On compactness of operators from Banach spaces of holomorphic functions to Banach spaces

We investigate a widely used application of compactness of bounded linear operators T: X(\BbbB) → Y, where X(\BbbB) is a Banach space of holomorphic functions on the open unit ball \BbbB ⊂ C ^N and Y is a Banach space. In particular, we show that compactness of the operator when X(\BbbB) is not reflexive, is not a sufficient condition for the property that every bounded sequence (fn) _n∈ _N in X(\BbbB) such that fn → 0 with respect to the compact open topology as n → ∞, implies that T(fn) → 0 with respect to the norm of Y as n → ∞.