Integral Equations and Operator Theory

, Volume 61, Issue 4, pp 493–509

Convolution-Dominated Operators on Discrete Groups

  • Gero Fendler
  • Karlheinz Gröchenig
  • Michael Leinert
Article

DOI: 10.1007/s00020-008-1604-7

Cite this article as:
Fendler, G., Gröchenig, K. & Leinert, M. Integr. equ. oper. theory (2008) 61: 493. doi:10.1007/s00020-008-1604-7

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 xG 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 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 L1-algebras and the symmetry of group algebras.

Keywords.

Groups of polynomial growthconvolutionsymmetric Banach algebrasinverse-closedgeneralized L1-algebra

Mathematics Subject Classification (2000).

Primary 47B35Secondary 43A20

Copyright information

© Birkhaueser 2008

Authors and Affiliations

  • Gero Fendler
    • 1
  • Karlheinz Gröchenig
    • 2
  • Michael Leinert
    • 3
  1. 1.LaudenbachGermany
  2. 2.Fakultät für MathematikUniversität WienWienAustria
  3. 3.Institut für Angewandte MathematikUniversität HeidelbergHeidelbergGermany