On the generators of exponentially bounded C-semigroups
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.
scientific article; zbMATH DE number 4000819