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 B π of Bessel vectors for π is dense in H , then for any vector x ∈ H the analysis operator Θ x makes sense as a densely defined operator from B π to ℓ 2 ( G )‐space. Two vectors x and y are called π‐orthogonal if the range spaces of Θ x and Θ y are orthogonal, and they are π‐weakly equivalent if the closures of the ranges of Θ x and Θ 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 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 2 (ℝ d ) if and only if the corresponding adjoint Gabor sequence is ℓ 2 ‐linearly independent. Some other applications are also discussed.