On \(B\)-bounded semigroups as a generalization of \(C_0\)-semigroups
Let \(X,Z\) be Banach spaces, \(A:D(A)\subset X\to X\), \(B:D(B) \subset X\to Z\) be linear operators. The author obtains a new characterization theorem for the so-called \(B\)-quasi bounded semigroup of linear operators \(\{y (t)\}_{t\geq 0}\) with \(y(0)=B\) in the sense of the work of \textit{A. Belleni-Morante} [Ann. Mat. Pura Appl., IV. Ser. 170, 359-376 (1996; Zbl 0882.47012), \textit{J. Banasiak}, Ann. Mat. Pura Appl. IV. Ser. 175, 307-326 (1998; Zbl 1005.47042)], under the condition (i): \(D(A)\subset D(B)\subset X\) and the condition (ii)': for some \(\omega\in\mathbb{R}\) the map \((\lambda I-A):D (A)\to D(B)\) is bijective for all \(\lambda >\omega\) where \(D_B(A):=\{f\in D(A): Af\in D(B)\}\). The author gives his proof via a constructive procedure which essentially coincides with Yosida's procedure when \(B=I\), and thus allows to construct this \(\{y(t)\}_{t\geq 0}\) under weaker assumptions than those of J. Banasiak (loc. cit.). Secondly, the author proves a new generation theorem for the representation \(y(t)f= \exp(t\overline {\mathcal A})Bf\). From this theorem he deduces Banasiak's generation theorem (loc. cit.) but under the weaker assumption (ii)'.
scientific article; zbMATH DE number 1453202