Approximate Birkhoff-James orthogonality preserver on Lebesgue-Bochner spaces

TL;DR

This paper investigates the approximate Birkhoff-James orthogonality preservers in Lebesgue-Bochner spaces, demonstrating under certain conditions that these spaces do not possess Property P, which relates to approximate orthogonality preservation.

Contribution

It extends the understanding of approximate orthogonality preservation in vector-valued function spaces by establishing non-existence of Property P in Lebesgue-Bochner spaces under specific conditions.

Findings

Lebesgue-Bochner spaces $L^p(\mu,X)$ lack Property P under certain conditions.

The study clarifies the behavior of approximate orthogonality preservers in vector-valued integrable function spaces.

Abstract

In this article, we examine an approximate version of Koldobsky-Blanco-Turn\v{s}ek theorem (namely, Property P) in the space of vector-valued integrable functions. More precisely, we prove that the Lebesgue-Bochner spaces $L^p(\mu,X),\;(1\leq p<\infty)$, do not have Property P under certain conditions on $\mu$ and the Banach space $X$.