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

2019/04/23 by G. Pérez-Villalón, Perez-Villalon, Gerardo
Mathematics · Physics and Astronomy · #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics #Quantum Mechanics and Applications

paper · pdf · doi:10.48550/arxiv.1904.10457

Abstract

We study the bounded endomorphisms of \ℓ_N²(G)=\ℓ²(G)\× \…\n\×\ℓ²(G) that commute with translations, where G is a discrete\nabelian group. It is shown that they form a C*-algebra isomorphic to the\nC*-algebra of N\× N matrices with entries in L^\∞( widehatG),\nwhere widehatG is the dual space of G. Characterizations of when these\nendomorphisms are invertible, and expressions for their norms and for the norms\nof their inverses, are given. These results allow us to study Riesz systems\nthat arise from the action of G on a finite set of elements of a Hilbert\nspace.\n

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