The Sz.-Nagy-Foiaş model
theory uses generalized restricted shifts as canonical models for contractions in
Hubert space. This paper considers a class of contractive and unitary perturbations
of a generalized restricted shift acting on a Sz.-Nagy-Foiaş space generated by an
analytic operator-valued function S(z) whose values are contractions on a separable
Hubert space. The spectra and characteristic functions of the perturbations are
computed and related to the original operator. When the perturbation is unitary, a
unitary equivalence to multiplication by ei𝜃 on L2(μ), for an operator-valued measure
μ, is given.
In [2], D. N. Clark studied the one-dimensional unitary perturbations of restricted
shifts in H2, i.e. S(z) a scalar inner function, and in [3], he announced results for the
case where S(z) is an arbitrary scalar (characteristic) function. The general
unitary perturbations are implicit in work of de Branges and Rovnyak [1],
though in the context of the de Branges-Rovnyak model theory rather than
the Sz.-Nagy-Foiaş. P. A. Fuhrmann [5] considered a class of completely
nonunitary and unitary perturbations for the case of S(z) an inner function
on a finite-dimensional space. In this case, the maps considered are always
compact perturbations. Here we generalize results of [5] and [2]. We will
follow the general outline of [5], and we correct a minor error occurring
there so our description of the perturbations in the general case is actually
as sharp as in the finite-dimensional case. As was pointed out in [5], these
perturbations have applications to the theory of stability of linear control
systems.
