## Journal of the Mathematical Society of Japan

### Perturbation of non-exponentially-bounded $a$-times integrated $C$-semigroups

#### Abstract

Let $T(\cdot)$ be a (not necessarily exponentially bounded, not necessarily nondegenerate)$\alpha$-times integrated $C$-semigroup and let $-B$ be the generator of a $(C_{0})-$ group $S(\cdot)$ commuting with $T(\cdot)$ and $C$. Under suitable conditions on $T(\cdot)$ and $S(\cdot)$ we prove the existence of an $\alpha$-times integrated $C$-semigroup $V(\cdot)$, which has generator $\overline{A+B}$ provided that $T(\cdot)$ is nondegenerate and has generator $A$. Explicit expressions of $V(\cdot)$ in terms of $T(\cdot)$ and $S(\cdot)$ are obtained. In particular, when $B$ is bounded, $V(\cdot)$ can be constructed by means of a series in terms of $T(\cdot)$ and powers of $B$.

#### Article information

Source
J. Math. Soc. Japan, Volume 55, Number 4 (2003), 1115-1136.

Dates
First available in Project Euclid: 3 October 2007

https://projecteuclid.org/euclid.jmsj/1191418767

Digital Object Identifier
doi:10.2969/jmsj/1191418767

Mathematical Reviews number (MathSciNet)
MR2003763

Zentralblatt MATH identifier
1067.47057

#### Citation

LI, Yuan-Chuan; SHAW, Sen-Yen. Perturbation of non-exponentially-bounded $a$ -times integrated $C$ -semigroups. J. Math. Soc. Japan 55 (2003), no. 4, 1115--1136. doi:10.2969/jmsj/1191418767. https://projecteuclid.org/euclid.jmsj/1191418767