Title

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 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 ℓ2(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 L2( d) if and only if the corresponding adjoint Gabor sequence is ℓ2- linearly independent. Some other applications are also discussed. © 2008 London Mathematical Society.

Publication Date

1-1-2008

Publication Title

Bulletin of the London Mathematical Society

Volume

40

Issue

4

Number of Pages

685-695

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1112/blms/bdn049

Socpus ID

47949084497 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/47949084497

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