Geodesic
Extensions, dilations and functional models of infinite Jacobi matrix
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 593-609
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A space of boundary values is constructed for the minimal symmetric operator generated by an infinite Jacobi matrix in the limit-circle case. A description of all maximal dissipative, accretive and selfadjoint extensions of such a symmetric operator is given in terms of boundary conditions at infinity. We construct a selfadjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of dilation. We construct a functional model of the dissipative operator and define its characteristic function. We prove a theorem on the completeness of the system of eigenvectors and associated vectors of dissipative operators.
A space of boundary values is constructed for the minimal symmetric operator generated by an infinite Jacobi matrix in the limit-circle case. A description of all maximal dissipative, accretive and selfadjoint extensions of such a symmetric operator is given in terms of boundary conditions at infinity. We construct a selfadjoint dilation of maximal dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of dilation. We construct a functional model of the dissipative operator and define its characteristic function. We prove a theorem on the completeness of the system of eigenvectors and associated vectors of dissipative operators.
Classification : 47A20, 47A40, 47A45, 47B25, 47B36, 47B39, 47B44
Keywords: infinite Jacobi matrix; symmetric operator; selfadjoint and nonselfadjoint extensions; maximal dissipative operator; selfadjoint dilation; scattering matrix; functional model; characteristic function; completeness of the system of eigenvectors and associated vectors 
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Allahverdiev, B. P. Extensions, dilations and functional models of infinite Jacobi matrix. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 593-609. http://geodesic.mathdoc.fr/item/CMJ_2005_55_3_a2/

[1] N. I. Akhiezer: The Classical Moment Problem and Some Related Questions in Analysis. Fizmatgiz, Moscow, 1961.

[2] F. V. Atkinson: Discrete and Continuous Boundary Problems. Academic Press, New York, 1964. | MR | Zbl

[3] M. Benammar, W.  D. Evans: On the Friedrichs extension of semi-bounded difference operators. Math. Proc. Camb. Philos. Soc. 116 (1994), 167–177. | DOI | MR

[4] Yu. M. Berezanskij: Expansion in Eigenfunctions of Selfadjoint Operators. Naukova Dumka, Kiev, 1965.

[5] V. M. Bruk: On a class of boundary-value problems with a spectral parameter in the boundary conditions. Mat. Sb. 100 (1976), 210–216. | MR

[6] L. S. Clark: A spectral analysis for self-adjoint operators generated by a class of second order difference equations. J.  Math. Anal. Appl. 197 (1996), 267–285. | DOI | MR | Zbl

[7] M. L Gorbachuk, V. I. Gorbachuk and A. N. Kochubei: The theory of extensions of symmetric operators and boundary-value problems for differential equations. Ukrain. Mat. Zh. 41 (1989), 1299–1312. | MR

[8] V. I. Gorbachuk and M. L. Gorbachuk: Boundary Value Problems for Operator Differential Equations. Naukova Dumka, Kiev, 1984. | MR

[9] A. N. Kochubei: Extensions of symmetric operators and symmetric binary relations. Mat. Zametki 17 (1975), 41–48. | MR

[10] A. Kuzhel: Characteristic Functions and Models of Nonself-adjoint Operators. Kluwer, Boston-London-Dordrecht, 1996. | Zbl

[11] P. D. Lax and R. S. Phillips: Scattering Theory. Academic Press, New York, 1967.

[12] B. Sz.-Nagy and C. Foias: Analyse Harmonique des Operateurs de l’espace de Hilbert. Masson and Akad Kiadó, Paris and Budapest, 1967. | MR

[13] J. von Neumann: Allgemeine Eigenwerttheorie Hermitischer Funktionaloperatoren. Math. Ann. 102 (1929), 49–131. | DOI

[14] F. S. Rofe-Beketov: Self-adjoint extensions of differential operators in space of vector-valued functions. Dokl. Akad. Nauk SSSR 184 (1969), 1034–1037. | MR

[15] M. M. Stone: Linear Transformations in Hilbert Space and Their Applications to Analysis, Vol.  15. Amer. Math. Soc. Coll. Publ., Providence, 1932. | MR

[16] S. T. Welstead: Boundary conditions at infinity for difference equations of limit-circle type. J.  Math. Anal. Appl. 89 (1982), 442–461. | DOI | MR | Zbl

[17] H. Weyl: Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Functionen. Math. Ann. 68 (1910), 222–269.