Abstract:
For selfadjoint operators A 1 and A 2 in a Pontryagin space Πκ such that the resolvent difference of A 1 and A 2 is n-dimensional it is shown that the dimensions of the spectral subspaces corresponding to open intervals in gaps of the essential spectrum differ at most by n+2κ. This is a natural extension of a classical result on finite rank perturbations of selfadjoint operators in Hilbert spaces to the indefinite setting.With the help of an explicit operator model for scalar rational functions it is shown that the estimate is sharp. Furthermore, the general perturbation result and the operator model are illustrated with an application to a singular Sturm–Liouville problem, where the boundary condition depends rationally on the eigenparameter. © 2018, Springer International Publishing AG.
Registro:
Documento: |
Artículo
|
Título: | Finite rank perturbations in Pontryagin Spaces and a Sturm–Liouville problem with λ-rational boundary conditions |
Autor: | Behrndt, J.; Philipp, F. |
Filiación: | Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, Graz, A-8010, Austria Departamento de Matemática Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires Ciudad Universitaria, Pabellón I, Buenos Aires, 1428, Argentina
|
Año: | 2018
|
Volumen: | 263
|
Página de inicio: | 163
|
Página de fin: | 189
|
DOI: |
http://dx.doi.org/10.1007/978-3-319-68849-7_6 |
Título revista: | Operator Theory: Advances and Applications
|
Título revista abreviado: | Oper. Theory
|
ISSN: | 02550156
|
Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_02550156_v263_n_p163_Behrndt |
Referencias:
- Alpay, D., Bruinsma, P., Dijksma, A., De Snoo, H.S.V., A Hilbert space associated with a Nevanlinna function (1990) Signal Processing, Scattering and Operator Theory, and Numerical Methods, Proceedings International Symposium MTNS 89, 3, pp. 115-122. , Progress in Systems and Control Theory, Birkhäuser, Basel
- Alpay, D., Dijksma, A., Rovnyak, J., De Snoo, H.S.V., Schur Functions, Operator Colligations, and Pontryagin Spaces (1997) Oper. Theory Adv. Appl., 96. , Birkhäuser, Basel
- Arlinskii, Y., Belyi, S., Derkach, V.A., Tsekanovskii, E., On realization of the Krein–Langer class Nκ of matrix-valued functions in Pontryagin spaces (2008) Math. Nachr., 281, pp. 1380-1399
- Azizov, T.Y., Extensions of J-isometric and J-symmetric operators (1984) Funktsional. Anal. I Prilozhen, 18, pp. 57-58. , (Russian)English translation: Functional Anal. Appl., 18 (1984), 46–48
- Azizov, T.Y., Ćurgus, B., Dijksma, A., Standard symmetric operators in Pontryagin spaces: A generalized von Neumann formula and minimality of boundary coefficients (2003) J. Funct. Anal., 198, pp. 361-412
- Azizov, T.Y., Iokhvidov, I.S., (1989) Linear Operators in Spaces with an Indefinite Metric, , John Wiley Sons, Ltd., Chichester
- Behrndt, J., Boundary value problems with eigenvalue depending boundary conditions (2009) Math. Nachr., 282, pp. 659-689
- Behrndt, J., Derkach, V.A., Hassi, S., De Snoo, H.S.V., A realization theorem for generalized Nevanlinna families (2011) Oper. Matrices, 5, pp. 679-706
- Behrndt, J., Jonas, P., Boundary value problems with local generalized Nevanlinna functions in the boundary condition (2006) Integral Equations Operator Theory, 55, pp. 453-475
- Behrndt, J., Leben, L., Martinez-Peria, F., Möws, R., Trunk, C., Sharp eigenvalue estimates for rank one perturbations of nonnegative operators in Krein spaces (2016) J. Math. Anal. Appl., 439, pp. 864-895
