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

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