Toeplitz operators from a higher point of view: contraction-structured operators

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.