  We prove several results asserting that the action of a Banach-Lie group on
Hilbert spaces of holomorphic sections of a holomorphic Hilbert space bundle
over a complex Banach manifold is multiplicity free. These results require the
existence of compatible antiholomorphic bundle maps and certain multiplicity
freeness assumptions for stabilizer groups. For the group action on the base,
the notion of an $(S,\sigma)$-weakly visible action (generalizing T.Koboyashi's
visible actions) provides an effective way to express the assumptions in an
economical fashion. In particular, we derive a version for group actions on
homogeneous bundles for larger groups. We illustrate these general results by
several examples related to operator groups and von Neumann algebras.
