Toeplitz Operators With Different Symbols On The Weighted Bergman Spaces Of The Unit Ball

Operator theory in function spaces has being the hot problem for discussion in the field of operator theory and analysis, and it is closely related with operator theory, operator algebra, function theory, differential equation, complex analysis, differential topology and so on. On the other hand, it has many important applications in control, quantum me-chanics, probability and statistics, etc. Toeplitz operators and Hankel operators, the most important operators on the function spaces, which have the extremely profound affect to operator theory, operator algebra and complex analysis, have attracted the attentions of many scholars.This thesis mainly focuses on the algebraic properties of Toeplitz operators with differ-ent symbols on the weighted Bergman space of the unit ball, and the algebraic properties of Toeplitz products and Haplitz products. More precisely, This thesis contains the following items.1. We mainly deal with boundedness and compactness of Toeplitz operator with a BMO symbol on the weighted Bergman space of the unit ball.By the definition of BMOa1(Bn), we get the relationship between |f|(z) and |f(z)|, where f∈BMOa1(Bn),z∈Bn. Under this condition, we give necessary and sufficient conditions for boundedness of Toeplitz operator with a BMO symbol. Using the relation between Hankel operator and little Hankel operator, together with the properties of Carleson measure, we prove that the Bergman projection Pa:BMOap(Bn)→B is bounded for any p≥1 and furthermore give necessary and sufficient conditions for compactness of Toeplitz operator with a BMO symbol.2. We discuss the algebraic properties of Toeplitz operators with positive symbol on the weighted Bergman space of the unit ball.By the definition of r-lattice for Bn and Berezin transform, we give necessary and sufficient conditions for boundedness, compactness and Schatten Toeplitz operator with positive symbol. By the definition of SMO, we prove that the Schatten class positive Toeplitz operator is equivalent to Schatten class little Hankel operator.3. We study the algebraic properties of Toeplitz products with square integrable holo-morphic symbol on the weighted Bergman space of the unit ball.By the property of mean value of the holomorphic functions and the property of the re-producing kernel kza, we obtain necessary and sufficient condition for boundedness, compact-ness and bounded invertible Toeplitz products with square-integrable holomorphic symbol. We also give a necessary and sufficient condition for bounded Haplitz products. Finally, we obtain a necessary condition for bounded Fredholm Toeplitz products.