Zero-two law for cosine families

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    Abstract

    Abstract. For $(C(t))_{t\geq 0}$ being a strongly continuous cosine family on a Banach space, we show that the estimate $\lim \sup_{t\rightarrow 0+} \|C(t) - I\| < 2$ implies that $C(t)$ converges to $I$ in the operator norm. This implication has become known as the zero-two law. We further prove that the stronger assumption of $sup_{t\geq 0} \|C(t) - I\| < 2$ yields that $C(t) = I$ for all $t \geq 0 $. For discrete cosine families, the assumption $\sup_{n∈N} \|C(n) - I\| \leq r < \frac{3}{2}$ yields that $C(n) = I$ for all $n\in N$. For $r \geq \frac{3}{2}$, this assertion does no longer hold.
    Original languageUndefined
    Pages (from-to)559-569
    Number of pages11
    JournalJournal of evolution equations
    Volume15
    Issue number3
    DOIs
    Publication statusPublished - Sep 2015

    Keywords

    • EWI-26531
    • MSC-47D09 MSC-47D06
    • Cosine families
    • IR-98683
    • Zero-two law
    • Semigroup of operators
    • METIS-315075
    • Zero-one law

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