Local Spectral Theory for Multipliers and Convolution Operators

This note centers around the class D(G) of decomposable measures on a locally compact abelian group G. This class is a large subalgebra of the measure algebra M(G), has excellent spectral properties, and is related to a number of concepts from commutative harmonic analysis. The discussion of D(G) in Section 1 is to illustrate this point. The main features of this class are collected in Theorem 1.1, which improves recent results from [19] and includes some new properties related to the involution of M(G). We shall present a different approach, which avoids previous tools like the hull-kernel topology [19], [23] or the spectral theory of several commuting operators [2], [10]. Theorem 1.1 is an immediate consequence of the spectral theory for multipliers on Banach algebras in Section 3. The emphasis is here on multipliers with the decomposition property (δ) from [3], which characterizes the quotients of decomposable operators. We show that multipliers with property (δ) behave very nicely and coincide with the strongly decomposable multipliers under fairly mild conditions on the underlying Banach algebra. Our results on multipliers require some new results on general local spectral theory in Section 2, which should be of independent interest. In particular, some basic results on decomposable operators from [8] and [28] will be extended to the more flexible case of quotients and restrictions of decomposable operators in the spirit of [3].