Tauberian convolution operators acting on L1(G)

We study the convolution operators TμTμ which are tauberian as operators acting on the group algebras L1(G)L1(G), 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 R(Tμ)R(Tμ) is not assumed to be closed, we prove that TμTμ is Fredholm when the singular continuous part of μ with respect to the Haar measure on G is zero.