LAUSR

Abstract We study the convolution operators T μ which are tauberian as operators acting on the group algebras L 1 ( 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 ( T μ ) is not assumed to be closed, we prove that T μ is Fredholm when the singular continuous part of μ with respect to the Haar measure on G is zero.