Spectral decompositions and decomposable multipliers

A perturbation result for spectral decompositions of several commuting operators is developed. Roughly speaking, we prove that a composed system (S,T) consisting of commuting continuous linear Banach space operators admits spectral decompositions, if the restrictions of S onto and the quotients of S modulo the spectral subspaces of T admit spectral decompositions. In the second part it is shown that this result has a natural application to systems of multipliers on complex Banach algebras with bounded approximate identity.Dedicated to my teacher Professor H. G. Tillmann on his 60th birthday