## The Cauchy problem and a generalization of the Hille-Yosida theorem.(English)Zbl 0651.47026

Let C be an injective bounded operator with dense range, on a Banach space. A family $$(S_ t)_{t\geq 0}$$ is an exponentially bounded C- semigroup if $$(S_ t)_{t\geq 0}$$ is a strongly continuous family of bounded operators satisfying $$S_ 0=C$$, $$S_{p+q}=S_ pS_ q(p,q\geq 0)$$, $$\| S_ t\| \leq Me^{\omega t}(t\geq 0).$$
In the first part of the paper the authors derive properties of the generator of an exponentially bounded C-semigroup, show in what sense the Cauchy problem may be solved for the generator, and prove a generation theorem, i.e., a generalization of the Hille-Yosida theorem.
In the second part the relation of exponentially bounded C-semigroups to other types of one-parameter semigroups is investigated. In particular it is shown that semigroups of order $$\alpha (>0)$$ are of this type, and that resolvent positive operators (on ordered Banach spaces) are generators of C-semigroups.
Reviewer: J.Voigt

### MSC:

 47D03 Groups and semigroups of linear operators 47B60 Linear operators on ordered spaces 34G10 Linear differential equations in abstract spaces
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