Convolution-Dominated Operators on Discrete Groups

Author： Fendler Gero  

Publisher： Springer Publishing Company

ISSN： 0378-620X

Source： Integral Equations and Operator Theory, Vol.61, Iss.4, 2008-08, pp. : 493-509

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Abstract

We study infinite matrices A indexed by a discrete group G that are dominated by a convolution operator in the sense that $$|(Ac)(x)| leq (a ast |c|)(x)$$ for x ∈ G and some $$a in ell^1(G)$$ . This class of “convolution-dominated” matrices forms a Banach-*-algebra contained in the algebra of bounded operators on l 2(G). Our main result shows that the inverse of a convolution-dominated matrix is again convolution-dominated, provided that G is amenable and rigidly symmetric. For abelian groups this result goes back to Gohberg, Baskakov, and others, for non-abelian groups completely different techniques are required, such as generalized L 1-algebras and the symmetry of group algebras.

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