Vol. 63, No. 2, 1976

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On a class of contractive perturbations of restricted shifts

Joseph Anthony Ball and Arthur R. Lubin

Vol. 63 (1976), No. 2, 309–323

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.

Mathematical Subject Classification 2000
Primary: 47A45
Received: 4 December 1975
Revised: 11 February 1976
Published: 1 April 1976
Joseph Anthony Ball
Department of Mathematics
Virginia Tech
Blacksburg VA 24061
United States
Arthur R. Lubin