Realization of nonstrict matrix Nevanlinna functions as Weyl functions of symmetric operators in Pontryagin spaces
HTML articles powered by AMS MathViewer
- by Jussi Behrndt
- Proc. Amer. Math. Soc. 137 (2009), 2685-2696
- DOI: https://doi.org/10.1090/S0002-9939-09-09812-8
- Published electronically: February 3, 2009
- PDF | Request permission
Abstract:
Matrix-valued Nevanlinna functions with possibly noninvertible imaginary part are realized as $Q$-functions or Weyl functions of symmetric operators in Pontryagin spaces. The functions are decomposed into a constant part, which gives rise to a realization in a finite dimensional Pontryagin space $\mathcal {K}$, and a strict or uniformly strict part, which gives rise to a realization in a Hilbert space $\mathcal {H}$. A coupling procedure then leads to a symmetric operator in the product space $\mathcal {H}\times \mathcal {K}$ and to the realization of the given Nevanlinna function.References
- Tomas Azizov, Branko Ćurgus, and Aad Dijksma, Standard symmetric operators in Pontryagin spaces: a generalized von Neumann formula and minimality of boundary coefficients, J. Funct. Anal. 198 (2003), no. 2, 361–412. MR 1964544, DOI 10.1016/S0022-1236(02)00041-1
- J. Behrndt, Boundary value problems with eigenvalue depending boundary conditions, Math. Nachr., to appear.
- Jussi Behrndt, Seppo Hassi, and Henk de Snoo, Functional models for Nevanlinna families, Opuscula Math. 28 (2008), no. 3, 233–245. MR 2425117
- Jussi Behrndt and Peter Jonas, Boundary value problems with local generalized Nevanlinna functions in the boundary condition, Integral Equations Operator Theory 55 (2006), no. 4, 453–475. MR 2250158, DOI 10.1007/s00020-005-1400-6
- Ju. M. Berezans′kiĭ, Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. MR 0222718
- V. Derkach, On Weyl function and generalized resolvents of a Hermitian operator in a Kreĭn space, Integral Equations Operator Theory 23 (1995), no. 4, 387–415. MR 1361051, DOI 10.1007/BF01203914
- V. A. Derkach, On generalized resolvents of Hermitian relations in Krein spaces, J. Math. Sci. (New York) 97 (1999), no. 5, 4420–4460. Functional analysis, 5. MR 1728871, DOI 10.1007/BF02366102
- V. A. Derkach and M. M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 (1991), no. 1, 1–95. MR 1087947, DOI 10.1016/0022-1236(91)90024-Y
- V. A. Derkach and M. M. Malamud, The extension theory of Hermitian operators and the moment problem, J. Math. Sci. 73 (1995), no. 2, 141–242. Analysis. 3. MR 1318517, DOI 10.1007/BF02367240
- V. A. Derkach, S. Hassi, M. M. Malamud, and H. S. V. de Snoo, Generalized resolvents of symmetric operators and admissibility, Methods Funct. Anal. Topology 6 (2000), no. 3, 24–55. MR 1903120
- Vladimir Derkach, Seppo Hassi, Mark Malamud, and Henk de Snoo, Boundary relations and orthogonal coupling of symmetric operators, Proceedings of the Algorithmic Information Theory Conference, Vaasan Yliop. Julk. Selvityksiä Rap., vol. 124, Vaasan Yliopisto, Vaasa, 2005, pp. 41–56. MR 2222445
- Vladimir Derkach, Seppo Hassi, Mark Malamud, and Henk de Snoo, Boundary relations and their Weyl families, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5351–5400. MR 2238919, DOI 10.1090/S0002-9947-06-04033-5
- V.A. Derkach, S. Hassi, M.M. Malamud, H.S.V. de Snoo, Boundary relations and generalized resolvents of symmetric operators, preprint, arXiv:math/0610299. To appear in Russ. J. Math. Phys.
