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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; selfadjoint dilation; scattering matrix; functional model; characteristic function; completeness of the system of eigenvectors and associated vectors
Summary:
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.
References:
[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 0176141 | Zbl 0117.05806
[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 10.1017/S0305004100072467 | MR 1274167
[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 0415381
[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 10.1006/jmaa.1996.0020 | MR 1371289 | Zbl 0857.39008
[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 1034669
[8] V. I. Gorbachuk and M. L. Gorbachuk: Boundary Value Problems for Operator Differential Equations. Naukova Dumka, Kiev, 1984. MR 1190695
[9] A. N. Kochubei: Extensions of symmetric operators and symmetric binary relations. Mat. Zametki 17 (1975), 41–48. MR 0365218
[10] A. Kuzhel: Characteristic Functions and Models of Nonself-adjoint Operators. Kluwer, Boston-London-Dordrecht, 1996. Zbl 0927.47005
[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 0225183
[13] J. von Neumann: Allgemeine Eigenwerttheorie Hermitischer Funktionaloperatoren. Math. Ann. 102 (1929), 49–131. DOI 10.1007/BF01782338
[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 0244808
[15] M. M. Stone: Linear Transformations in Hilbert Space and Their Applications to Analysis, Vol. 15. Amer. Math. Soc. Coll. Publ., Providence, 1932. MR 1451877
[16] S. T. Welstead: Boundary conditions at infinity for difference equations of limit-circle type. J. Math. Anal. Appl. 89 (1982), 442–461. DOI 10.1016/0022-247X(82)90112-3 | MR 0677740 | Zbl 0493.39002
[17] H. Weyl: Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Functionen. Math. Ann. 68 (1910), 222–269.
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