- Behrndt, J., Möws, R., Trunk, C., Eigenvalue estimates for operators with finitely many negative squares (2016) Opuscula Math, 36, pp. 717-734
- Binding, P.A., Browne, P.J., Seddighi, K., Sturm–Liouville problems with eigenparameter dependent boundary conditions Proc. Edinburgh Math. Soc., 37 (1993), pp. 57-72
- Binding, P.A., Browne, P.J., Watson, B.A., Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter. I (2002) Proc. Edinb. Math. Soc, 45, pp. 631-645
- Binding, P.A., Browne, P.J., Watson, B.A., Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter. II (2002) J. Comput. Appl. Math, 148, pp. 147-168
- Binding, P.A., Hryniv, R., Langer, H., Najman, B., Elliptic eigenvalue problems with eigenparameter dependent boundary conditions (2001) J. Differential Equations, 174, pp. 30-54
- Birman, M.S., Solomjak, M.Z., (1987) Spectral Theory of Selfadjoint Operators in Hilbert Space, Mathematics and Its Applications, , D. Reidel Publishing Company
- Bognar, J., (1974) Indefinite Inner Product Spaces, , Springer
- Cross, R., (1998) Multivalued Linear Operators, Monographs and Textbooks in Pure and Applied Mathematics 213, , Marcel Dekker, Inc., New York
- Ćurgus, B., Dijksma, A., Read, T., The linearization of boundary eigenvalue problems and reproducing kernel Hilbert spaces (2001) Linear Algebra Appl, 329, pp. 97-136
- Derkach, V.A., On generalized resolvents of Hermitian relations in Krein spaces (1999) J. Math. Sci. (New York), 97, pp. 4420-4460
- Derkach, V.A., Hassi, S., A reproducing kernel space model for Nκ-functions (2003) Proc. Amer. Math. Soc, 131, pp. 3795-3806
- Derkach, V.A., Hassi, S., Malamud, M.M., De Snoo, H.S.V., Generalized resolvents of symmetric operators and admissibility (2000) Methods Funct. Anal. Topology, 6, pp. 24-53
- Derkach, V.A., Malamud, M.M., Generalized resolvents and the boundary value problems for Hermitian operators with gaps (1991) J. Funct. Anal, 95, pp. 1-95
- Derkach, V.A., Malamud, M.M., The extension theory of Hermitian operators and the moment problem (1995) J. Math. Sci. (New York), 73, pp. 141-242
- Dijksma, A., Langer, H., (1996) Operator Theory and Ordinary Differential Operators, Lectures on Operator Theory and Its Applications, pp. 73-139. , Fields Inst. Monogr. 3, Amer. Math. Soc., Providence, RI
- Dijksma, A., Langer, H., De Snoo, H.S.V., Representations of holomorphic operator functions by means of resolvents of unitary or selfadjoint operators in Krein spaces (1987) Oper. Theory Adv. Appl., 24, pp. 123-143
- Dijksma, A., Langer, H., De Snoo, H.S.V., Symmetric Sturm–Liouville operators with eigenvalue depending boundary conditions (1987) CMS Conf. Proc., 8, pp. 87-116
- Dijksma, A., Langer, H., De Snoo, H.S.V., Hamiltonian systems with eigenvalue depending boundary conditions (1988) Oper. Theory Adv. Appl., 35, pp. 37-83
- Dijksma, A., Langer, H., De Snoo, H.S.V., Eigenvalues and pole functions of Hamiltonian systems with eigenvalue depending boundary condition (1993) Math. Nachr., 161, pp. 107-154
- Dijksma, A., De Snoo, H.S.V., Symmetric and selfadjoint relations in Krein spaces I (1987) Oper. Theory Adv. Appl., 24, pp. 145-166
- Dijksma, A., De Snoo, H.S.V., Symmetric and selfadjoint relations in Krein spaces II (1987) Ann. Acad. Sci. Fenn. Ser. a I Math., 12, pp. 199-216
