Decomposable multipliers and applications to harmonic analysis
For a multiplier on a semisimple commutative Banach algebra, the decomposability in the sense of Foiaş will be related to certain continuity properties and growth conditions of its Gelfand transform on the spectrum of the multiplier algebra. If the multiplier algebra is regular, then all multipliers will be seen to be decomposable. In general, an important tool will be the hull-kernel topology on the spectrum of the typically nonregular multiplier algebra. Our investigation involves various closed subalgebras of the multiplier algebra and includes perturbation results of Wiener-Pitt type for the invertibility of multipliers. Under suitable topological assumptions on the spectrum of the given Banach algebra, we shall characterize decomposable multipliers, Riesz multipliers, and multipliers with natural or countable spectrum. Most of these results are new even in the case of convolution operators given by measures on a locally compact abelian group. We shall obtain various classes of measures for which the corresponding convolution operators are decomposable both on the measure algebra and on the group algebra. Moreover, the spectral properties of a convolution operator will be related to the behavior of the Fourier-Stieltjes transform of the underlying measure on the dual group and on the spectrum of the measure algebra. Finally, it will be shown that the decomposability of convolution operators behaves nicely with respect to absolute continuity and singularity of measures.