SPECTRAL DISTRIBUTIONS AND GENERALIZATION OF STONE THEOREM

In this paper, we introduce the notion of spectral distribution which is a generalization of the spectral measure. This notion is closely related to distribution semigroups and generalized scalar operators. The associated operator (called the momentum of the spectral distribution) has a functional calculus defined for infinitely differentiable functions on the real line. Our main result says that A generating a smooth distribution group of order k is equivalent to having a k-times integrated group that are O(\t\k) or iA being the momentum of a spectral distribution of degree k. We obtain the standard version of Stone's theorem as a special case of this result. The standard properties of a functional calculus together with spectral mapping theorem are derived. Finally, we show how the degree of a spectral distribution is related to the degree of the nilpotent operators which separate its momentum from its scalar part