Inverse operator of the generator of a C₀-semigroup

Volume 198 (2007)
Number 8
Pages 1095-1110

A M Gomilko^*, ^a H Zwart, ^b Yu Tomilov^c

^a Institute of Hydromechanics of NAS of Ukraine, Kiev, Ukraine
^b Twente University, Enschede, Netherlands
^c Nikolaus Copernicus University, Torun, Poland

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 $X=\ell^p$ , $p\in(1,2)\cup(2,\infty)$ . In the case when $ X$ is a Hilbert space it is proved that there exist $ C_0$ -semigroups $(e^{tA})$ , $t\ge0$ , of arbitrarily slow growth at infinity such that the densely defined operator $A^{-1}$ is not the generator of a $ C_0$ -semigroup.

Bibliography: 19 titles.

DOI	10.1070/SM2007v198n08ABEH003874
Citation	A M Gomilko, H Zwart, Yu Tomilov, "Inverse operator of the generator of a C₀-semigroup", SB MATH, 2007, 198 (8), 1095-1110.
Classification	AMS MSC: Primary: 47D60, Secondary: 47D06
Full Text:	PDF file (357 kB)
References:	PDF file (131 kB) HTML file

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