Operator ideal norms on $L^{p}$

Let $p$ be a real number such that $p \in (1,+\infty )$ and its conjugate exponent $q\not =4,6,8\ldots$. We prove that for an operator $T$ defined on $L^{p}(\lambda )$ with values in a Banach space, the image of the unit ball determines whether $T$ belongs to any operator ideal and its operator ideal norm. We also show that this result fails to be true in the remaining cases of $p$. Finally we prove that when the result holds in finite dimension, the map which associates to the image of the unit ball the operator ideal norm is continuous with respect to the Hausdorff metric.