A finiteness theorem for holomorphic Banach bundles

Let E be a holomorphic Banach bundle over a compact complex manifold, which can be defined by a cocycle of holomorphic transition functions with values of the form id+K where K is compact. Assume that the characteristic fiber of E has the compact approximation property. Let n be the complex dimension of X and 0 <= q <= n. Then: If V -> X is a holomorphic vector bundle (of finite rank) with H-q (X, V)=0, then dim H-q (X, V circle times E)<infinity. In particular, if dim H-q (X, O)=0, then dim H-q (X, E)<infinity.