The Commutativity And Hyponormality Of K-order Slant Toeplitz Operators On The Bergman Space

Due to the development of the theory of wavelet analysis and its widely application in quantum mechanics,signal processing,image processing,the differential equation and so on,the slant Toeplitz operators on the function space have attracted the attention of many scholars and have been researched in depth,and many achievements have been obtained.But there are still many problems to be solved,this thesis mainly studies the commutativity and the hyponormality of the k-order slant Toeplitz operator on the Bergman space of polydisk.In the first chapter and the second chapter we mainly introduce the general research situation at home and abroad of slant Toeplitz operators and professional basic knowledge required for this thesis in brief.The commutativity of k-order slant Toeplitz operators on the Bergman space of bidisk is mainly studied in the third chapter.we obtain the necessary and sufficient condition for the commutativity of k-order slant Toeplitz operators with bounded co-analytic symbols or biharmonic symbols is such that their symbol functions are linearly dependent,k-order slant Toeplitz operators with bounded co-analytic symbols commute with Toeplitz operators with such symbols if and only if the symbol function of slant Toeplitz operators is zero or the symbol function of Toeplitz operators is constant,and get that the necessary and sufficient condition for the product of two k-order slant Toeplitz operators with bounded co-analytic symbols to be zero is such that at least one of their symbol functions is zero.In the fourth chapter,we mainly study the hyponormality of k-order slant Toeplitz operators on the Bergman space of polydisk.First of all,we give the necessary and sufficient condition for the hyponormality of k-order slant Toeplitz operators with bounded co-analytic symbols,secondly we fully characterize the hyponormality of k-order slant Toeplitz operators with bounded analytic symbols and obtain k-order slant Toeplitz operators with bounded analytic polynomial symbols are hyponormal if and only if their symbols functions are zero.Lastyly we get the hyponormality of k-order slant Toeplitz operators with polyharmonic symbols.