The Quasinormality And Hyponormality Of Toeplitz Operators In Bergman Spaces On The Annulus

The theory of operators in function spaces is closely related to many mathematical research fields.The Toeplitz operators studied in this paper also play an important role in physics and quantum mechanics.Normal operators are the most basic class of operators in operators theory.They have been completely characterized,and the condition of normality can be weakened in different ways.Therefore,it is extended to the study of quasinormality and hyponormality of operators.The study of Toeplitz operators in Bergman spaces has been on the disk all the time,but few studies have been done on the quasinormality and hyponormality of Toeplitz operators in Bergman spaces with multiconnected regions such as annulus.In this paper,we study the quasinormality and hyponormality of Toeplitz operators in the Bergman spaces on the annulus.We prove that Toeplitz operators with certain harmonic polynomial symbols in the Bergman spaces on the annulus must be normal if they are quasinormal.On the other hand,we try to give the necessary conditions for the hyponormality of Toeplitz operators with some harmonic polynomial symbols.The main contents of this paper are as follows:The paper is divided into five parts.It introduces the background of the research work and the source of the work topics,the necessary preparatory knowledge,the derivation process and the relevant conclusions.Finally,it summarizes the conclusions and prospects for the future.In the first chapter,we mainly introduce the research background of normal Toeplitz operators,quasinormal Toeplitz operators and hyponormal Toeplitz operators in Bergman spaces on the annulus,and then introduce their development and research status at home and abroad.In the second chapter,we introduce the basic knowledge of Toeplitz operators,normal Toeplitz operators,quasinormal Toeplitz operators and hyponormal Toeplitz operators in the Bergman space on the annulus.In the third chapter,we prove the quasinormality of Toeplitz operators with certain harmonic function symbols in the Bergman space on the annulus.In the fourth chapter,we prove the hyponormality of Toeplitz operators with some polynomial symbols in the Bergman space on the annulus.The fifth chapter,we summarize and forecast the results of this study.