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On the generators of exponentially bounded C-semigroups. (English) Zbl 0617.47032
Let X be a Banach space and let $$C: X\to X$$ be an injective bounded linear operator with dense range. $$\{S(t): t\geq 0\}$$ is called an exponentially bounded C-semigroup if $$S(t)$$: $$X\to X$$, $$0\leq t<\infty$$, is a family of bounded linear operators satisfying
(1) $$S(t+s)C=S(t)S(s)$$ $$(t,s\geq 0)$$, $$S(0)=C,$$
(2) S(t)x is continuous in $$t\geq 0$$ for every $$x\in X$$ and
(3) there exist $$M\geq 0$$ and $$a\in (-\infty,\infty)$$ such that $$\| S(t)\| \leq M\exp (at)$$ $$(t\geq 0).$$
Define operators G and Z by $$Gx=\lim_{t-0+}(C^{-1}S(t)x-x)/t$$ for $$x\in D(G)=\{x\in R(C):\lim_{t+0+}(C^{-1}T(t)x-x)/t$$ exists$$\}$$ and $$Zx=(\lambda -L^{-1}C)x$$ for $$x\in D(Z)=\{x\in X:$$ $$Cx\in R(L_{\lambda})\}$$, where $$L_{\lambda}x= \int^{\infty}_{0}e^{- \lambda t}S(t)xdt$$ (x$$\in X)$$ and $$\lambda>a$$. G is densely defined in X and $$G\subset Z$$, and Z is a closed linear operator which is called the generator of $$\{S(t): t\geq 0\}$$. The following theorems are obtained:
Theorem 1. The closure $$\bar G$$ of G is a densely defined linear operator in X satisfying the following $$(a_ 1)-(a_ 4);$$
(a$$_ 1$$) $$\lambda-\bar G$$ is injective,
(a$$_ 2$$) $$D((\lambda-\bar G)^{-n})\supset R(C)$$ $$(n\geq 1,\lambda >a),$$
(a$$_ 3$$) $$\| (\lambda -\bar G)^{-n}C\| \leq M/(\lambda -a)^ n$$ $$(n\geq 1,\lambda >a),$$
(a$$_ 4)$$ $$(\lambda-\bar G)^{-1}Cx= C(\lambda-\bar G)^{-1}x$$ $$(x\in D((\lambda-\bar G)^{-1})$$, $$\lambda>a).$$
Theorem 2. If T is a densely defined closed linear operator in X satisfying (a$$_ 1$$)-(a$$_ 4$$) in Theorem 1, then $$C^{-1}TC$$ is the generator of an exponentially bounded C-semigroup $$\{S(t); t\geq 0\}$$ satisfying $$\| S(t)\| \leq M\exp(at)$$ $$(t\geq 0)$$. These theorems give a generalization of the Hille-Yosida theorem.

##### MSC:
 47D03 Groups and semigroups of linear operators
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##### References:
 [1] E. B. Davies and M. M. H. Pang: The Cauchy problem and a generalization of the Hille-Yosida theorem (to appear). · Zbl 0651.47026 [2] E. Hille and R. S. Phillips: Functional Analysis and Semi-Groups. Amer. Math. Soc. Coll. Publ., vol. 31 (1957). · Zbl 0078.10004 [3] K. Yosida: Functional Analysis. 6th ed., Springer-Verlag (1980).
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