## 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.

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 language | English |
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Pages (from-to) | 409-413 |

Journal | Systems and Control Letters |

Volume | 12 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1989 |