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A new characterization of generators of differentiable semigroups. (English) Zbl 1028.47030
Summary: Let $$A:D(A)\subseteq X\to X$$ be the infinitesimal generator of a $$C_0$$-semigroup of contractions $$\{S(t); t\geq 0\}$$. We prove that $$\{S(t); t\geq 0\}$$ is differentiable if and only if, for each $$\alpha(0,1)$$, there exists $$\lim_{n\to\infty}A(I-{t\over n}A)^{-n}$$ in the norm topology of $$C([\alpha,1/ \alpha];{\mathfrak L}(X))$$. A consequence concerning analytic $$C_0$$-semigroups of contractions is also included.
##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 47B44 Linear accretive operators, dissipative operators, etc.
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