  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 $
(\epsilon,A)- $approximate orthogonality and $ (\epsilon,A)- $approximate
orthogonality in the sense of Chmieli$\acute{n}$ski and establish a relation
between them. We also characterize $ (\epsilon,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.
