# Inverse operator of the generator of a C0-semigroup

A.M. Gomilko, Heiko J. Zwart, Y Tomilov

4 Citations (Scopus)

## 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 1095-1110 16 Sbornik : mathematics 198 8 https://doi.org/10.1070/SM2007v198n08ABEH003874 Published - 2007

## Keywords

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