A GENERALIZATION OF THE WIDDER–ARENDT THEOREM

Abstract

We establish a generalization of the Widder–Arendt theorem from Laplace transform theory. Given a Banach space $E$, a non-negative Borel measure $\mea$ on the set $\Rplus$ of all non-negative numbers, and an element $\bnd$ of $\R\cup\{-\infty\}$ such that $\natres{-\coefl}$ is $\mea$-integrable for all $\coefl>\bnd$, where $\natres{-\coefl}$ is defined by $\natres{-\coefl}(t)=\exp(-\coefl t)$ for all $t\in\Rplus$, our generalization gives an intrinsic description of functions $\f\colon\Set\to E$ that can be represented as $\f(\coefl)=T(\natres{-\coefl})$ for some bounded linear operator $T\colon\Ma\to E$ and all $\coefl> \bnd$; here $\Ma$ denotes the Lebesgue space based on $\mea$. We use this result to characterize pseudo-resolvents with values in a Banach algebra, satisfying a growth condition of Hille–Yosida type.

AMS 2000 Mathematics subject classification: Primary 44A10; 47A10. Secondary 43A20; 46B22; 46G10; 46J25; 47D06