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

Abstract

This paper explores the concept of approximate Birkhoff–Jamesorthogonality in the context of operators on semi-Hilbert spaces. These spacesare generated by positive semi-definite sesquilinear forms. We delve into the fundamentalproperties of this concept and provide several characterizations of it.Using innovative arguments, we extend a widely known result initially proposedby Magajna [17]. Additionally, we improve a recent result by Sen and Paul [24] regardinga characterization of approximate numerical radius orthogonality of twosemi-Hilbert space operators, such that one of them is A-positive. Here, A isassumed to be a positive semi-definite operator.