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Discrete convolution operators and Riesz systems generated by actions of abelian groups

  • G. Perez-VillalonEmail author
Original Paper

Abstract

We study the bounded endomorphisms of \(\ell ^2(G)\times \dots \times \ell ^2(G)=\ell _{N}^2(G)\) that commute with translations, where G is a discrete abelian group. It is shown that they form a C*-algebra isomorphic to the C*-algebra of \(N\times N\) matrices with entries in \(L^\infty ({\widehat{G}})\), where \({\widehat{G}}\) is the dual space of G. Characterizations of when these endomorphisms are invertible, and expressions for their norms and for the norms of their inverses, are given. These results allow us to study Riesz systems that arise from the action of G on a finite set of elements of a Hilbert space.

Keywords

Discrete convolution C*-algebra Multiplier Shift-invariant space Discrete abelian group and Riesz basis 

Mathematics Subject Classification

47L25 43A99 46L99 

Notes

Acknowledgements

The author wishes to thank Antonio García and Miguel Angel Hernández Medina for the stimulating conversations on this work, their suggestions and constructive comments.

References

  1. 1.
    Aldroubi, A.: Oblique proyections in atomic spaces. Proc. Am. Math. Soc. 124, 2051–2060 (1996)CrossRefGoogle Scholar
  2. 2.
    Barbieri, D., Hernández, E., Parcet, J.: Riesz and frame systems generated by unitary actions of discrete groups. Appl. Comput. Harmon. Anal. 39, 369–399 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Barbieri, D., Hernández, E., Paternostro, V.: Invariant spaces under unitary representations of discrete groups. arXiv: 1811.02993 (2018)
  4. 4.
    Benedetto, J.J.: Harmonic Analysis and Applications. CRC Press, Boca Raton (1996)zbMATHGoogle Scholar
  5. 5.
    Benzi, M., Boito, P.: Decay properties for functions of matrices over C*-algebras. Linear Algebra Appl. 456, 174–198 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Blackadar, B.: Operator Algebras: Theory of C*-Algebras and von Neumann Algebras. Encyclopaedia of Mathematical Sciences, vol. 122. Springer, Berlin (2006)CrossRefGoogle Scholar
  7. 7.
    Cabrelli, C., Paternostro, V.: Shift-invariant spaces on LCA groups. J. Funct. Anal. 258, 2034–2059 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2003)CrossRefGoogle Scholar
  9. 9.
    Folland, G.B.: A Course in Abstract Harmonic Analysis. CRC Press, Boca Raton (1995)zbMATHGoogle Scholar
  10. 10.
    García, A.G., Pérez-Villalón, G.: Riesz bases associated with regular representations of semidirect product groups. Banach J. Math. Anal. (To appear) Google Scholar
  11. 11.
    García, A.G., Hernández-Medina, M.A., Pérez-Villalón, G.: Convolution systems on discrete Abelian groups as a unifying strategy in sampling theory (Preprint) Google Scholar
  12. 12.
    Goodman, T.N., Lee, S.L., Tang, W.S.: Wavelet bases for a set of commuting unitary operators. Adv. Comput. Math. 1, 109–126 (1993)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hernández, E., Sikic, H., Weiss, G., Wilson, E.: Cyclic subspaces for unitary representations of LCA groups; generalized Zak transform. Colloq. Math. 118, 313–332 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jia, R.Q., Micchelli, C.A.: Using the Refinement Equations for the Construction of Pre-waveles II: Powers of Two. In Curves and Surfaces, pp. 209–246. Academic Press, Boston (1991)Google Scholar
  15. 15.
    Kailath, T.: Linear Systems. Prentice Hall, Berlin (1980)zbMATHGoogle Scholar
  16. 16.
    Larsen, R.: An Introduction to the Theory of Multipliers. Springer, New York (1971)CrossRefGoogle Scholar
  17. 17.
    Ron, A., Shen, Z.: Frames and stable bases for shift-invariant subspaces of \(L^2({\mathbb{R}}^d)\). Can. J. Math. 47, 1051–1094 (1995)CrossRefGoogle Scholar
  18. 18.
    Rudin, W.: Fourier Analysis on Groups. Interscience Publishers, NewYork (1962)zbMATHGoogle Scholar

Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.Departamento de Matematica Aplicada a las TICUPMMadridSpain

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