Exponentially Bounded $C$-Semigroups and Integrated Semigroups
Isao Miyadera, Naoki Tanaka · Tokyo Journal of Mathematics · 1989
Let $X$ be a Banach space.We denote by $B(X)$ the set of all bounded linear operators from $X$ into itself.there are $M\geqq 0$ and $a\in R\equiv(-\infty, \infty)$ such thatSimilarly as in the case of $\overline{R(C)}=X$ (see [4]), we see that $L_{\lambda}$ is injective for $\lambda>a$ and the closed linear operator $Z$ defined by (0.4)Recently, Davies and Pang [4] introduced the notion of an exponen- tially bounded C-semigroup under the assumption that $R(C)$ is dense in $X$ and gave a characterization of the generator of an exponentially bounded C-semigroup.(See [3] also.) Later, the authors [6, 9, 11] gave a char- acterization of the complete infinitesimal generator of an exponentially