Multipliers with Natural Local Spectra on Commutative Banach Algebras

Several notions from the abstract spectral theory of bounded linear operators on Banach spaces are investigated and characterized in the context of multipliers on a semi-simple commutative Banach algebra. Particular emphasis is given to the determination of the local spectra of such multipliers in connection with Dunfordʹs property (C), Bishopʹs property (β), and decomposability in the sense of Foiaş. The strongest results are obtained for regular Tauberian Banach algebras with approximate units and for multipliers whose Gelfand transforms on the spectrum of the Banach algebra vanish at infinity. The general theory is then applied to convolution operators induced by measures on a locally compact abelian groupG. Our results give new insight into the spectral theory of convolution operators on the group algebraL1(G) and on the measure algebraM(G). In particular, we identify large classes of measures for which the corresponding convolution operators have excellent spectral properties. We also obtain a number of negative results such as examples of convolution operators onL1(G) without natural local spectra, but with natural spectrum in the sense of Zafran.