## 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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