Approximate Roberts directional orthogonalities

Abstract

We define two types of approximate Roberts orthogonality with direction in the framework of a complex normed space. We examine their geometrical properties and demonstrate that the notion of \(\epsilon \)-approximate directional orthogonality is weaker than that of \(\epsilon \)-approximate orthogonality. Concerning the approximate Birkhoff orthogonality, we talk about the connection between them. Also, we provide the notion of an approximation Roberts directional orthogonality set and analyze the geometric characteristics of these sets. Furthermore, we discuss approximate orthogonality preserving mapping.