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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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