Miyadera, Isao A generalization of the Hille-Yosida theorem. (English) Zbl 0683.47027 Proc. Japan Acad., Ser. A 64, No. 7, 223-226 (1988). 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 \(C_ 1=C|_{\overline{D(A)}}\). Cited in 1 ReviewCited in 6 Documents MSC: 47D03 Groups and semigroups of linear operators Keywords:Hille-Yosida theorem; integrated C-semigroup PDF BibTeX XML Cite \textit{I. Miyadera}, Proc. Japan Acad., Ser. A 64, No. 7, 223--226 (1988; Zbl 0683.47027) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.