Weak compactness of operators acting on o–O type spaces

perfekt, karl-mikael;

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 oscillation, Lipschitz–H¨older spaces, and many others. The main result characterizes the weakly compact operators T in terms of a certain norm naturally attached to M, weaker than the Mnorm, and shows that weakly compact operators T : M0 → Z are already quite close to being completely continuous. Further, we develop a method to extract c0-subsequences from sequences in M0. Applications are given to the characterizations of the weakly compact composition and Volterra-type integral operators on weighted spaces of analytic functions, BMOA, VMOA, and
the Bloch space.