Zero Product Problems Of Block Toeplitz Operators In Vector-Valued Bergman Spaces

Block Toeplitz operators have important practical applications.They exist in great many directions of physics and mathematics and play an important role in the theory of quantum mechanics.On the basis of previous studies,this paper discusses the zero product problems of Toeplitz operators on vector-valued Bergman spaces,whose symbols are essential bounded measurable functions and some harmonic polynomial functions.Zero product problems are special forms of product problems in the operator theory of function spaces,that is,whether the product of two operators is zero operator or not is an important algebraic property of operator.By consulting literature resources,up to now,there has been no result on the problem that the product of block Toeplitz operators on vector-valued Bergman spaces is zero,which is symbolized by essential bounded functions and harmonic polynomial functions.In this paper,starting from the finite sum of the products of Toeplitz operator on Bergman spaces,we discuss the necessary and sufficient conditions for finite sum of products,the zero products of two block Toeplitz operators on vector-valued Bergman spaces with the symbols of the essential bounded measurable function and the polynomial function.The paper is divided into five parts.It introduces the relevant knowledge background of the research work and the source of topic selection,the required preparatory knowledge,the deduction processes and related conclusions.Finally,it summarizes and prospects the conclusions.In the first chapter,we focus on the research background of Toeplitz operators.And then introduces the development of zero product problems in various spaces and research status at home and abroad.In the second chapter,we introduce the basic knowledge related to this article.In the third chapter,we give a sufficient and necessary condition for the finite sum of the product of Toeplitz operators on Bergman spaces to be zero operators for the symbols of essential bounded measurable functions and some harmonic polynomial functions,respectively.In the fourth chapter,we characterize the zero product problem of block Toeplitz operattors in the vector-valued Bergman space on the basis of the previous chapter.In the fifth chapter,we summarize and prospect the research results of this paper.