Concerning the domains of generators of linear semigroups
J. W. Spellmann · Pacific Journal of Mathematics · 1970
Let S denote a Banach space over the real numbers.Let A denote the infinitesimal generator of a strongly continuous semigroup T of bounded linear transformations on S. It is T(x)pdx is in the domain a of A (denoted by D(A)) for each p in S and each nonnegative number interval [α, b].This paper develops sufficient conditions on nonnegative continuous functions / and on elements S b T(f(x))pdx be an a element of the^domain of A. 2. A change of variable technique* A change of variable theorem may sometimes be used to transform b T(f(x))pdx to \ d T(x)(f-y(x)pdx a Jcwhere f~ι denotes the inverse of /.This motivates the first theorem.THEOREM 1. Suppose pe S, 0 ^ c < d and h is a real valued function which has a continuous derivative on [c, d\.Then )T(d)p -h(c)T(c)p -['τ(x)h f (x)pdx .