Convolution-dominated integral operators

Gero Fendler, Karlheinz Gröchenig, and Michael Leinert. "Convolution-dominated integral operators." Banach Center Publications 89.1 (2010): 121-127. <http://eudml.org/doc/286343>.

@article{GeroFendler2010,
abstract = {For a locally compact group G we consider the algebra CD(G) of convolution-dominated operators on L²(G), where an operator A: L²(G) → L²(G) is called convolution-dominated if there exists a ∈ L¹(G) such that for all f ∈ L²(G) |Af(x)| ≤ a⋆|f|(x), for almost all x ∈ G. (1) The case of discrete groups was treated in previous publications [fgl08a, fgl08]. For non-discrete groups we investigate a subalgebra of regular convolution-dominated operators generated by product convolution operators, where the products are restricted to those given by multiplication with left uniformly continuous functions. This algebra, $CD_\{reg\}(G)$, is canonically isomorphic to a twisted L¹-algebra. For amenable G that is rigidly symmetric as a discrete group we show the following result: An element of $CD_\{reg\}(G)$ is invertible in $CD_\{reg\}(G)$ if and only if it is invertible as a bounded operator on L²(G). This report is about work in progress. Complete details and further results will be given in a paper still in preparation.},
author = {Gero Fendler, Karlheinz Gröchenig, Michael Leinert},
journal = {Banach Center Publications},
keywords = {convolution-dominated operators; inverse-closed subalgebras; symmetry},
language = {eng},
number = {1},
pages = {121-127},
title = {Convolution-dominated integral operators},
url = {http://eudml.org/doc/286343},
volume = {89},
year = {2010},
}

TY - JOUR
AU - Gero Fendler
AU - Karlheinz Gröchenig
AU - Michael Leinert
TI - Convolution-dominated integral operators
JO - Banach Center Publications
PY - 2010
VL - 89
IS - 1
SP - 121
EP - 127
AB - For a locally compact group G we consider the algebra CD(G) of convolution-dominated operators on L²(G), where an operator A: L²(G) → L²(G) is called convolution-dominated if there exists a ∈ L¹(G) such that for all f ∈ L²(G) |Af(x)| ≤ a⋆|f|(x), for almost all x ∈ G. (1) The case of discrete groups was treated in previous publications [fgl08a, fgl08]. For non-discrete groups we investigate a subalgebra of regular convolution-dominated operators generated by product convolution operators, where the products are restricted to those given by multiplication with left uniformly continuous functions. This algebra, $CD_{reg}(G)$, is canonically isomorphic to a twisted L¹-algebra. For amenable G that is rigidly symmetric as a discrete group we show the following result: An element of $CD_{reg}(G)$ is invertible in $CD_{reg}(G)$ if and only if it is invertible as a bounded operator on L²(G). This report is about work in progress. Complete details and further results will be given in a paper still in preparation.
LA - eng
KW - convolution-dominated operators; inverse-closed subalgebras; symmetry
UR - http://eudml.org/doc/286343
ER -