文摘
We study the convolution operators class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304899&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304899&_rdoc=1&_issn=0022247X&md5=a25c443017cb970514c9aa2ea22b29fd" title="Click to view the MathML source">Tμclass="mathContainer hidden">class="mathCode"> which are tauberian as operators acting on the group algebras class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304899&_mathId=si2.gif&_user=111111111&_pii=S0022247X16304899&_rdoc=1&_issn=0022247X&md5=6fdcc134bd3ce5599b5bd288d5270311" title="Click to view the MathML source">L1(G)class="mathContainer hidden">class="mathCode">, where G is a locally compact abelian group and μ is a complex Borel measure on G. We show that these operators are invertible when G is non-compact, and that they are Fredholm when they have closed range and G is compact. In the remaining case, when G is compact and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304899&_mathId=si3.gif&_user=111111111&_pii=S0022247X16304899&_rdoc=1&_issn=0022247X&md5=4ff70627db1d28b75f268bc03ecd8d7d" title="Click to view the MathML source">R(Tμ)class="mathContainer hidden">class="mathCode"> is not assumed to be closed, we prove that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304899&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304899&_rdoc=1&_issn=0022247X&md5=a25c443017cb970514c9aa2ea22b29fd" title="Click to view the MathML source">Tμclass="mathContainer hidden">class="mathCode"> is Fredholm when the singular continuous part of μ with respect to the Haar measure on G is zero.