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2003 ◽  
Vol 46 (2) ◽  
pp. 357-372 ◽  
Author(s):  
Valentin Keyantuo ◽  
Claus Müller ◽  
Peter Vieten

AbstractWe establish two characterizations of local Laplace transforms in Banach spaces. The first result follows the classic approach of Widder, while the second is in terms of vector-valued moment sequences. As a consequence, we derive characterizations of nilpotent semigroups.AMS 2000 Mathematics subject classification: Primary 44A10; 47D03


2002 ◽  
Vol 45 (1) ◽  
pp. 161-179 ◽  
Author(s):  
Wojciech Chojnacki

AbstractWe 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


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