Small composition operators on analytic vector-valued function spaces

Abstract

Let φ be an analytic mapping of the unit disk D into itself. We characterize the weak compactness of the composition operator Cφ : f 7→ f ◦ φ on the vector-valued Hardy space H(X) (= H(D,X)) and on the Bergman space B1(X), where X is a Banach space. Reflexivity of X is a necessary condition for the weak compactness of Cφ in each case. Assuming this, the operator Cφ : H1(X)→ H(X) is weakly compact if and only if φ satisfies the Shapiro condition: Nφ(w) = o(1−|w|) as |w| → 1−, where Nφ stands for the Nevanlinna counting function of φ. This extends a previous result of Sarason in the scalar case. Similarly, Cφ is weakly compact on B1(X) if and only if the angular derivative condition lim|w|→1−(1− |φ(w)|)/(1− |w|) =∞ is satisfied. We also characterize the weak compactness of Cφ on vector-valued (little and big) Bloch spaces and on H∞(X). Finally, we find conditions for weak conditional compactness of Cφ on the above mentioned spaces of analytic vector-valued functions.