### Abstract

Let $A$ be the generator of a uniformly bounded $C_0$-semigroup in a Banach space $X$ such that $A$ has a trivial kernel and a dense range. The question whether $A^{-1}$ is a generator of a $C_0$-semigroup is considered. It is shown that the answer is negative in general for $X = \ell_p$, $p \in (1, 2) \cap (2,\infty)$. In the case when $X$ is a Hilbert space it is proved that there exist $C_0$-semigroups ($e^{tA})$, $t > 0$, of arbitrarily slow growth at infinity such that the densely defined operator $A^{-1}$ is not the generator of a $C_0$-semigroup.

Original language | Undefined |
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Pages (from-to) | 1095-1110 |

Number of pages | 16 |

Journal | Sbornik : mathematics |

Volume | 198 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2007 |

### Keywords

- EWI-11685
- METIS-247093
- MSC-47D06

## Cite this

Gomilko, A. M., Zwart, H. J., & Tomilov, Y. (2007). Inverse operator of the generator of a C

_{0}-semigroup.*Sbornik : mathematics*,*198*(8), 1095-1110. https://doi.org/10.1070/SM2007v198n08ABEH003874