Frechet differentiability and quasi-polyhedrality in spaces of operators

Let $X, Y$ be infinite dimensional, Banach spaces. Let $\mathcal{L}(X, Y)$ be the space of bounded operators . Motivated by the fact that smoothness of norm in the higher duals of even order of a Banach space can lead to Frechet differentiability, we exhibit classes of Banach spaces $X, Y$ where very smooth points (i.e., smooth points that remain smooth in the bidual) in the space of compact operators $\mathcal{K}(X, Y)$ are Frechet smooth in $\mathcal{L}(X, Y)$ and hence in $\mathcal{K}(X, Y)$.