On approximate $A$-seminorm and $A$-numerical radius orthogonality of operators

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.