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

TL;DR

This paper generalizes approximate orthogonality concepts in semi-Hilbertian spaces, characterizes these notions for bounded operators, and extends symmetry properties of operators, thereby broadening the theoretical framework of operator orthogonality.

Contribution

It introduces generalized approximate orthogonality in semi-Hilbertian structures and characterizes it for specific classes of operators, extending existing Hilbert space results.

Findings

Established relations between different notions of approximate orthogonality.

Characterized approximate orthogonality for A-bounded and compact operators.

Extended symmetry properties of operators in semi-Hilbertian spaces.

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 $ \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.