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$$C$$-semigroups and the Cauchy problem. (English) Zbl 0895.47029
Summary: We extend the definition of generator to $$C$$-semigroups that may not be exponentially bounded, where the range of $$C$$ may not be dense. We then characterize linear operators, $$A$$, for which the associated abstract Cauchy problem has a unique solution, for every initial value in the domain of another operator, $$B$$, without assuming that the domain of $$A$$ is dense, or the solutions are exponentially bounded. We also give Hille-Yosida type characterizations of generators that may not be densely defined.

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations
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