Study on Birkhoff orthogonality and symmetry of matrix operators

Abstract

We focus on the problem of generalized orthogonality of matrix operators in operator spaces. Especially, on ℬ(l1n,lpn)(1≤p≤∞){\mathcal{ {\mathcal B} }}\left({l}_{1}^{n},{l}_{p}^{n})\left(1\le p\le \infty ), we characterize Birkhoff orthogonal elements of a certain class of matrix operators and point out the conditions for matrix operators which satisfy the Bhatia-Šemrl property. Furthermore, we give some conclusions which are related to the Bhatia-Šemrl property. In a certain class of matrix operator space, such as ℬ(l∞n){\mathcal{ {\mathcal B} }}\left({l}_{\infty }^{n}), the properties of the left and right symmetry are discussed. Moreover, the equivalence condition for the left symmetry of Birkhoff orthogonality of matrix operators on ℬ(lpn)(1<p<∞){\mathcal{ {\mathcal B} }}\left({l}_{p}^{n})\left(1\lt p\lt \infty ) is obtained.

Keywords

• birkhoff orthogonality
• matrix operator
• bhatia-šemrl property
• left symmetry
• right symmetry
• 47a30
• 47a99
• 46e15