Abstract

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

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