Abstract
Given a discrete group and a unitary representation on a Hilbert space \(\mathcal {H}\), we prove that the notions of operator Bracket map and Gramian coincide on a dense set of \(\mathcal {H}\). As a consequence, combining this result with known frame theory, we can recover all previous Bracket characterizations of Riesz and frame sequences generated by a single element under a unitary representation.
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Acknowledgements
D. Barbieri was supported by a Marie Curie Intra European Fellowship (Prop. N. 626055) within the 7th European Community Framework Programme. D. Barbieri and E. Hernández were supported by Grants MTM2013-40945-P and MTM2016-76566-P (Ministerio de Economía y Competitividad, Spain). V. Paternostro by Grants UBACyT 2002013010022BA and 20020150200037BA, and CONICET-PIP 11220110101018.
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Barbieri, D., Hernández, E. & Paternostro, V. Group Riesz and frame sequences: the Bracket and the Gramian. Collect. Math. 69, 221–236 (2018). https://doi.org/10.1007/s13348-017-0202-x
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DOI: https://doi.org/10.1007/s13348-017-0202-x