The Characterization And Representation For The Generalized Inverse A_T,S⁽²⁾ Over Banach Space

The theory of operators originated from the beginning of 20th century and devel-oped rapidly in the early 30s of 20th century since its extensive applications in math-ematics and other fields. As one of important branches of the operators, the theory ofgeneralized inverse of operators possessed well developed and perfected from 30s in lastcentury. Recently, the generalized inverse A_T,S² of operators over Banach spaces has be-come a hot topic. Since such known several generalized inverses as the Moore-Penroseinverse, the group inverse, the Drazin inverse and so on, are the generalized inverseA_T,S², we can obtain the properties of every concrete generalized inverse and find thecommonness of every known generalized inverse.In this thesis, we mainly consider the characterization and representation of thegeneralized inverse AT(2,)S over Banach spaces. The thesis is divided into three chapters.In chapter 1, we give some required preparative knowledge including basic concepts,lemmas and general symbols. Also, the main theorems of the thesis are presented.In chapter 2, we first give some characterizations of the generalized inverse A_T,S²over Banach spaces. We obtain two equivalent characterizations of the generalizedinverse AT(2,)S from the {1,5} inverse of AG, and uniformly get a representation of A_T,S²related to the {1,5} inverse. Secondly, using the operator block technique, we give acharacterization of AT(2,)S over Hilbert space.In chapter 3, we mainly present some representations of the generalized inverseA_T,S² over Banach spaces. Utilizing the operator block technique and some concreteknowledge of operators, we give the integral and limit representation of the generalizedinverse AT(2,)S over Banach spaces. Also, a representation and application of A_T,S² overHilbert space is presented.