On wandering vector multipliers for unitary groups

A wandering vector multiplier is a unitary operator which maps the set of wandering vectors for a unitary system acting on a separable Hilbert space $\mathcal H$ into itself. It is proved that the wandering vector multipliers for a unitary group form a group, which gives a positive answer for a problem of Han and Larson. Furthermore, non-abelian unitary groups of order 6 are considered. We prove that the wandering vector multipliers of such a unitary group can not generate $\mathcal B(\mathcal H)$. This negatively answers another of their problems.