Wandering vector multipliers for unitary groups

A wandering vector multiplier is a unitary operator which maps the set of w andering vectors for a unitary system into itself. A special case of unitar y system is a discrete unitary group. We prove that for many (and perhaps a ll) discrete unitary groups, the set of wandering vector multipliers is its elf a group. We completely characterize the wandering vector multipliers fo r abelian and ICC unitary groups. Some characterizations of special wanderi ng vector multipliers are obtained for other cases. In particular, there ar e simple characterizations for diagonal and permutation wandering vector mu ltipliers. Similar results remain valid for irrational rotation unitary sys tems. We also obtain some results concerning the wandering vector multiplie rs for those unitary systems which are the ordered products of two unitary groups. There are applications to wavelet systems.