Weyl's theorems for some classes of operators

We introduce the class of operators on Banach spaces having property (H) and study Weyl's theorems, and related results for operators which satisfy this property. We show that a-Weyl's theorem holds for every decomposable operator having property (H). We also show that a-Weyl's theorem hold,; for every multiplier T of a commutative semi-simple regular Tauberian Banach algebra. In particular every convolution operator T, of a group algebra L-1(G), G a locally compact abelian group, satisfies a-Weyl's theorem. Similar results are given for multipliers of other important commutative Banach algebras.