## Abstract

This paper is concerned with self-adjoint Hilbert space operators T

that have a special displacement structure depending on a given contraction A.

It allows one to view the operator T as a compression of a self-adjoint Toeplitz

operator. The main issue is the following problem. Given a contraction A under

what conditions is a self-adjoint operator T such an A-structured operator.

The problem is solved for the case when the underlying Hilbert space is finite

dimensional and the contraction is stable, using the fact that in that case the

Sz.-Nagy-Foias characteristic function defined by the contraction A is a bi-inner

rational function. As an application, a formula is derived for the inverse of a

finite dimensional invertible self-adjoint A-structured operator.

that have a special displacement structure depending on a given contraction A.

It allows one to view the operator T as a compression of a self-adjoint Toeplitz

operator. The main issue is the following problem. Given a contraction A under

what conditions is a self-adjoint operator T such an A-structured operator.

The problem is solved for the case when the underlying Hilbert space is finite

dimensional and the contraction is stable, using the fact that in that case the

Sz.-Nagy-Foias characteristic function defined by the contraction A is a bi-inner

rational function. As an application, a formula is derived for the inverse of a

finite dimensional invertible self-adjoint A-structured operator.

Original language | English |
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Pages (from-to) | 253-270 |

Number of pages | 18 |

Journal | Pure and Applied Functional Analysis |

Volume | 7 |

Issue number | 1 |

Publication status | Published - Feb 2022 |

## Keywords

- Toeplitz operators, stable contractions, displacement structure, contraction-structured operators, finite dimensional solutions, Sz.-Nagy-Foias characteristic function, inverse formula.