Study on Birkhoff orthogonality and symmetry of matrix operators

Abstract

Abstract We focus on the problem of generalized orthogonality of matrix operators in operator spaces. Especially, on ℬ ( l 1 n , l p n ) ( 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 ( l p n ) ( 1 < p < ∞ ) {\mathcal{ {\mathcal B} }}\left({l}_{p}^{n})\left(1\lt p\lt \infty ) is obtained.