On an abstract evolution equation with a spectral operator of scalar type

Abstract

It is shown that the weak solutions of the evolution equation y′(t)=Ay(t), t∈[0,T) (0<T≤∞), where A is a spectral operator of scalar type in a complex Banach space X, defined by Ball (1977), are given by the formula y(t)=e tAf, t∈[0,T), with the exponentials understood in the sense of the operational calculus for such operators and the set of the initial values, f's, being ∩ 0≤t<TD(e tA), that is, the largest possible such a set in X.