This is an announcement for the paper "Weak compactness of operators acting on o-O type spaces" by Karl-Mikael Perfekt. 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. The main result characterizes the weakly compact operators T in terms of a certain norm naturally attached to M, weaker than the M-norm. 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. Archive classification: math.FA math.CV Remarks: 12 pages Submitted from: karlmikp@math.ntnu.no The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1405.0502 or http://arXiv.org/abs/1405.0502