Operator ideal norms on L-p

Let p be a real number such that p is an element of (1, +infinity) and its conjugate exponent q not equal 4, 6, 8... We prove that for an operator T d efined 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 id eal norm. We also show that this result fails to be true in the remaining c ases 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.