Abstract
A boundary differential operator generated by the Sturm-Liouville differential expression with bounded operator potential and nonlocal boundary conditions is considered. The conditions for a considered operator to be a Fredholm and solvable operator are established and its resolvent is constructed.
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References
T. Kato,Theory of Perturbations of Linear Operators [Russian translation], Mir, Moscow (1972).
M. A. Naimark,Linear Differential Operators [in Russian], Nauka, Moscow (1969).
F. Z. Ziatdinov, “On linear second-order differential operators in the Hilbert space of vector functions from an abstract separable Hilbert space,”Izv. Vyssh. Uchebn. Zaved., Mat., No. 4, 89–100 (1960).
F. S. Rofe-Beketov, “Self-adjoint extensions of differential operators in a space of vector functions,”Teor. Funkts., Funkts. Anal. Prilozhen.,8, 3–24 (1969).
V. I. Gorbachuk and M. L. Gorbachuk,Boundary-Value Problems for Operator Differential Equations [in Russian], Naukova Dumka, Kiev (1984).
R. C. Brown, “The operator theory of generalized boundary value problems,”Can. J. Math.,28, No. 3, 486–512 (1976).
E. A. Coddington, “Self-adjoint subspace extensions of nondensely defined symmetric operator,”Adv. Math.,14, No. 3, 309–332 (1974).
A. M. Krall, “Differential-boundary operators,”Trans. Amer. Math. Soc.,154, 429–456 (1971).
V. E. Lyantse, “On some relations between closed operators,”Dokl. Akad. Nauk SSSR,204, No. 3, 542–545 (1972).
O. G. Storozh, “Some spectral properties of an operator related to a differential operator,”Differents. Uravn.,11, No. 6, 1141–1143 (1975).
V. E. Lyantse and O. G. Storozh,Methods of the Theory of Unbounded Operators [in Russian], Naukova Dumka, Kiev (1983).
Yu. L. Daletskii and M. G. Krein,Stability of Solutions of Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1970).
L. I. Vainerman, “On extensions of closed operators in a Hilbert space,”Mat. Zametki,28, No. 6, 833–842 (1980).
A. N. Kochubei, “On extensions of a positive definite symmetric operator,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 3, 168–171 (1979).
V. A. Mikhailets, “Spectra of operators and boundary-value problems,” in:Spectral Analysis of Differential Operators [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1980), pp. 106–131.
V. M. Bruk, “On invertible restrictions of closed operators in Banach spaces,”Funkts. Anal.,28, 17–22 (1988).
M. I. Vishik, “On general boundary-value problems for elliptic differential equations,”Tr. Mosk. Mat. Obshch.,1, 187–246 (1952).
O. G. Storozh, “On solvability of extensions of a closed operator,” in:Methods for Investigation of Differential and Integral Operators [in Russian], Naukova Dumka, Kiev (1989), pp. 171–175.
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 517–524, April, 1995.
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Storozh, O.G. On the conditions of solvability and the resolvent of a second-order deferential boundary operator in a space of vector functions. Ukr Math J 47, 599–608 (1995). https://doi.org/10.1007/BF01056046
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DOI: https://doi.org/10.1007/BF01056046