Abstract
The aim of this paper is to develop a general approach to function models of Hilbert space contractions. This approach has been drafted (mainly by the second author) in [1], [2] and in a latent form already in [3] , [4], [5]. The main idea of the method is to stop the standard construction of the function model half-way from a unitary dilation to final formulae of the model. In other words we do not fix a concrete spectral representation of the unitary dilation of a given contraction but work directly with an (abstract) dilation equipped with a special “function imbedding operator”. We hope you find such model more flexible to be adapting to various problems of spectral theory because as a “free parameter” for such an adaption it contains your choice of a spectral representation of the minimal unitary dilation. To demonstrate a few possibilities we consider as partial cases of our “coordinate-free” model the well-known models due to Sz.-Nagy — Foiaş (both in the original and Pavlov’s forms) and to de Branges — Rovnyak. Some other possibilities are considered.
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References
N.G. Makarov, V.I. Vasyunin. A model for noncontractions and stability of the continuous spectrum. — Lect.Notes in Math., 1981, v.864, 365-412.
N.K. Nikolskii, V.I. Vasyunin. Notes on two function models. — Proc.Confer.on the Occasion of the proof of the Bieberbach conjecture, 1986, 113–141.
B. Szökefalvi-Nagy, C. Foiaş. Harmonic Analysis of Operators on Hilbert Space. North-Holland / Akadémiai Kiado, Amsterdam / Budapest, 1970.
V.M. Adamyan, D.Z. Arov. Unitary couplings of semi-unitary operators. — Mat.Issled., 1966, v.1, N 2, 3–64 (Russian).
V.I. Vasyunin. Construction of the functional model of B.Sz.-Nagy and C.Foiaş. — In: Investigations on linear operators and the theory of functions. VIII, Zap.Nauch.Sem.Leningrad. Otdel.Mat.Inst.Steklov (LOMI), 1977, v.73, 16–23. (Russian).
R.G. Douglas. Canonical Models. — Math.Surveys, v.13, AMS, Providence, 1974, 161–218.
B.S. Pavlov. Conditions for separation of the spectral components of a dissipative operator. — Izv.Akad.Nauk SSSR Ser. Mat., 1975, v.39, N 1, 123–148. (Russian)
D. Sarason. On spectral sets having connected complement. — Acta Sci.Math. (Szeged), 1965, t.26, 289–299.
B. Szökefalvi-Nagy, C. Foiaş Sur les contractions de l’espace de Hilbert. IX. Factorisations de la fonction caracteristique. Sous-espaces invariants. — Acta Sci.Math. (Szeged), 1964, t.25, 283–316.
L. de Branges, J. Rovnyak. Canonical models i n quantum scattering theory. In: Perturbation Theory and its applications in Quantum Mechanics, Wiley, New York, 1966, 295–392.
L. de Branges. Square summable power series. Heidelberg, Springer-Verlag, to appear.
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Dedicated to the 60th anniversary of Professor I. Gohberg
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© 1989 Birkhäuser Verlag Basel
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Nikolskii, N.K., Vasyunin, V.I. (1989). A Unified Approach to Function Models, and the Transcription Problem. In: Dym, H., Goldberg, S., Kaashoek, M.A., Lancaster, P. (eds) The Gohberg Anniversary Collection. Operator Theory: Advances and Applications, vol 41. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9278-0_23
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DOI: https://doi.org/10.1007/978-3-0348-9278-0_23
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