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A generalization of the Hille-Yosida theorem. (English) Zbl 0683.47027
Summary: Let $$C\in B(X)$$ be injective. In this paper we introduce the notion of integrated C-semigroups [cf. W. Arendt, Isr. J. Math. 59, 327-352 (1987; Zbl 0637.44001)] and prove the following theorems.
Theorem 1. An operator A is the generator of an integrated C-semigroup $$\{$$ U(t); $$t\geq 0\}$$ on X satisfying $(1.1)\quad \| U(t+h)- U(t)\| \leq M he^{a(t+h)}\quad for\quad t,h\geq 0,$ where $$M>0$$ and $$a\geq 0$$ are constants, if and only if A satisfies the following properties (A1)-(A3) and it is maximal with respect to (A1)-(3):
(A1) A is a closed linear operator and $$\lambda$$-A is injective for $$\lambda >a;$$
(A2) $$D((\lambda -A)^{-m})\supset R(C)$$ and $$\| (\lambda -A)^{-m} C\| \leq M/(\lambda -a)^ m$$ for $$\lambda >a$$ and $$m\geq 1;$$
(A3) Cx$$\in D(A)$$ and $$ACx=CAx$$ for $$x\in D(A).$$
Theorem 2. If A satisfies the equivalent conditions of Theorem 1, then the part of A in $$\overline{D(A)}$$ is the generator of a $$C_ 1$$- semigroup $$\{S_ 1(t)$$; $$t\geq 0\}$$ on $$\overline{D(A)}$$ satisfying $$\| S_ 1(t)x\| \leq Me^{at}\| x\|$$ for $$x\in \overline{D(A)}$$ and $$t\geq 0$$, where $$C_ 1=C|_{\overline{D(A)}}$$.

##### MSC:
 47D03 Groups and semigroups of linear operators
##### Keywords:
Hille-Yosida theorem; integrated C-semigroup
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##### References:
 [1] W. Arendt: Israel J. Math., 59, 327-352 (1987). · Zbl 0637.44001 · doi:10.1007/BF02774144 [2] E. B. Davies and M. M. Pang: Proc. London Math. Soc, 55, 181-208 (1987). · Zbl 0651.47026 · doi:10.1112/plms/s3-55.1.181 [3] E. Hille and R. S. Phillips: Amer. Math. Soc. Colloq. Publ., vol. 31 (1957). · Zbl 0078.10004 · www.ams.org [4] N. Tanaka and I. Miyadera: Exponentially bounded C-semigroups and integrated semigroups (preprint). · Zbl 0702.47028 · doi:10.3836/tjm/1270133551
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