Frame Representations For Group-Like Unitary Operator Systems

A group-like unitary system U is a set of unitary operators such that the group generated by the system is contained in double-strok T signU, where double-strok T sign denotes the unit circle. Every frame representation for a group-like unitary system is (unitarily equivalent to) a subrepresentation of its left regular representation and the norm of a normalized tight frame vector determines the redundancy of the representation. In the case that a group-like unitary system admits enough Bessel vectors, the commutant of the system can be characterized in terms of the analysis operators associated with all the Bessel vectors. This allows us to define a natural quantity (the frame redundancy) for the system which will determine when the system admits a cyclic vector. A simple application of this leads to an elementary proof to the well-known time-frequency density theorem in Gabor analysis