Frame duality properties for projective unitary representations
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Let be a projective unitary representation of a countable group G on a separable Hilbert space H. If the set B of Bessel vectors for is dense in H, then for any vector x H the analysis operator x makes sense as a densely defined operator from B to 2(G)-space. Two vectors x and y are called -orthogonal if the range spaces of x and y are orthogonal, and they are -weakly equivalent if the closures of the ranges of x and 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 (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 L2( d) if and only if the corresponding adjoint Gabor sequence is 2- linearly independent. Some other applications are also discussed. 2008 London Mathematical Society.