Bergman space.Firstly,we will introduce this kind of operator and its hyponormality.It is well-known that the normal operator must be hyponormal,but the hyponormal Toeplitz operator on the Bergman space may be not normal.Fortunately,with the help of orthogonal projection onto the harmonic Bergman space,the normality and the hyponormality of Toeplitz operators on the harmonic Bergman space are equivalent.Since people have done the work of normality of the Toeplitz operators with symbols of the bounded harmonic functions,we are not necessary to study the same symbols.And we are going to consider the hyponormality of the Toeplitz operators with the symbols of non-harmonic functions on the harmonic Bergman space.For the second part,we will introduce the Weyl spectrum of Toeplitz operators on the harmonic Bergman space and the corresponding Weyl's theorem.In Chapter 3,we are going to explore the dual Toeplitz operator on the orthogonal complement of the harmonic Bergman space.Similarly,we will introduce the hoponomality of this kind of operators.Fortunately,the hyponormal dual Toeplitz operator on the orthogonal complement of harmonic Bergman space is also normal,so we only need to consider normal dual Toeplitz operators.Moreover,we are going to study dual Toeplitz operators with the symbols of non-harmonic functions,since others have done the same works of the dual Toeplitz operators with the symbols of bounded harmonic functions.We will introduce dual truncated Toeplitz operators on the orthogonal complement of model spaceH~2?u H~2 in Chapter 4.On the one hand,the dual truncated Toeplitz operator is a new concept which is studied naturally as classical dual Toeplitz operators on Hardy space and Bergman space,etc.On the other hand,the study of dual truncated Toeplitz operators is helpful for exploring the properties of truncated Toeplitz operators on model space.The main idea is to transfer the problems into the corresponding problems of classical operators on the Hardy space.In Chapter 5,we completely describe the essentially semi-commuting and essentially commuting dual truncated Toeplitz operators via the theory of function algebras.The semi-commuting and commuting problems for Toeplitz operators are still active in the field of operator theory.Many complete works about above problems have been done on different spaces,such as Hardy space,Bergman space and so on.For the dual truncated Toeplitz operators on the orthogonal complement of model space,there are also complete descriptions of semi-commuting problem and commuting problem.So we are going to study the essentially semi-commuting and essentially commuting dual truncated Toeplitz operators.Our main idea is to study dual truncated Toeplitz operators via Toeplitz operator and Hankel operator on the classical Hardy space,where we have lots of methods to use.In Chapter 6,we will make an overview of all contents in the thesis.Besides,we will consider some unsolved problems and try to work it out.