Toeplitz Operators On Hardy Type Spaces Of Harmonic Functions

Toeplitz operators are of importance in connection with a variety of problems in physics,probability theory,information and control theory,and several other fields.Besides the differential operators,Toeplitz operators constitute one of the most important classes of non-selfadjoint operators and they are a fascinating example of the fruitful interplay between such topics as operator theory,function theory,and the theory of Banach algebras.In this thesis,we promoted the study of Toeplitz opertors on the pluriharmonic Hardy space,the dual truncated Toeplitz operator is introduced and the algebra problems related to this operator are studied.The results of the study show that there is a huge difference between the Toeplitz operators on Hardy type spaces of harmonic functions and the Toeplitz operators on Hardy space,Bergman space,Dirichlet space,harmonic Bergman space,and harmonic Dirichlet space.The thesis in this paper consists of six parts.In Chapter 1,Toeplitz operator on the Hardy-type harmonic function space is introduced to compare with the Toeplitz operator in the classical function space.In Chapter 2,we completely characterize the(semi-)commutativity of Topelitz operators with pluriharmonic symbols on pluriharmonic Hardy space of bidisc,these results are quite different from the corresponding properties of Toeplitz operators on Hardy spaces,harmonic Bergman spaces and harmonic Dirichlet spaces.On Hardy space,two analytic Toeplitz operators must be commutative or semi-commutative,and on harmonic Bergman spaces and harmonic Dirichlet spaces,the(semi-)commutativity of Topelitz operators only have trivial cases.But on pluriharmonic Hardy space of bidisc,two analytic Toeplitz operators do not always commute,and there exist two nontrivial analytic Toeplitz operators commute.In Chapter 3,we introduce dual truncated Toeplitz operators.For each nonconstant inner function u,letH~2 be the classical Hardy space of open unit disk,we denote the model space by (?).We study the dual truncated Toeplitz operators defined on the orthogonal complement of model space (?).(?) is a harmonic function space which has an asymmetric structure.This feature is different from the structure of the harmonic Bergman space and the harmonic Dirichlet space.On Hardy space,the product of two Toeplitz operators is zero operator is equivalent to the product is a finite rank operator,but on (?),the situation is completely different and depends on u.We give the necessary and sufficient conditions for the product of two dual truncated Toeplitz operators to be zero and a finite rank operator,and fully characterize the conditions of the semi-commutativity of the dual truncated Toeplitz operators.The main results of this chapter are published in JMAA,461(1),929-946,2018。In Chapter 4,We study the commutativity of dual truncated Toeplitz operators.We find that the problem of studying the commutativity of dual truncated Toeplitz operators can be reduced to the commutativity of analytic dual truncated Toeplitz operators,the case of analytic dual truncated Toeplitz operators is equivalent to the problem of mixed commutativity of three Hankel operators on a Hardy space,this is a difficult question.We give the conditions for the commuting dual truncated Toeplitz operators for two types of analytic symbols.In Chapter 5,we study several types of algebras related to dual truncated Toeplitz operators and obtain two short exact sequences.The result is similar to the dual Toeplitz operators on the orthogonal complement of Bergman space.In addition,we fully characterize the commutant of the dual truncated Toeplitz operator with symbol z and find that there are a large number of non-dual truncated Toeplitz operators in the commutant.The result is in stark contrast to the Toeplitz operator and the truncated Toeplitz operator in Hardy space on unit disk.We also use the above conclusions to study the Fredholmness and spectral set structure of dual truncated Toeplitz operators.In Chapter 6,We sum up the conclusions of the full text,find the difficulties of unsolved problems,and look forward to the next step of research.