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On the exponentially bounded C-semigroups. (English) Zbl 0631.47029
If C is an injective bounded operator with dense range on a Banach space, one defines an exponentially bounded C-semigroup as a strongly continuous family of bounded operators, S(t), $$t\geq 0$$, such that $$S(t+s)C=S(t)S(s)$$, $$S(0)=C$$, and $$\| S(t)\| \leq Me^{at}$$ [E. B. Davies and M. M. N. Pang, The Cauchy problem and a generalization of the Hille-Yosida theorem (to appear)]. The C-complete infinitesimal generator (C-c.i.g.) of S(t) is the closure $$\bar G$$ of the “derivative” at $$t=0$$ of $$C^{-1}S(t)$$. Representation theorems of S(t) in terms of $$\bar G$$ are proved. Also, necessary and sufficient conditions (generalizing those for $$(C_ 0)$$-semigroups) for a closed operator to be a C-c.i.g. are given. It is shown that, if the abstract Cauchy problem: $$u'(t)=Au(t)$$, $$u(0)=x$$, where A and C commute, has a unique solution with $$\| u(t)\| \leq Me^{at}\| C^{-1}x\|$$ for all $$x\in CD(A)$$, and if CD(A) is a core of A, then A is a C-c.i.g. (The converse of a result by Davies-Pang.) Finally, connections with semigroups of growth order $$\alpha$$ are established.
Reviewer: N.Angelescu

MSC:
 47D03 Groups and semigroups of linear operators
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