Extensions, dilations and functional models of infinite Jacobi matrix

B. P. Allahverdiev

Czechoslovak Mathematical Journal (2005)

  • Volume: 55, Issue: 3, page 593-609
  • ISSN: 0011-4642

Abstract

top
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.

How to cite

top

Allahverdiev, B. P.. "Extensions, dilations and functional models of infinite Jacobi matrix." Czechoslovak Mathematical Journal 55.3 (2005): 593-609. <http://eudml.org/doc/30971>.

@article{Allahverdiev2005,
abstract = {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.},
author = {Allahverdiev, B. P.},
journal = {Czechoslovak Mathematical Journal},
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; infinite Jacobi matrix; symmetric operator; selfadjoint and nonselfadjoint extensions; maximal dissipative operator},
language = {eng},
number = {3},
pages = {593-609},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Extensions, dilations and functional models of infinite Jacobi matrix},
url = {http://eudml.org/doc/30971},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Allahverdiev, B. P.
TI - Extensions, dilations and functional models of infinite Jacobi matrix
JO - Czechoslovak Mathematical Journal
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 3
SP - 593
EP - 609
AB - 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.
LA - eng
KW - 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; infinite Jacobi matrix; symmetric operator; selfadjoint and nonselfadjoint extensions; maximal dissipative operator
UR - http://eudml.org/doc/30971
ER -

References

top
  1. The Classical Moment Problem and Some Related Questions in Analysis, Fizmatgiz, Moscow, 1961. (1961) 
  2. Discrete and Continuous Boundary Problems, Academic Press, New York, 1964. (1964) Zbl0117.05806MR0176141
  3. 10.1017/S0305004100072467, Math. Proc. Camb. Philos. Soc. 116 (1994), 167–177. (1994) MR1274167DOI10.1017/S0305004100072467
  4. Expansion in Eigenfunctions of Selfadjoint Operators, Naukova Dumka, Kiev, 1965. (1965) 
  5. On a class of boundary-value problems with a spectral parameter in the boundary conditions, Mat. Sb. 100 (1976), 210–216. (1976) MR0415381
  6. 10.1006/jmaa.1996.0020, J.  Math. Anal. Appl. 197 (1996), 267–285. (1996) Zbl0857.39008MR1371289DOI10.1006/jmaa.1996.0020
  7. The theory of extensions of symmetric operators and boundary-value problems for differential equations, Ukrain. Mat. Zh. 41 (1989), 1299–1312. (1989) MR1034669
  8. Boundary Value Problems for Operator Differential Equations, Naukova Dumka, Kiev, 1984. (1984) MR1190695
  9. Extensions of symmetric operators and symmetric binary relations, Mat. Zametki 17 (1975), 41–48. (1975) MR0365218
  10. Characteristic Functions and Models of Nonself-adjoint Operators, Kluwer, Boston-London-Dordrecht, 1996. (1996) Zbl0927.47005
  11. Scattering Theory, Academic Press, New York, 1967. (1967) 
  12. Analyse Harmonique des Operateurs de l’espace de Hilbert, Masson and Akad Kiadó, Paris and Budapest, 1967. (1967) MR0225183
  13. 10.1007/BF01782338, Math. Ann. 102 (1929), 49–131. (1929) DOI10.1007/BF01782338
  14. Self-adjoint extensions of differential operators in space of vector-valued functions, Dokl. Akad. Nauk SSSR 184 (1969), 1034–1037. (1969) MR0244808
  15. Linear Transformations in Hilbert Space and Their Applications to Analysis, Vol.  15, Amer. Math. Soc. Coll. Publ., Providence, 1932. (1932) MR1451877
  16. 10.1016/0022-247X(82)90112-3, J.  Math. Anal. Appl. 89 (1982), 442–461. (1982) Zbl0493.39002MR0677740DOI10.1016/0022-247X(82)90112-3
  17. Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Functionen, Math. Ann. 68 (1910), 222–269. (1910) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.