A generalization of the Widder-Arendt theorem

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 an element ω of R∪{−∞} such that −λ is m-integrable for all λ > ω, where −λ is defined by −λ(t) = exp(−λt) 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 : L1(R+,m) → E and all λ > ω; here L1(R+,m) denotes the Lebesgue space based on m. We use this result to characterize pseudo-resolvents with values in a Banach algebra, satisfying a growth condition of Hille–Yosida type.Wojciech Chojnack