In page 67 of Topology and Analysis by Booss and Bleecker, it is claimed that any Hilbert bundle is topologically trivial. Clearly, any smooth Hilbert bundle over a smooth manifold is topologically trivial, but it appears to be no reason to believe that this trivialization is smooth. So, my questions are:

• Are there known conditions on the manifold and on the Hilbert space to guarantee that such topological trivialization is actually smooth?

• Are there known counterexamples showing that such smooth trivialization is impossible in the general case?

EDIT: The definition of smooth Hilbert bundle is the one defined in 2.1 of the following paper: https://arxiv.org/pdf/1004.4863.pdf