Abstract
We develop techniques which allow one to describe in simple terms the set of operators on Hilbert space of the form M* (∞) |M, where M is multiplication by z on a Hilbert space of analytic functions satisfying certain technical assumptions, M* (∞) is the direct sum of a countably infinite number of copies of M*, andM is invariant for M* (∞). One of the main ingredients in our technique is the Arveson Extension Theorem and this paper illustrates the great power and tractability of that theorem in a concrete setting.
Similar content being viewed by others
References
Agler, J., "Subjordan operators",Jour. of Func. Anal., (to appear).
Agler, J., "Subjordan operators: Bishop's theorem, spectral inclusion, and spectral sets",Jour. of Op. Th., (to appear).
Arveson, W.B., "Subalgebras of C*-algebras",Acta Math. 123 (1969).
Ball, J.A., "Rota's Theorem for general functional Hilbert spaces",P.A.M.S. 64 (1977), 55–61.
de Branges, L. and Rovnyak, J., "Appendix on square summable power series, Canonical models in quantum scattering theory,Perturbation Theory and its Applications in Quantum Mechanics, 347–392, Wiley, New York, 1966.
Bunce, J. and Salinas, N., "Completely positive maps of C*-algebras and the left matricial spectra of an operator",Duke Math. Jour. 43 (1976), 747–774.
Choi, M.-D., "Positive linear maps on C*-algebras", Univ. of Toronto, Dissertation, 1972.
Foias, C., "Some applications of spectral sets I: harmonic spectral measure",Acad. R.P. Romine Stud. Cerc. Math., 10 (1959), 365–401; English transl.,Amer. Math. Soc. Trans. (2) 61, (1967), 25–62
Lebow, A., "On von Neumann's theory of spectral sets",J. Math. Anal. and Appl. 7 (1963), 64–90.
Rota, G.-C., "On models for linear operators",Comm. Pure Appl. Math. 13 (1960), 469–472.
Rovnyak, J., "Some Hilbert spaces of analytic functions", Yale Dissertation, 1963.
Rudin, W.,Functional Analysis, McGraw-Hill, 1973.
Stinespring, W.F., "Positive functions on C*-algebras", P.A.M.S. 6 (1955), 211–216.
Sz.-Nagy, B. and Foiaş, C., "Sur les contractions de l'espace de Hilbert. VIII",Acta Sci. Math. 25 (1964), 38–71.
Author information
Authors and Affiliations
Additional information
Research partially supported by NSF grant MCS 81-02518
Rights and permissions
About this article
Cite this article
Agler, J. The Arveson Extension Theorem and coanalytic models. Integr equ oper theory 5, 608–631 (1982). https://doi.org/10.1007/BF01694057
Issue Date:
DOI: https://doi.org/10.1007/BF01694057