Frame duality properties for projective unitary representations

Abstract

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.