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

  • G. Perez-VillalonEmail author
Original Paper


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.


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

Mathematics Subject Classification

47L25 43A99 46L99 



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


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Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.Departamento de Matematica Aplicada a las TICUPMMadridSpain

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