Research on operators with natural spectrum

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We gave a sufficient condition for operating functions defined on a certain Banach function space to be only Lipschitz functions. Operating functions on non-trivial Banach function al-gebras or spaces need not be Lipschitz, but sutisifies strong continuity property. This means implicitly that it is hard to characterize the Gelfand space for a Banach function algebra in terms of operating functions.Let M(G) be the measure algebra on a non-discrete locally compact abelian group G and NS(G) denote the set of all measures in M(G) with natura1 spectrum. Then, NS(G) is not closed under addition and NS(G) + L^1(G) = M(G) holds if G is not compact. Let M_0(G) be a closed subalgebra of M(G) which consisit of all measurs whose Fuorier-Stieltjes transforms vanish at infinity and NS_0(G) denotes the subset of M_0(G) whose element have natura1 spectra. If G is compact NS_0(G) coincides with the Apostol algebra of M_0(G), which is not the case for non-compact G.There exists a measure mu in NS(G) such that mu is not decomposable as an operator on L^1(G). In particular, NS(G)+ NS(G)+ NS(G) = M(G) holds.The Apostol algebra of a Douglas algebra coincides with the algebra of all Q-continuous functions. Let H^* be the algebra of all bounded analytic functions on the open unit disk. Then NSH^*+ NSH^* = H^* holds, thus NSH^* is not closed under addition and it is rather large subset of H^*.