Collectanea Mathematica

, Volume 69, Issue 2, pp 221–236 | Cite as

Group Riesz and frame sequences: the Bracket and the Gramian

  • Davide Barbieri
  • Eugenio Hernández
  • Victoria Paternostro
Article
  • 41 Downloads

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.

Keywords

Riesz and frame sequences Group von Neumann algebras Invariant subspaces Bracket map Gramian operator 

Mathematics Subject Classification

42C15 43A30 47C15 

Notes

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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Copyright information

© Universitat de Barcelona 2017

Authors and Affiliations

  • Davide Barbieri
    • 1
  • Eugenio Hernández
    • 1
  • Victoria Paternostro
    • 2
  1. 1.Universidad Autónoma de MadridMadridSpain
  2. 2.Universidad de Buenos Aires and IMAS-CONICET, Consejo Nacional de Investigaciones Científicas y TécnicasBuenos AiresArgentina

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