Abstract
For a locally compact group G we consider the algebra CD(G) of convolution dominated operators on \({L^{2}(G)}\): An operator \({A:L^2(G) \to L^2(G)}\) is called convolution dominated if there exists \({a\in L^1(G)}\) such that for all \({f \in L^2(G)}\)
In the case of discrete groups those operators can be dealt with quite sufficiently if the group in question is rigidly symmetric. For non-discrete groups we investigate the subalgebra of regular convolution dominated operators \({CD_{reg}(G)}\). For amenable G which is rigidly symmetric as a discrete group we show that any element of \({CD_{reg}(G)}\) is invertible in \({CD_{reg}(G)}\) if it is invertible as a bounded operator on \({L^2(G)}\). We give an example of a symmetric group E for which the convolution dominated operators are not inverse-closed in the bounded operators on \({L^2(E)}\).
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Fendler, G., Leinert, M. On Convolution Dominated Operators. Integr. Equ. Oper. Theory 86, 209–230 (2016). https://doi.org/10.1007/s00020-016-2319-9
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DOI: https://doi.org/10.1007/s00020-016-2319-9