An $\mathcal{H}_\infty$ calculus of admissible operators

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    Abstract

    Given a Hilbert space and the generator A of a strongly continuous, exponentially stable, semigroup on this Hilbert space. For any $g(-s) \in H_{\infty}$ we show that there exists an infinite-time admissible output operator $g(A)$. If $g$ is rational, then this operator is bounded, and equals the “normal‿ definition of $g(A)$. In particular, when $g(s) = 1/(s + a), a \in {\mathbb C}_0^+$ , then this admissible output operator equals $(I - A)^{-1}$. Although in general $g(A)$ may be unbounded, we always have that $g(A)$ multiplied by the semigroup is a bounded operator for every (strictly) positive time instant. Furthermore, when there exists an admissible output operator $C$ such that $(C, A)$ is exactly observable, then $g(A)$ is bounded for all $g$’s with $g(-s) \in H_{\infty}$.
    Original languageUndefined
    Title of host publicationProceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems, MTNS 2010
    Place of PublicationBudapest
    PublisherEötvös Loránd University
    Pages1679-1683
    Number of pages5
    ISBN (Print)978-963-311-370-7
    Publication statusPublished - Jul 2010
    Event19th International Symposium on Mathematical Theory of Networks and Systems, MTNS 2010 - Budapest, Hungary
    Duration: 5 Jul 20109 Jul 2010
    Conference number: 19

    Publication series

    Name
    PublisherEötvös Loránd University

    Conference

    Conference19th International Symposium on Mathematical Theory of Networks and Systems, MTNS 2010
    Abbreviated titleMTNS
    CountryHungary
    CityBudapest
    Period5/07/109/07/10

    Keywords

    • IR-75751
    • EWI-19383
    • MSC-93A25
    • METIS-275862

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