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


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