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References
N. I. Achiezer and I. M. Glazman: Theorie der Linearen Operatoren im Hilbert Raum. Akademie-Verlag, Berlin, 1981.
F. V. Atkinson: Discrete and continuous boundary problems. Academic Press, New York, 1968.
A. I. Benedek and R. Panzone: On Sturm-Liouville problems with the square-root of the eigenvalue parameter contained in the boundary conditions. Notas de Algebra y Analisis 10, Universidad Nacional del Sur, Bahia Blanca, Argentina, 1981.
J. Bognar: Indefinite inner product spaces. Springer-Verlag, Berlin, 1974.
E. A. Coddington: Generalized Sturm-Liouville theory. Lectures delivered at the Instructional Conference on Differential Equations, Edinburgh, Scotland, 1967.
E. A. Coddington: Extension theory of formally normal and symmetric subspaces. Mem. Amer. Math. Soc. 134, 1973.
E. A. Coddington and R. C. Gilbert: Generalized resolvents of ordinary differential operators. Trans. Amer. Math. Soc. 93 (1959) 216–241.
A. Dijksma: Eigenfunction expansions for a class of J-selfadjoint ordinary differential operators with boundary conditions containing the eigenvalue parameter. Proc. Roy. Soc. Edinburgh 86 A (1980) 1–27.
A. Dijksma and H. S. V. de Snoo: Self-adjoint extensions of symmetric subspaces. Pacific J. Math. 54 (1974) 71–100.
A. Dijksma and H. S. V. de Snoo: Symmetric and selfadjoint relations in Krein spaces, in preparation.
C. T. Fulton: Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions. Proc. Roy. Soc. Edinburgh 87 A (1980) 1–34.
A. H. Gelig, V. A. Jakubovič and G. A. Leonov: The stability of nonlinear systems with a nonunique equilibrium state. Nauka, Moscow, 1970 (Russian).
I. C. Gohberg and M. G. Krein: Introduction to the theory of linear nonselfadjoint operators. Amer. Math. Soc., Transl. of Math. Monographs 18, 1969.
I. S. Iohvidov, M. G. Krein and H. Langer: Introduction to the spectral theory of operators in spaces with an indefinite metric. Mathematical Research 9, Akademie-Verlag, Berlin, 1982.
M. G. Krein: On the resolvents of an Hermitian operator with defect-index (m, m). Dokl. Akad. Nauk. SSSR 52 (1946) 657–660 (Russian).
M. G. Krein and H. Langer: Defect subspaces and generalized resolvents of a hermitian operator in the space ΠK. Functional Anal. Appl. 5 (1971/72) 136–146, 217–228.
M. G. Krein and H. Langer: Über die verallgemeinerten Resolventen und die charakterische Funktion eines isometrischen Operators im Raume ΠK. “Colloquia mathematica societatis Janos Bolyai 5: Hilbert space operators and operator algebras (Tihany 1970)”, North Holland Publishing Company, Amsterdam-London 1972, 353–399.
M. G. Krein and H. Langer: Über die Q-Funktion eines π-hermiteschen Operators im Raume ΠK. Acta Sci. Math. (Szeged) 34 (1973) 191–220.
M. G. Krein and H. Langer: Some propositions on analytic matrix functions related to the theory of operators in the space ΠK. Acta Sci. Math. (Szeged) 43 (1981) 181–205.
M. G. Krein and Sh. N. Saakyan: The resolvent matrix of an Hermitian operator and the characteristic functions associated with it. Functional Anal. Appl. 4 (1970) 258–259.
H. Langer: Spectral functions of definitizable operators in Krein spaces, “Functional Analysis Proceedings Dubrovnik”, Lecture Notes in Mathematics 948, Springer Verlag, Berlin, 1982, 1–46.
H. Langer and P. Sorjonen: Verallgemeinerte Resolventen hermitescher und isometrischer Operatoren im Pontrijaginraum. Ann. Acad. Sci. Fenn. AI, 561 (1974) 3–45.
H. Langer and B. Textorius: On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space. Pacific J. Math. 72 (1977) 135–165.
H. Langer and B. Textorius: L-resolvent matrices of symmetric linear relations with equal defect numbers; applications to canonical differential relations. Integral equations and operator theory 5 (1982) 208–243.
R. McKelvey: Spectral measures, generalized resolvents and functions of positive type. J. Math. Anal. Appl. 11 (1965) 447–477.
B. Orcutt: Canonical differential equations. University of Virginia Ph.D. Thesis, 1969.
R. S. Phillips: On dissipative operators. “Lecture series in Differential Equations Vol. II, Van Nostrand Mathematical Studies 19”, Van Nostrand, New York (1969) 65–113.
H. Röh: Self-adjoint subspace extensions satisfying λ-linear boundary conditions. Proc. Roy. Soc. Edinburgh 90 A (1981) 107–124.
E. M. Russakovskij: Operator treatment of boundary problems with spectral parameters entering via polynomials in the boundary conditions. Functional Anal. Appl. 9 (1975) 358–359.
E. M. Russakovskij: Operator treatment of a boundary-value problem with the spectral parameter appearing rationally in the boundary conditions. “Theory of functions, functional analysis and its applications 30”, Kharkov 1978, 120–128 (Russian).
A. Schneider: On spectral theory for the linear selfadjoint equation Fy=λGy. “Ordinary and Partial Differential Equations, Proceedings, Dundee, Scotland 1980”, Lecture Notes in Mathematics 846, Springer-Verlag, Berlin, 1981, 306–332.
P. Sorjonen: Extensions of isometric and symmetric linear relations in a Krein space. Ann. Acad. Sci. Fenn. AI, 5 (1980) 355–376.
A. V. Štraus: Extensions and generalized resolvents of a non-densely defined symmetric operator. Math. USSR-Izv. 4 (1970) 179–208.
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Dijksma, A., Langer, H. & de Snoo, H.S.V. Selfadjoint ΠK of symmetric subspaces: An abstract approach to boundary problems with spectral parameter in the boundary conditions. Integr equ oper theory 7, 459–515 (1984). https://doi.org/10.1007/BF01238863
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DOI: https://doi.org/10.1007/BF01238863