Study On Generalized Orthogonality And Symmetry Of Matrix Operators

The theory of orthogonality has greatly promoted the development of the geometric theory on inner product spaces.Scholars have introduced generalized orthogonality in general normed spaces,such as Roberts orthogonality,Birkhoff orthogonality,etc,so that the methods and ideas of orthogonal theory in inner product spaces to be extended to general normed spaces.In 1999,Bhatia R and Semrl P pointed out that for a matrix operator S on B(H)(H is a Hilbert space),if we have S orthogonal to an operator T,then there exists x such that S attains its norm at x and Sx is orthogonal to Tx.Later on,this property of operators on a certain space was called the Bhatia-Semrl property.Scholars have carried out extensive research on the Bhatia-Semrl properties in some classical operator spaces and have obtained rich results.In addition,in the sense of Birkhoff orthogonality,the left and right symmetry of operators has been a extensively concerned in recent years,but the results in the study of matrix operators still can be deepened.This paper focuses on the Bhatia-Semrl property and the left and right symmetry of matrix operators,which are studied as follows:The orthogonality and the Bhatia-Semrl property of matrix operators are studied with the help of matrix theory and generalized orthogonality theory.First,on B(lln,lpn)(1≤p≤∞),we characterize Birkhoff orthogonal elements of matrix operators,and obtain the conditions for a matrix operators satisfy the Bhatia-Semrl property.Furthermore,we get the relations between the vector orthogonality on lpn(1≤p≤∞)and the operator orthogonality on B(lln,lpn)(1≤p≤∞)both in the sense of Birkhoff orthogonality.Secondly,when S Birkhoff orthogonal to T is not necessarily hold,the sufficient conditions for Tx Birkhoff orthogonal to Sx and Tx ∈ Sx+(Tx∈Sx-)are given respectively,thus expanding the Bhatia-Semrl property.Moreover,the equivalence condition for the Birkhoff orthogonality of the matrix operator are obtained on the m×n real matrix space with Frobenius norm.Finally,the symmetry of matrix operators is studied in the Birkhoff orthogonal sense through the utilizing of operator theory.On B(X)and B(l∞n),where X is a strictly convex space,we explore the properties of the left and right symmetry of operators,and discuss the equivalence condition for left symmetry of matrix operators on B(lpn)(1<p<∞).Also,a sufficient condition for left symmetry of matrix operators on B(l∞2)is obtained.