×

Contractions as restricted shifts. (English) Zbl 0588.47013

In the ’60s, B. Sz.-Nagy and C. Foiaş developed a functional model for a completely nonunitary contraction on a complex separable Hilbert space [Harmonic analysis of operators on Hilbert space (1967; Zbl 0157.432)]. It says that such an operator is unitarily equivalent to a restriction of the orthogonal sum of a backward shift and a bilateral shift. In this note, the author constructs explicitly the shift operators in question, the space of the restriction as well as the implementing unitary operators in terms of T. The method is elementary and self-contained. It generalizes Rota’s contruction for the case when \(T^ n\) converges strongly to 0 as n approaches \(\infty\).
Reviewer: Wu Pei Yuan

MSC:

47A45 Canonical models for contractions and nonselfadjoint linear operators
47A20 Dilations, extensions, compressions of linear operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

Citations:

Zbl 0157.432
PDF BibTeX XML Cite