  This paper explores the concept of approximate Birkhoff-James orthogonality
in the context of operators on semi-Hilbert spaces. These spaces are generated
by positive semi-definite sesquilinear forms. We delve into the fundamental
properties of this concept and provide several characterizations of it. Using
innovative arguments, we extend a widely known result initially proposed by
Magajna in [J. London. Math. Soc., 1993]. Additionally, we improve a recent
result by Sen and Paul in [Math. Slovaca, 2023] regarding a characterization of
approximate numerical radius orthogonality of two semi-Hilbert space operators,
such that one of them is $A$-positive. Here, $A$ is assumed to be a positive
semi-definite operator.

Functional Analysis