zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Dilation and functional model of dissipative operator generated by an infinite Jacobi matrix. (English) Zbl 1047.47023
Summary: We consider the maximal dissipative operators acting in the Hilbert space $\ell_\bbfC^2(\Bbb N;E)$ $(\Bbb N = \{0,1,2,\dots\}$, $\dim E = n < \infty$) that are extensions of a minimal symmetric operator with maximal deficiency indices $(n,n)$ (in the completely indeterminate case or the limit-circle case) generated by an infinite Jacobi matrix with matrix entries. We construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, making it possible to determine the scattering matrix of the dilation. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of the scattering matrix of the dilation. We prove the completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators.

MSC:
47B36Jacobi (tridiagonal) operators (matrices) and generalizations
47A20Dilations, extensions and compressions of linear operators
47A40Scattering theory of linear operators
47A45Canonical models for contractions and nonselfadjoint operators
WorldCat.org
Full Text: DOI
References:
[1] Nagy, B. Sz.; Foiaş, C.: Analyse harmonique des operateurs de l’espace de Hilbert. (1970) · Zbl 0202.13102
[2] Kuzhel, A.: Characteristic functions and models of nonself-adjoint operators. (1996) · Zbl 0927.47005
[3] Lax, P. D.; Phillips, R. S.: Scattering theory. (1967) · Zbl 0214.12002
[4] Krein, M. G.: Infinite J-matrices and the matrix moment problem (in russian). Dokl. akad. Nauk SSSR 69, 125-128 (1949)
[5] Krein, M. G.: Basic statement of the theory of representations of Hermitian operators with deficiency indices (m, m). Ukr. matem. Zh. 2, 3-66 (1949)
[6] Berezanskii, Yu.M.: Expansion in eigenfunctions of selfadjoint operators. (1968)
[7] Kostyuchenko, A. G.; Mirzoev, K. A.: Three-term recursion relations with matrix coefficients. The completely indeterminate case. Math. notes 63, 624-630 (1998) · Zbl 0923.47015
[8] Kostyuchenko, A. G.; Mirzoev, K. A.: Generalized Jacobi matrices and deficiency numbers of ordinary differential operators with polynomial coefficients. Funct. anal. Appl 33, 25-37 (1999) · Zbl 0953.47031
[9] Kostyuchenko, A. G.; Mirzoev, K. A.: Complete indefiniteness tests for Jacobi matrices with matrix entries. Funct. anal. Appl. 35, 265-269 (2001) · Zbl 1009.47016
[10] Agarwal, R. P.: Difference equations and inequalities. (1992) · Zbl 0925.39001
[11] Elaydi, S. N.: An introduction to difference equations. (1996) · Zbl 0840.39002
[12] Allahverdiev, B. P.: Extensions of the symmetric operator generated by an infinite Jacobi matrix. Mathl. comput. Modelling 37, No. 9/10, 1093-1098 (2003) · Zbl 1047.47024
[13] Nikolśkii, N. K.: Treatise on the shift operator. (1986)
[14] Ginzburg, Yu.P.; Talyush, N. A.: Exceptional sets of analytical matrix-functions, contracting and dissipative operators. Izv. vyssh. Uchebn. zaved mat. 267, 9-14 (1984) · Zbl 0577.47017
[15] Ronkin, L. I.: Introduction to the theory of entire functions of several variables. (1974) · Zbl 0286.32004