Frame duality properties for projective unitary representations

Let pi be a projective unitary representation of a countable group G on a separable Hilbert space H. If the set B-pi of Bessel vectors for pi is dense in H, then for any vector x is an element of H the analysis operator Theta(x) makes sense as a densely defined operator from B-pi to l(2)(G)-space. Two vectors x and y are called pi-orthogonal if the range spaces of Theta(x) and Theta(y) are orthogonal, and they are pi-weakly equivalent if the closures of the ranges of Theta(x) and Theta(y) are the same. These properties are characterized in terms of the commutant of the representation. It is proved that a natural geometric invariant ( the orthogonality index) of the representation agrees with the cyclic multiplicity of the commutant of pi(G). These results are then applied to Gabor systems. A sample result is an alternate proof of the known theorem that a Gabor sequence is complete in L-2(R-d) if and only if the corresponding adjoint Gabor sequence is l(2)-linearly independent. Some other applications are also discussed.