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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.

Weak compactness of operators acting on o–O type spaces