Let E and F be Banach spaces. The semigroup Φ+(E,F) of semi-Fredholm operators consists of the bounded linear mappings E→F with closed image and finite-dimensional kernel. By a well known result of Yood we have that T∈Φ+(E,F) if and only if for any bounded set B⊂E the condition TB relatively compact implies that B is relatively compact. Lebow and Schechter[10] gave a quantitative version of the above qualitative characterization, namely the operator T belongs to Φ+(E,F) if and only if there is c ≥ 0 such thatfor all bounded B⊂E. Here γ is the well known Hausdorff measure of non-compactnesswith BE the closed unit ball of E.

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 H 1 (X) (= H 1 (D, X)) and on the Bergman space B 1 (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 :

Any analytic map ϕ of the unit disc D into itself induces a composition operator Cϕ on BMOA, mapping f → f • ϕ, where BMOA is the Banach space of analytic functions f : D → C whose boundary values have bounded mean oscillation on the unit circle. We show that Cϕ is weakly compact on BMOA precisely when it is compact on BMOA, thus solving a question initially posed by Tjani and by Bourdon, Cima and Matheson in the special case of VMOA. As a crucial step of our argument we simplify the compactness criterion due to Smith for Cϕ on BMOA and show that his condition on the Nevanlinna counting function alone characterizes compactness. Additional equivalent compactness criteria are established. Furthermore, we prove the unexpected result that compactness of Cϕ on VMOA implies compactness even from the Bloch space into VMOA.

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