Concave and positive summing norms for operators on Lp

Abstract.

Let p be a real number such that $p \in [1,+ \infty]$ and its conjugate exponent p' is not an even integer and let T be an operator defined 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 determines the representability of the operator.