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.
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机译：我们根据拉普拉斯变换理论建立了Widder-Arendt定理的推广。给定一个Banach空间E，对所有非负数的集合R +进行非负Borel测度m，并对R∪{-∞}的元素ω使得-λ对于所有λ>ω都是m可积的，其中对于所有t∈R +，−λ由−λ（t）= exp（-λt）定义，我们的概括给出了函数r的内在描述：（ω，∞）→E可以表示为r（λ）= T （−λ）对于一些有界线性算子T：L1（R +，m）→E且所有λ>ω; L1（R +，m）表示基于m的勒贝格空间。我们使用此结果来表征满足Banach代数生长条件的Banach代数中的伪溶剂。
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