Geodesic
Extensions, dilations and functional models of infinite Jacobi matrix
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 593-609

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
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 
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/
@article{CMJ_2005_55_3_a2,
     author = {Allahverdiev, B. P.},
     title = {Extensions, dilations and functional models of infinite {Jacobi} matrix},
     journal = {Czechoslovak Mathematical Journal},
     pages = {593--609},
     year = {2005},
     volume = {55},
     number = {3},
     mrnumber = {2153085},
     zbl = {1081.47036},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2005_55_3_a2/}
}
TY  - JOUR
AU  - Allahverdiev, B. P.
TI  - Extensions, dilations and functional models of infinite Jacobi matrix
JO  - Czechoslovak Mathematical Journal
PY  - 2005
SP  - 593
EP  - 609
VL  - 55
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/CMJ_2005_55_3_a2/
LA  - en
ID  - CMJ_2005_55_3_a2
ER  - 
%0 Journal Article
%A Allahverdiev, B. P.
%T Extensions, dilations and functional models of infinite Jacobi matrix
%J Czechoslovak Mathematical Journal
%D 2005
%P 593-609
%V 55
%N 3
%U http://geodesic.mathdoc.fr/item/CMJ_2005_55_3_a2/
%G en
%F 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.