Multi-frame vectors for unitary systems

In this paper, the set of all complete multi-normalized tight frame vectors NF ^r (U) with multiplicity r and the set of all complete multi-frame vectors F ^r (U) with multiplicity r for a system U of unitary operators acting on a separable Hilbert space are characterized in terms of co-isometries and surjective operators in (U), the set of all operators which locally commute with U at Ψ ^r , a fixed complete wandering r-tuple for U. Then we study the linear combinations of multi-frame vectors for U and establish some conditions under which these combinations are still the same type of multi-frame vectors for U. Finally, we establish some interesting properties for multi-frame vectors when U is a unitary group. All these results have potential applications in the theory of multi-Gabor systems and multi-wavelet systems.