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

Abstract

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 H, we define (ϵ,A)-approximate orthogonality and (ϵ,A)-approximate orthogonality in the sense of Chmieliński and establish a relation between them. We also characterize (ϵ,A)-approximate orthogonality in the sense of Chmieliński 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 is illustrated by extending some of the previous results obtained by various authors in the setting of Hilbert spaces.