- Etkin, A., On an abstract boundary value problem with the eigenvalue parameter in the boundary condition (2000) Fields Inst. Commun., 25, pp. 257-266
- Gorbachuk, V.I., Gorbachuk, M.L., (1991) Boundary Value Problems for Operator Differential Equations, , Kluwer Academic Publishers, Dordrecht
- Jonas, P., Operator representations of definitizable functions (2000) Ann. Acad. Sci. Fenn., Ser. A. I. Mathematica, 25, pp. 41-72
- Jonas, P., On operator representations of locally definitizable functions (2005) Oper. Theory Adv. Appl., 162, pp. 165-190
- Hassi, S., De Snoo, H.S.V., Szafraniec, F.H., Componentwise and canonical de-compositions of linear relations (2009) Dissertationes Math., 465. , (59 pages)
- Hassi, S., De Snoo, H.S.V., Woracek, H., Some interpolation problems of Nevanlinna–Pick type. The Krein–Langer method (1998) Oper. Theory Adv. Appl., 106, pp. 201-216
- Iohvidov, I.S., Krein, M.G., Langer, H., (1982) Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric, , Akademie-Verlag
- Krein, M.G., Langer, H., On the spectral function of a self-adjoint operator in a space with indefinite metric (Russian) (1963) Dokl. Akad. Nauk SSSR, 152, pp. 39-42
- Krein, M.G., Langer, H., Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raum Πκ zusammenhängen, I. Einige Funktionenklassen und ihre Darstellungen (1977) Math. Nachr., 77, pp. 187-236
- Langer, H., (1965) Spektraltheorie Linearer Operatoren in J-Räumen Und Einige Anwendungen Auf Die Schar L(λ) = λ2 I + λB + C, , Habilitationsschrift, Technische Universität Dresden
- Langer, H., Spectral functions of definitizable operators in Krein spaces (1982) Functional Analysis (Dubrovnik, 1981), Lect. Notes Math., 948, pp. 1-46
- Langer, H., Möller, M., Linearization of boundary eigenvalue problems (1991) Integral Equations Operator Theory, 14, pp. 105-119
- Pivovarchik, V.N., Direct and inverse three-point Sturm–Liouville problem with parameter-dependent boundary conditions (2001) Asymptotic Anal, 26, pp. 219-238
- Russakovskii, E.M., The matrix Sturm–Liouville problem with spectral parameter in the boundary condition. Algebraic and operator aspects (1997) Trans. Moscow Math. Soc., 1996, pp. 159-184
- Shkalikov, A.A., Boundary problems for ordinary differential equations with parameter in the boundary conditions (1986) J. Soviet Math., 33, pp. 1311-1342
Citas:
---------- APA ----------
Behrndt, J. & Philipp, F.
(2018)
. Finite rank perturbations in Pontryagin Spaces and a Sturm–Liouville problem with λ-rational boundary conditions. Operator Theory: Advances and Applications, 263, 163-189.
http://dx.doi.org/10.1007/978-3-319-68849-7_6---------- CHICAGO ----------
Behrndt, J., Philipp, F.
"Finite rank perturbations in Pontryagin Spaces and a Sturm–Liouville problem with λ-rational boundary conditions"
. Operator Theory: Advances and Applications 263
(2018) : 163-189.
http://dx.doi.org/10.1007/978-3-319-68849-7_6---------- MLA ----------
Behrndt, J., Philipp, F.
"Finite rank perturbations in Pontryagin Spaces and a Sturm–Liouville problem with λ-rational boundary conditions"
. Operator Theory: Advances and Applications, vol. 263, 2018, pp. 163-189.
http://dx.doi.org/10.1007/978-3-319-68849-7_6---------- VANCOUVER ----------
Behrndt, J., Philipp, F. Finite rank perturbations in Pontryagin Spaces and a Sturm–Liouville problem with λ-rational boundary conditions. Oper. Theory. 2018;263:163-189.
http://dx.doi.org/10.1007/978-3-319-68849-7_6