Some remarks on orthogonality of bounded linear operators

TL;DR

This paper investigates the relationship between operator orthogonality and element orthogonality in normed spaces, introduces Property Pn for Banach spaces, and examines its implications for polyhedral Banach spaces.

Contribution

It introduces Property Pn for Banach spaces and explores its connection with operator orthogonality, providing new insights into the structure of polyhedral Banach spaces.

Findings

Established conditions under which operator orthogonality implies element orthogonality.

Connected Property Pn with the orthogonality of bounded linear operators.

Analyzed Property Pn in various polyhedral Banach spaces.

Abstract

We explore the relation between the orthogonality of bounded linear operators in the space of operators and that of elements in the ground space. To be precise, we study if $ T, A \in \mathbb{L}(\mathbb{X}, \mathbb{Y}) $ satisfy $ T \bot_B A,$ then whether there exists $ x \in \mathbb{X} $ such that $ Tx\bot_B Ax$ with $ \|x\| =1, \|Tx\| = \|T\|$, where $\mathbb{X}, \mathbb{Y} $ are normed linear spaces. In this context, we introduce the notion of Property $ P_n $ for a Banach space and illustrate its connection with orthogonality of a bounded linear operator between Banach spaces. We further study Property $ P_n $ for various polyhedral Banach spaces.