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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
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[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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