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A new characterization of \(B\)-bounded semigroups with application to implicit evolution equations. (English) Zbl 1002.47018

Summary: We consider the one-parameter family of linear operators that A. Belleni Morante recently introduced and called \(B\)-bounded semigroups. We first determine all the properties possessed by a couple \((A,B)\) of operators if they generate a \(B\)-bounded semigroup \((Y(t))_{t\geq 0}\). Then we determine the simplest further property of the couple \((A,B)\) which can assure the existence of a \(C_0\)-semigroup \((T(t))_{t \geq 0}\) such that for all \(t \geq 0\), \(f \in D(B)\) we can write \(Y(t)f = T(t)Bf\). Furthermore, we compare our result with the previous ones and finally we show how our method allows to improve the theory developed by Banasiak for solving implicit evolution equations.

MSC:

47D06 One-parameter semigroups and linear evolution equations
34G10 Linear differential equations in abstract spaces
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