Frame Duality Properties for Projective Unitary Representations

Let π be a projective unitary representation of a countable group G on a separable Hilbert space H. If the set Bπ of Bessel vectors for π is dense in H, then for any vector x ∈ H the analysis operator Θx makes sense as a densely defined operator from Bπ to ` (G)-space. Two vectors x and y are called π-orthogonal if the range spaces of Θx and Θy are orthogonal, and they are π weakly equivalent if the closures of the ranges of Θx and Θy 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 π(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(R ) if and only if the corresponding adjoint Gabor sequence is `-linearly independent. Some other applications are also discussed.