For any natural numberk, we characterize once-integrated Laplace transforms ofO((1+t)k) andO(t k) Banach-space-valued functions. We use this to give Hille-Yosida type characterizations of generators of polynomially bounded strongly continuous semigroups, related families of operators, and solutions of the abstract Cauchy problem.
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Much of this work was done while the third author was visiting Ohio University. He would like to thank Ohio University, and Professors deLaubenfels and Swardson, for their hospitality and support.
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DeLaubenfels, R., Huang, Z., Wang, S. et al. Laplace transforms of polynomially bounded vector-valued functions and semigroups of operators. Israel J. Math. 98, 189–207 (1997). https://doi.org/10.1007/BF02937334
- Banach Space
- Nonnegative Integer
- Mild Solution
- Closed Operator
- Functional Calculus