On α-nuclear operators with applications to vector-valued function spaces

Let X be a Banach space and let Y be a closed subspace of a Banach space Z. Let α   be a tensor norm. Our main result is as follows. Assume that X⁎X⁎ or Z⁎Z⁎ has the approximation property. If there is a bounded linear extension operator from Y⁎Y⁎ to Z⁎Z⁎, then any bounded linear operator T:X→YT:X→Y is α-nuclear whenever T is α-nuclear from X to Z  . Using this result, we characterize the space Cp(Ω,X)Cp(Ω,X) (respectively, UCp(Ω,X)UCp(Ω,X)) of continuous functions from a compact Hausdorff space Ω into a Banach space X whose range is p-compact (respectively, unconditionally p-compact).