- Aad Dijksma, Heinz Langer, and Henk de Snoo, Eigenvalues and pole functions of Hamiltonian systems with eigenvalue depending boundary conditions, Math. Nachr. 161 (1993), 107–154. MR 1251013, DOI 10.1002/mana.19931610110
- A. Dijksma and H. S. V. de Snoo, Symmetric and selfadjoint relations in Kreĭn spaces. II, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), no. 2, 199–216. MR 951970, DOI 10.5186/aasfm.1987.1208
- M. L. Gorbachuk (ed.), Granichnye zadachi dlya differentsial′no-operatornykh uravneniĭ, Akad. Nauk Ukrainy, Inst. Mat., Kiev, 1991 (Russian). MR 1190695
- S. Hassi, M. Kaltenbäck, and H. S. V. de Snoo, The sum of matrix Nevanlinna functions and self-adjoint extensions in exit spaces, Recent progress in operator theory (Regensburg, 1995) Oper. Theory Adv. Appl., vol. 103, Birkhäuser, Basel, 1998, pp. 137–154. MR 1635009
- S. Hassi, H.S.V. de Snoo, H. Woracek, Some interpolation problems of Nevanlinna-Pick type. The Kreĭn-Langer method, Oper. Theory Adv. Appl. 106, Birkhäuser, Basel, 1998, 201–216.
- Michael Kaltenbäck and Harald Woracek, On representations of matrix valued Nevanlinna functions by $u$-resolvents, Math. Nachr. 205 (1999), 115–130. MR 1709165, DOI 10.1002/mana.3212050106
- M. G. Kreĭn and G. K. Langer, The defect subspaces and generalized resolvents of a Hermitian operator in the space $\Pi _{\kappa }$, Funkcional. Anal. i Priložen. 5 (1971), no. 2, 59–71 (Russian). MR 0282238
- M. G. Kreĭn and H. Langer, Über die $Q$-Funktion eines $\pi$-hermiteschen Operators im Raume $\Pi _{\kappa }$, Acta Sci. Math. (Szeged) 34 (1973), 191–230 (German). MR 318958
- M. G. Kreĭn and H. Langer, Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume $\Pi _{\kappa }$ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr. 77 (1977), 187–236. MR 461188, DOI 10.1002/mana.19770770116
- M. G. Kreĭn 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), no. 1-2, 181–205. MR 621369
- H. Langer and B. Textorius, On generalized resolvents and $Q$-functions of symmetric linear relations (subspaces) in Hilbert space, Pacific J. Math. 72 (1977), no. 1, 135–165. MR 463964
- Heinz Langer, Spectral functions of definitizable operators in Kreĭn spaces, Functional analysis (Dubrovnik, 1981) Lecture Notes in Math., vol. 948, Springer, Berlin-New York, 1982, pp. 1–46. MR 672791
- Heinz Langer, Branko Najman, and Christiane Tretter, Spectral theory of the Klein-Gordon equation in Pontryagin spaces, Comm. Math. Phys. 267 (2006), no. 1, 159–180. MR 2238908, DOI 10.1007/s00220-006-0022-4
- M. M. Malamud and S. M. Malamud, Spectral theory of operator measures in a Hilbert space, Algebra i Analiz 15 (2003), no. 3, 1–77 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 3, 323–373. MR 2052164, DOI 10.1090/S1061-0022-04-00812-X
Bibliographic Information
- Jussi Behrndt
- Affiliation: Department of Mathematics MA 6–4, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany
- MR Author ID: 760074
- Email: behrndt@math.tu-berlin.de
- Received by editor(s): January 30, 2008
- Received by editor(s) in revised form: October 20, 2008
- Published electronically: February 3, 2009
- Communicated by: Marius Junge
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 2685-2696
- MSC (2000): Primary 47B50, 30E99; Secondary 47B25, 47A56, 47A48
- DOI: https://doi.org/10.1090/S0002-9939-09-09812-8
- MathSciNet review: 2497481