On approximate orthogonality and symmetry of operators in semi-Hilbertian structure

The purpose of the article is to generalize the concept of approximate Birkhoff-James orthogonality, in the semi-Hilbertian structure. Given a positive operator $A$ on a Hilbert space $\mathbb{H},$ we define $(ϵ,A)-$approximate orthogonality and $(ϵ,A)-$approximate orthogonality in the sense of Chmieli$\acute{n}$ski and establish a relation between them. We also characterize $(ϵ,A)-$approximate orthogonality in the sense of Chmieli$\acute{n}$ski for $A$-bounded and $A$-bounded compact operators. We further generalize the concept of right symmetric and left symmetric operators on a Hilbert space. The utility of these notions are illustrated by extending some of the previous results obtained by various authors in the setting of Hilbert spaces.