Let p ∈ [1, +∞] such that its conjugate exponent q is not an even integer and let T be an operator defined on Lp(λ) with values in a Banach space. In this note we discuss how the image of the unit ball determines whether T belongs to some classes of operators such as operator ideals or the class of representable operators. We also study the monotonicity of these properties, proving that a Banach space is Cisomorphic to a subspace of an Lq
space if and only if the representability of every operator on Lp is monotone with respect to the image of the unit ball.Plan Nacional I+D+I (Ministerio de Ciencia y Tecnología)Junta de Andalucí
