Growth of semigroups in discrete and continuous time

Alexander Gomilko, Hans Zwart, Niels Besseling

    Research output: Contribution to journalArticleAcademicpeer-review

    3 Citations (Scopus)
    17 Downloads (Pure)


    We show that the growth rates of solutions of the abstract differential equations $\dot{x}(t)=Ax(t), \dot{x}(t)=A^{-1}x(t)$, and the difference equation $x_d(n+1)=(A+I)(A-1)^{-1}x_d(n)$ are closely related. Assuming that $A$ generates an exponentially stable semigroup, we show that on a general Banach space the lowest growth rate of the semigroup $\left(e^{A^{-1}t}\right)_{t\geq 0}$ is $O(\sqrt[4]{t})$, and for $\left((A+I)(A-I)^{-1}\right)^n$ it is $O(\sqrt[4]{n})$. The similarity in growth holds for all Banach spaces. In particular, for Hilbert spaces the best estimates are $O(\log(t))$ and $O(\log(n))$, respectively. Furthermore, we give conditions on $A$ such that the growth rate of $\left((A+I)(A-I)^{-1}\right)^n$ is $O(1)$, i.e., the operator is power bounded.
    Original languageEnglish
    Pages (from-to)273-292
    Number of pages20
    JournalStudia mathematica
    Issue number3
    Publication statusPublished - 2011


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