References
N.G. Askerov, S.G. Krein, G.I. Laptev. On a class of non-self-adjoint boundary problems. Dokl. Akad. nauk SSSR, 1964, 155, Nr. 3, 499–502. (Russian)
Yu. P. Ginzburg. On projections in a Hilbert space with bilinear matric. Dokl. Akad. nauk SSSR, 1961, 139, Nr. 4, 775–778 (Russian)
Yu. P. Ginzburg. On subspaces of a Hilbert space with indefinite metric. Naucn. zap. kafedr mat., fis. i estestvozn. Odessk. ped. in-ta, 1960, 25, Nr. 2, 3–9. (Russian)
Yu. P. Ginzburg, I.S. Iohvidov. Investigations on the geometry of infinite spaces with bilinear metric. Usp. mat. nauk, 1962, 17, Nr. 4 (106), 3–56. (Russian)
I.C. Gohberg, M.G. Krein. Fundamental statements on defect numbers, root numbers and indices of linear operators. Usp. mat. nauk, 1957, 12, Nr. 2 (74), 43–118. (Russian)
I.C. Gohberg, M.G. Krein. Introduction to the theory of linear non-self-adjoint operators in a Hilbert space. Moskva, Izd-vo "Nauka", 1965. (Russian)
N. Dunford, J.T. Schwartz. Linear Operators, 1 – 2. New York-London, Interscience Publ., 1958–1963.
R.J. Duffin. A Mimimax Theory for Overdamped Networks. J. Rational Mech. and Analysis, 1955, 4, Nr. 2, 221–233.
I.S. Iohvidov. On some classes of operators in a space with total indefinite metric. In: Functional analysis and its applications (Trudy V Vses. konf., 1959). Baku, Izd-vo Akad. nauk Azerb SSR, 1961, str. 20–25 (Russian)
I.S. Iohvidov, M.G. Krein. Spectral theory of operators in spaces with indefinite metric, 1. Trudy Mosk. mat. ob-va, 1956, 5, 367–432. (Russian)
M.V. Keldys. On eigen-values and eigen-functions of some classes of non-self-adjoint equations. Dokl. Akad. nauk SSSR, 1951, 77, Nr. 1, 11–14. (Russian)
M.V. Keldys, V.B. Lidskii. Questions of the spectral theory of non-self-adjoint operators. Trudy IV Vses. mat. s'ezda, 1, Leningrad, Izd.-vo Akad. nauk SSSR, 1963, str. 101–120. (Russian)
M.G. Krein. Theory of self-adjoint extensions of semi-bounded Hermitian operators and its applications, 1. Mat. sb., 1947, 20 (62), Nr. 3, 431–495. (Russian)
M.G. Krein. On a new application of the fixed-point principle in the theory of operators on a space with indefinite metric. Dokl. Akad. nauk SSSR, 1964, 154, Nr. 5, 1023–1026. (Russian)
M.G. Krein. On the theory of linear non-self-adjoint operators. Dokl. Akad. nauk SSSR, 1960, 130, Nr. 2, 2540256. (Russian)
M.G Krein. On linear compact operators in functional spaces with two norms. Zb. prac' In-tu mat Akad. nauk URSR, 1947 (1948), Nr. 9, 104–129. (Ukrainian)
M.G. Krein, H. Langer. On the theory of quadratic pencils of self-adjoint operators. Dokl. Akad. nauk SSSR, 1964, 154, Nr. 6, 1258–1261. (Russian)
S.G. Krein. On vibrations of a viscous fluid in a vessel. Dokl. Akad. nauk SSSR, 1964, 159, Nr. 2, 262–266. (Russian)
H. Langer. On J-Hermitian operators. Dokl. Akad. nauk SSSR, 1960, 134, Nr. 2, 263–266. (Russian)
H. Langer. Zur Spektraltheorie J-selbstadjungierter Operatoren. Math. Ann., 1962, 146, Nr. 1, 60–85.
H. Langer. Eine Verallgemeinerung eines Satzes von L.S. Pontrjagin. Math. Ann., 1963, 152, Nr. 5, 434–436.
G.I. Laptev. Boundary problems for differential equations in a Banach space and their applications. Dissertation. University of Voronez, 1964. (Russian)
M.S. Livsic. On the spectral expansion of linear non-self-adjoint operators. Mat. sb. 1954, 34 (76), Nr. 1, 144–199. (Russian)
V.B. Lidskii. On the summability of series by principal vectors of non-self-adjoint operators. Trudy Mosk. mat. ob-va, 1962, 11, 3–35. (Russian)
A.I. Mal'cev. Foundations of linear algebra. Izd. 2 Moskva, Gostehizdat, 1956. (Russian)
S.G. Mihlin. A minimum problem of a quadratic functional. Moskva-Leningrad, Gostehizdat, 1962. (Russian)
B.R. Mukminov. On the expansion by eigen functions of dissipative kernels. Dokl. Akad. nauk SSSR, 1954, 99, Nr. 4, 499–502. (Russian)
P.H. Müller. Eigenwertabschätzungen für Gleichungen vom Typ (λ2I − λA − B)x = 0. Arch. Math., 1961, 12, Nr. 4, 307–310.
P.H. Müller. Über eine Klasse von Eigenwertaufgaben mit nichtlinearer Parameterabhängigkeit. Math. Nachr. 1954, 12, Nr. 3–4, 173–181.
L.S. Pontryagin. Hermitian operators in a space with indefinite metric. Izv. Akad. nauk SSSR, seriya mat. 1944, 8, Nr. 6, 243–280. (Russian)
R.S. Phillips. Dissipative Operators and Hyperbolic Systems of Partial Differential Equations. Trans. Amer. Math. Soc., 1959, 90, Nr. 2, 193–254.
Author information
Authors and Affiliations
Additional information
Note of the editor:
This paper is the first part of a paper which was originally published in Russian in "Proc. Int. Sympos. on Applications of the Theory of Functions in Continuum Mechanics, Tbilisi 1963, vol II: Fluid and Gas Mechanics, Math. Methods, Nauka, Moskow, 1965, pp. 283 – 322." The second part will appear in one of the forth-coming issues of this journal. The division into two parts is made formally. The editor is grateful to R. Troelstra of the Wiskundig Seminarium of the Vrije Universiteit at Amsterdam for making the translation.
Rights and permissions
About this article
Cite this article
Krein, M.G., Langer, H. & Troelstra, R. On some mathematical principles in the linear theory of damped oscillations of continua I. Integr equ oper theory 1, 364–399 (1978). https://doi.org/10.1007/BF01682844
Issue Date:
DOI: https://doi.org/10.1007/BF01682844