Inverse operator of the generator of a C0-semigroup

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

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    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 languageUndefined
    Pages (from-to)1095-1110
    Number of pages16
    JournalSbornik : mathematics
    Issue number8
    Publication statusPublished - 2007


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

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