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Notes on analytical convoluted \(C\)-semigroups. (English) Zbl 1258.47061
The author presents some structural properties of exponentially bounded analytic convoluted \(C\)-semigroups and states a version of Kato’s analyticity criterion for such a class of operator semigroups. His results are a refinement of Proposition 3.7(a) of R. deLaubenfels [Tokyo J. Math. 15, No. 1, 17–38 (1992; Zbl 0782.47036)] and Theorem 10 of M. Kostić and S. Pilipović [J. Math. Anal. Appl. 338, No. 2, 1224–1242 (2008; Zbl 1171.47036)]. The main result establishes spectral characterizations of the integral generator of an analytic convoluted \(C\)-semigroup and shows that such characterizations still hold for an arbitrary subgenerator under a certain condition.

MSC:
47D06 One-parameter semigroups and linear evolution equations
47D09 Operator sine and cosine functions and higher-order Cauchy problems
47D60 \(C\)-semigroups, regularized semigroups
47D62 Integrated semigroups
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