We establish a generalization of the Widder-Arendt theorem from Laplace transform theory. Given a Banach space E, a non-negative Borel measure m on the set R + of all non-negative numbers, and for all t ∈ R + , our generalization gives an intrinsic description of functions r : (ω, ∞) → E that can be represented as r(λ) = T ( −λ ) for some bounded linear operator T : L 1 (R + , m) → E and all λ > ω; here L 1 (R + , m) denotes the Lebesgue space based on m. We use this result to characterize

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... o-resolvents with values in a Banach algebra, satisfying a growth condition of Hille-Yosida type.