Generation results for \(B\)-bounded semigroups
\(B\)-semigroups, first defined by \textit{A. Bellini-Morante} [Ann. Mod. Pura Appl., IV. Ser. 170, 359-376 (1996; Zbl 0882.47012)] and \(B\)-quasi-semigroups are solutions of evolution equation \[ Y(t) f= Bf+ \int^t_0 Y(s) Af ds, \] where \(A\), \(B\) are (in general unbounded) linear operators on a Banach space \(X\), fulfilling some regularity conditions. The aim is to characterize \(B\)-quasi-semigroups and their generators by generators of a related operator semigroup on a suitable completion \(X_B\) of \(X\) defined by \(B\). Having constructed the completion \(X_B\) and suitable extensions of \(A\) to the closure of \(D(A)\) in \(X_B\) the characterization of the generators is achieved in Theorem 4.1. These results are applied to semigroups defined by the adjoint operator \(A^*\), where the underlying Banach spaces are assumed to be reflexive and \(B\) is bounded.
scientific article; zbMATH DE number 1471609