Commutators And Semi-commutators Of Toeplitz Operators On The Pluriharmonic Hardy Spaces

Toeplitz operators are a class of important operators on function spaces.In recent studies,except for Hardy spaces of the unit disk,the algebraic properties of Toeplitz operators on high-dimensional spaces are far from being solved.In this paper,we mainly study the problem of finite ranks of(semi)-commutators of monomial Toeplitz operators on pluriharmonic function spaces.In the first chapter,we first introduce the research background of Toeplitz operators on function spaces,followed by the development process and current situation,then review some related concepts,and finally give the main conclusions of this paper.In the second chapter,we first study the properties of Toeplitz operators on pluriharmonic Hardy spaces of the torus,and find many interesting higher-dimensional phenomena,then obtain sufficient and necessary conditions for the finite ranks of(semi)-commutators composed of monomial Toeplitz operators.In particular,we give sufficient and necessary conditions for(semi)-commutative of monomial Toeplitz operators.In the third chapter,we study the properties of Toeplitz operators on pluriharmonic Hardy spaces of the unit sphere,and obtain sufficient and necessary conditions for the finite ranks of(semi)-commutators composed of monomial Toeplitz operators.In the fourth chapter,we compare the results in pluriharmonic Hardy spaces of the torus and the unit sphere,find many non-trivial examples of(semi)-commuting Toeplitz operators on the pluriharmonic Hardy spaces,and give some interesting examples.In particular,the proof method in the pluriharmonic Hardy spaces of the torus is different from that in other spaces,and our conclusion is perfect,that is,the rank of(semi)-commutator can be determined only by the monomial index.In addition,in the pluriharmonic Hardy spaces of the unit sphere,the study of Toeplitz operators has not yet begun,the monomial Toeplitz operators on this space are preliminarily explored.