Proceedings of the American Mathematical Society

Author(s): Vladimir Derkach; Seppo Hassi
Journal: Proc. Amer. Math. Soc. 131 (2003), 3795-3806.
MSC (2000): Primary 47B25, 47B50; Secondary 46C20, 46E22
Posted: March 25, 2003
Retrieve article in: PDF

Abstract: A new model for generalized Nevanlinna functions $Q\in\mathbf{N}_\kappa$ will be presented. It involves Bezoutians and companion operators associated with certain polynomials determined by the generalized zeros and poles of

. The model is obtained by coupling two operator models and expressed by means of abstract boundary mappings and the corresponding Weyl functions.

1.: D. Alpay, A. Dijksma, J. Rovnyak, and H.S.V. de Snoo, Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Oper. Theory: Adv. Appl., 96, Birkhäuser Verlag, Basel, 1997. MR 2000a:47024
2.: V. Derkach, ``On generalized resolvents of Hermitian relations'', J. Math. Sciences, 97 (1999) No. 5, 4420-4460. MR 2001c:47042
3.: V.A. Derkach, S. Hassi, M.M. Malamud, and H.S.V. de Snoo, ``Generalized resolvents of symmetric operators and admissibility'', Methods of Functional Analysis and Topology, 6 (2000), 24-55. MR 2003b:47042
4.: V.A. Derkach, S. Hassi, and H.S.V. de Snoo, ``Operator models associated with Kac subclasses of generalized Nevanlinna functions'', Methods of Functional Analysis and Topology, 5 (1999), 65-87. MR 2001f:47063
5.: V.A. Derkach, S. Hassi, and H.S.V. de Snoo, ``Operator models associated with singular perturbations of selfadjoint operators'', Methods of Functional Analysis and Topology, 7 (2001), 1-21. MR 2002m:47017
6.: V.A. Derkach, S. Hassi, and H.S.V. de Snoo, ``Rank one perturbations in a Pontryagin space with one negative square'', J. Funct. Anal., 188 (2002), 317-349. MR 2002k:47030
7.: V.A. Derkach, S. Hassi, and H.S.V. de Snoo, ``Singular perturbations of selfadjoint operators'', Mathematical Physics, Analysis and Geometry (to appear).
8.: V. Derkach and M. Malamud, ``Generalized resolvents and the boundary value problems for hermitian operators with gaps'', J. Funct. Anal., 95 (1991), 1-95. MR 93d:47046
9.: V. Derkach and M. Malamud, ``The extension theory of hermitian operators and the moment problem'', J. Math. Sciences, 73 (1995), 141-242. MR 95m:47009
10.: A. Dijksma, H. Langer, A. Luger, and Yu. Shondin, ``A factorization result for generalized Nevanlinna functions of the class ${\mathbf{N}}_\kappa$ '', Integral Equations Operator Theory, 36 (2000), 121-125. MR 2000i:47027
11.: A. Dijksma, H. Langer, Yu. Shondin, and C. Zeinstra, ``Self-adjoint operators with inner singularities and Pontryagin spaces'', Oper. Theory Adv. Appl., 117 (2000), 105-176. MR 2001f:47039
12.: A.Dijksma, H. Langer, H.S.V. de Snoo, ``Symmetric Sturm-Liouville operators with eigenvalue depending boundary conditions", Canadian Math. Society, Conference Proceedings, 8 (1987), 87-116. MR 88m:34022
13.: V.I. Gorbachuk and M.L. Gorbachuk, Boundary value problems for operator differential equations, Naukova Dumka, Kiev, 1984 (Russian) (English translation: Kluwer Academic Publishers, Dordrecht, Boston, London, 1990). MR 92m:34133
14.: P. Jonas and H. Langer, ``A model for $\pi_1$ -spaces and a special linear pencil'', Integral Equations Operator Theory, 8 (1985), 13-35. MR 86g:47047
15.: P. Jonas, H. Langer, and B. Textorius, ``Models and unitary equivalence of cyclic selfadjoint operators in Pontrjagin space'', Oper. Theory Adv. Appl., 59 (1992), 252-284. MR 94j:47052
16.: M.G. Krein and H. Langer, ``Über die -Function eines $\pi$ -hermiteschen Operators in Raume $\Pi_{\kappa}$ '', Acta. Sci. Math. (Szeged), 34 (1973), 191-230. MR 47:7504
17.: M.G. Krein and H. Langer, ``Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren in Raume $\Pi_{\kappa}$ zusammenhangen. I. Einige Funktionklassen und ihre Darstellungen'', Math. Nachr., 77 (1977), 187-236. MR 57:1173
18.: M.G. Krein and H. Langer, ``Some propositions on analytic matrix functions related to the theory of operators in the space $\Pi _\kappa$ '', Acta Sci. Math. (Szeged), 43 (1981), 181-205. MR 82i:47053
19.: P. Lancaster and M. Tismenetsky, The theory of matrices, Second edition. Academic Press, Inc., Orlando, FL, 1985. MR 87a:15001
20.: E.M. Russakovskii, ``An operator treatment of a boundary value problem with a spectral parameter that occurs polynomially in the boundary conditions'', Functional Anal. Appl., 9 (1975), 358-359. MR 54:3505
21.: P. Sorjonen, ``Pontryaginräume mit einem reproduzierenden Kern'', Ann. Acad. Sci. Fenn. A.I. Math., 594, (1975). MR 53:8875

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47B25, 47B50, 46C20, 46E22

Vladimir Derkach
Affiliation: Department of Mathematics, Donetsk National University, Universitetskaya str. 24, 83055 Donetsk, Ukraine
Address at time of publication: Department of Mathematics, Western Washington University, Bellingham, Washington 98225
Email: derkacv@cc.wwu.edu

Seppo Hassi
Affiliation: Department of Mathematics and Statistics, University of Vaasa, PL 700, 65101 Vaasa, Finland
Email: sha@uwasa.fi

DOI: 10.1090/S0002-9939-03-06946-6
PII: S 0002-9939(03)06946-6
Keywords: Generalized Nevanlinna function, symmetric operator, selfadjoint extension, Weyl function, boundary triplet, reproducing kernel Pontryagin space
Received by editor(s): December 7, 2001
Received by editor(s) in revised form: July 10, 2002
Posted: March 25, 2003
Additional Notes: The support of the Academy of Finland (projects 40362 and 52528) is gratefully acknowledged
Communicated by: Joseph A. Ball
Copyright of article: Copyright 2003, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS