Birkhoff-James orthogonality in certain tensor products of Banach spaces II

Abstract

In this article, we discuss the relationship between Birkhoff-James orthogonality of elementary tensors in the space $L^{p}(\mu)\otimes^{\Delta_{p}}X,\; (1\leq p<\infty)$ with the individual elements in their respective spaces, where $X$ is a Banach space whose norm is Fr$\acute{e}chet$ differentiable and $\Delta_{p}$ is the natural norm induced by $L^{p}(\mu,X)$. In order to study the said relationship, we first provide some characterizations of Birkhoff-James orthogonality of elements in the Lebesgue-Bochner space $L^{p}(\mu,X)$.