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.
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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.
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Communicated by Keith Taylor.
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Perez-Villalon, G. Discrete convolution operators and Riesz systems generated by actions of abelian groups. Ann. Funct. Anal. 11, 285–297 (2020). https://doi.org/10.1007/s43034-019-00003-6
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DOI: https://doi.org/10.1007/s43034-019-00003-6