Research On Generalized Orthogonal Problems In Operator Spaces

In the early 1980s,the concept of operator space gradually attracted the attention of peer scholars with the independent research of Wittstock and Paulsen.Therefore,to have a deeper and more detailed understanding of the geometric properties of operator space and to find the internal relationship between them has become a concerned research work in related fields for a period of time,with rich research contents and a lot of research achievements.In recent years,with the deepening of the research,the research results of generalized orthogonality in normed linear space have been very abundant and formed a system.The application of matrix theory in mathematics is very extensive,which attracts the attention of many scholars.At present,the generalized orthogonality of matrices is mostly discussed on the basis of the norm of matrices.However,as for operator space and matrix as elements in operator space,the relevant theories of Birkhoff orthogonality,Roberts orthogonality and isospheric orthogonality are seldom discussed,and need to be further explored.The related research work will provide us with the method and means to study the matrix operator from a special Angle.This paper explores some properties of generalized orthogonality in operator space by means of the related theories of generalized orthogonality in normed linear space.In particular,as an element in operator space,the matrix norm in this case is affected by the original space norm and image space norm,and then the generalized orthogonal problem of matrix is discussed,so as to provide different ideas and methods for the study of matrix in the future.By means of the research idea of generalized orthogonality in normed linear space,the generalized orthogonality problem of matrices as operators is discussed from the perspective of operator space.First,we study the necessary and sufficient conditions for two linear operators T₁ and T₂ on space(?)to satisfy the orthogonality of Birkhoff.We find the relationship between the Birkhoff orthogonality of two vectors in space(?)and the Birkhoff orthogonality of the matrix operator in space(?).Secondly,in the operator space formed by mapping between two lp Spaces,bothT₁ and T₂ are taken as n?n matrices,and the equivalent conditions for T₁ andT₂ to satisfy Birkhoff orthogonal,isospheric orthogonal and Roberts orthogonal are given.The equivalence conditions of Birkhoff orthogonality of operators in C^*-algebra are studied.