Toeplitz Operators On The Space Of Weighted Analytic Functions

Firstly,we study some characterizations of Toeplitz operators with positive operator-valued function as symbols on the vector-valued Doubling Fock spaces F_?² in C.The boundedness and compactness of the Toeplitz operators with posi-tive operator-valued function as symbols are characterized by Carleson conditions.Secondly,we study essential norm of Toeplitz operators with symbol of L^? on the Bergman spaces A_?² induced by regular-weight on unit disk,and obtains the approximate formula of the essential norm of non-compact Toeplitz operator by using the distance between Toeplitz operators and compact subset.Finally,we s-tudy the Schatten-p class properties of Toeplitz operators with positive measure symbols on Bergman Spaces induced by regular-weight on annulus,and obtain the equivalent characterization that Toeplitz operators with positive measure symbols belong to the Schatten-p class by using Berezin transformation.In Chapter 1,the historical background and research status of Toeplitz op-erators on Doubling Fock space and weighted Bergman spaces induced by regular weights are introduced,as well as some basic concepts,preliminary knowledge and main results of this paper.In Chapter 2,the boundedness and compactness of the Toeplitz operators with positive operator-valued function as symbols on the vector-valued Doubling Fock spaces F_?² in C are characterized by Carleson conditions.In Chapter 3,the approximate formula of the essential norm of non-compact Toeplitz operator are obtained by using the distance between Toeplitz operators and compact subset.In Chapter 4,the equivalent characterization that Toeplitz operators with positive measure symbols on Bergman Spaces induced by regular-weight on annu-lus belongs to the Schatten-p class are obtained by using Berezin transformation.