On the numerical radius parallelism and the numerical radius Birkhoff orthogonality

TL;DR

This paper extends the concepts of numerical radius parallelism and Birkhoff orthogonality from Hilbert spaces to normed spaces, analyzing their properties and interrelations.

Contribution

It introduces generalized definitions of these concepts for normed spaces and investigates their fundamental properties and connections.

Findings

Numerical radius parallelism is not transitive.

Numerical radius Birkhoff orthogonality is neither left nor right additive.

Characterizations of both concepts are provided.

Abstract

In this paper, we generalize the notions of numerical radius parallelism and numerical radius Birkhoff orthogonality, originally formulated for operators on Hilbert spaces, to operators on normed spaces. We then proceed to demonstrate their fundamental properties. Notably, our findings reveal that numerical radius parallelism lacks transitivity, and numerical radius Birkhoff orthogonality is neither left nor right additive. Additionally, we offer characterizations for both concepts. Furthermore, we establish a connection between numerical radius parallelism and numerical radius Birkhoff orthogonality.