We consider operators T : M0 → Z and T : M → Z, where Z is a Banach space and (M0,M) is a pair of Banach spaces belonging to a general construction in which M is defined by a ‘big-O’ condition and M0 is given by the corresponding ‘little-o’ condition. Prototype examples of such spaces M are given by ∞, weighted spaces of functions or their derivatives, bounded mean os…