Weak compactness of operators acting on o–O type spaces

Abstract

We consider operators T : M 0 → Z and T : M → Z , where Z is a Banach space and ( M 0 , M ) is a pair of Banach spaces belonging to a general construction in which M is defined by a ‘big- O’ condition and M 0 is given by the corresponding ‘little- o’ condition. Prototype examples of such spaces M are given by ℓ ∞ , weighted spaces of functions or their derivatives, bounded mean oscillation, Lipschitz–Hölder 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 M-norm, and shows that weakly compact operators T : M 0 → Z are already quite close to being completely continuous. Further, we develop a method to extract c 0 -subsequences from sequences in M 0 . 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.