A representation and the norm of Toeplitz operators

S.Q. Zhu, A.A. Stoorvogel

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    1 Citation (Scopus)

    Abstract

    The Toeplitz operator has been used in system and control theory for quite a long time. Recently, it played a conspoicuous in
    H∞-control
    theory. One of the important properties of the Toeplitz operator is that its norm is identical to the norm of the Laurent operator with the same symbol. The original proof of this property relies on some advanced tools in operator theory. In this paper, for Toeplitz operators with symbols consisting of an infinite-dimensional stable part and a finite-dimensional unstable part, an elementary and self-contained proof of this property is given. Our proof is based on a representation of the Toeplitz operator presented in this paper and the well known fact that an inner matrix defines an isometry. The representation presented in this paper gives insight into the structure of the Toeplitz operator. A further application of this representation is also presented.
    Original languageEnglish
    Pages (from-to)409-413
    JournalSystems and Control Letters
    Volume12
    Issue number5
    DOIs
    Publication statusPublished - 1989

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