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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

We introduce the notion of orthogonal sets for Birkhoff orthogonality, which we will call Birkhoff orthogonal sets in this paper. As a generalization of orthogonal sets in Hilbert spaces, Birkhoff orthogonal sets are not necessarily linearly independent sets in finite-dimensional real normed spaces. We prove that the Birkhoff orthogonal set A = { x 1 , x 2 , ... , x n } (n ≥ 3) containing n − 3 right symmetric points is linearly independent in smooth normed spaces. In particular, we obtain similar results in strictly convex normed spaces when n = 3 and in both smooth and strictly convex normed spaces when n = 4 . These obtained results can be applied to the mutually Birkhoff orthogonal sets studied in recently. [ABSTRACT FROM AUTHOR]