Concave and positive summing norms for operators on L-p

Let p be a real number such that p is an element of [1,+ infinity] and its conjugate exponent p' is not an even integer and let T be an operator defin ed on L-p(lambda) with values in a Banach space. We prove that the image of the unit ball determines if T belongs to the space of concave and positive summing operators. We also prove that the image of the unit ball determine s the representability of the operator.