Chen, Ze Qian; Liu, Hong Mei Vector-valued Laplace transforms and right continuous integral semigroups. (Chinese) Zbl 0910.47032 Acta Math. Sci. (Chin. Ed.) 16, No. 1, 15-22 (1996). Let \(\mu\) be an L-S measure defined on \({\mathcal B}(0,\infty)\). The Laplace transform of \(\mu\) is defined by \({\mathcal L}(\mu)(\lambda)= \int^\infty_0 e^{-\lambda t}\mu(dt)\). If any \(\nu\in C^\infty(0,\infty; E)\) satisfies: for some \(M\geq 0\) \[ \|\nu^{(k)}(\lambda)\|\leq (-1)^kM\cdot{\mathcal L}(\mu)^{(k)}(\lambda),\quad \lambda>0,\quad k=0,1,2,\dots, \] there exists \(\varphi\in L^\infty(\mu, E)\) such that \[ \nu(\lambda)= \int^\infty_0 e^{-\lambda t}\varphi(t)\mu(dt),\quad \lambda>0. \] Then \(E\) is called to have Widder property related to \(\mu\). If for every L-S measure on \({\mathcal B}(0,\infty)\) with finite-valued Laplace transform, \(E\) has Widder property, then \(E\) is called to have Widder property.The author proves: Let \(E\) be a Banach space. Suppose that \(\mu\) is an L–S measure on \({\mathcal B}(0,\infty)\) with \({\mathcal L}(\mu)(\lambda)<\infty\) \((\lambda>0)\).Then the following conditions are equivalent:i) \(E\) has Widder property related to \(\mu\);ii) \(E\) has Radon-Nikodým property related to \(\mu\). Reviewer: Wu Liangsen (Shanghai) Cited in 1 Document MSC: 47D06 One-parameter semigroups and linear evolution equations 44A10 Laplace transform 34G10 Linear differential equations in abstract spaces 47N20 Applications of operator theory to differential and integral equations 46B22 Radon-Nikodým, Kreĭn-Milman and related properties Keywords:right continuous integral semigroups; L-S measure; Laplace transform; Widder property; Radon-Nikodým property PDFBibTeX XMLCite \textit{Z. Q. Chen} and \textit{H. M. Liu}, Acta Math. Sci. (Chin. Ed.) 16, No. 1, 15--22 (1996; Zbl 0910.47032)