Abstract
This is a survey of the papers by M. G. Krein (and his disciples) devoted to the theory of operators in spaces with an indefinite metric and its applications.
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References
J. Bognar,Indefinite Inner Product Spaces, Springer, Berlin (1974).
T. Ya. Azizov and I. S. Lokhvidov,Foundations of the Theory of Linear Operator in Spaces-with Indefinite Metric [in Russian], Nauka, Moscow (1986).
T. Ya. Azizov and I. S. Iokhvidov, “Linear operators in spaces with indefinite metric and their applications,” in:Itogi VINITI, Ser. Mat. Analiz [in Russian], Vol. 17, VINITI, Moscow (1979), pp. 115–207.
M. Krein, “Sur les equations integrales chargees,”C. R. Acad. Sci.,201, 24–26 (1935).
M. G. Krein, “On loaded integral equations whose distribution functions are not monotone,” in:A Collection of Papers Dedicated to the Memory of Academician Grave [in Russian], Kiev (1940), pp. 88–103.
L. S. Pontryagin, “Hermitian operators in spaces with indefinite metric,”Izv, Akad. Nauk SSSR, Ser. Mat.,8, 243–280 (1944).
M. G. Krein and M. A. Rutman, “Linear operators preserving an invariant cone in a Banach space,”Usp. Mat. Nauk,3, No. 1, 3–95 (1948).
M. G. Krein, “An application of the fixed-point principle in the theory of linear transformations of spaces with indefinite metric,”Usp. Mat. Nauk,5, No. 2, 180–190 (1950).
I. S. Iokhvidov and M. G. Krein, “Spectral theory of operators in the spaces with indefinite metric, I”Trudy Mosk. Mat. Obshch.,5, 367–432 (1956).
I. S. Iokhvidov and M. G. Krein, “Spectral theory of operators in the spaces with indefinite metric, II”Trudy Mosk. Mat. Obshch.,8, 413–496 (1959).
I. S. Iohvidov. M. G. Krein: and H. Langer,Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric, Academie-Verlag, Berlin (1982).
M. G. Krein and Yu. L. Shmul'yan, “On linear-fractional transformations with operator coefficients,”Mat. Issled.,2, Issue 3, 64–96 (1967).
M. G. Krein, “On a new application of the fixed-point method in the theory of operators in spaces with indefinite metric,”Dokl. Akad. Nauk SSSR,154, No. 5, 1023–1026 (1964).
M. G. Krein and Yu. L. Shmul'yan, “On plus-operators in spaces with indefinite metric,”Mat. Issled.,1, No. 1, 131–161 (1986).
M. G. Krein and Yu. L. Shmul'yan, “J-polar representation of plus-operators,”Mat. Issled.,1, No. 2, 172–210 (1966).
M. G. Krein and Yu. L. Shmul'yan, “On stable plus-operators inJ-spaces,” in:Linear Operators [in Russian], Stiinta, Kishinev (1980), pp. 67–83.
M. G. Krein, “An introduction to the geometry of indefiniteJ-spaces and the theory of operators in these spaces,” in:Proceedings of the 2nd Summer Math. School, [in Russian], Vol. 1, Kiev (1965), pp. 15–92.
Yu. L. Daletskii and M. G. Krein,Stability of Solutions of Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1970).
M. G. Krein and H. Langer, “Spectral function of a self-adjoint operator in a space with indefinite metric,”Dokl. Akad. Nauk SSSR,152, No. 1, 39–42 (1963).
M. G. Krein and H. Langer, “To the theory of quadratic sheafs of self-adjoint operators,”Dokl. Akad. Nauk SSSR,154, No. 6, 1258–1261 (1964).
M. G. Krein and H. Langer, “On certain mathematical principles of the linear theory of damped oscillations of continua,” in:Proceedings of the Internat. Symposium in the Applications of the Theory of Functions in the Mechanics of Continua [in Russian], Vol. 2, (1966), pp. 283–322.
Yu. L. Lyubich and V. I. Matsaev, “Operators with separable spectrum,”Mat. Sb.,56, No. 4, 433–468 (1962).
R. J. Duffin, “A minimax theory for overdamped networks,”J. Rational Mech. Anal.,4, No. 2, 221–233 (1955).
I. Ts. Gokhberg and M. G. Krein,An Introduction to the Theory of Linear Nonself-Adjoint Operators in Hilbert Spaces [in Russian], Nauka, Moscow (1965).
V. M. Adamyan, D. Z. Arov, and M. G. Krein, “On infinite Hankel matrices and the generalized Carathéodory-Fejer and Riesz problems,”Funkts. Anal. Prilozhen.,2, No. 1, 1–19 (1968).
V. M. Adamyan, D. Z. Arov, and M. G. Krein, “Infinite Hankel matrices and generalized Carathéodory-Fejer and Schur problems,”Funkts. Anal. Prilozhen.,2, No. 4, 1–17 (1968).
V. M. Adamyan, D. Z. Arov, and M. G. Krein, “Infinite block-Hankel matrices and the related continuation problems,”Izv. Arm. SSR, Ser. Mat.,6, No. 2–3, 87–112 (1971).
M. G. Krein and H. Langer, “On definite subspaces and generalized resolvents of a Hermitian operator in the space Π K ,”Funkts. Anal. Prilozhen.,5, No. 2, 59–71; No. 3, 54–69 (1968).
M. G. Krein and H. Langer, “Über die verallgemeinerten Resolventen und charakteristische Funktion eines isometrischen Operators im Raume Π K ,” in:Colloquia Math. Soc. Janos Boyai, 5, Hilbert Space Operators and Operator Algebras, North-Holland, Amsterdam-London (1972), pp. 353–399.
M. G. Krein and H. Langer, “Über die Q-Funktion eines π-Hermiteschen Operators im Räume Π K ,”Acta Sci. Math.,34, 191–230 (1973).
M. G. Krein and H. Langer, “Über einige Forsetzungsprobleme die eng mit der Theorie Hermitescher Operatoren im Raume Π K zusammenhangen, I: Einige Funktionenklassen und ihre Darstellungen,”Math. Nachr.,77, 187–236 (1977).
M. G. Krein and H. Langer, “Über einige Forsetzungsprobleme die eng mit der Theorie Hermitescher Operatoren im Räume Π K zusammenhangen, II: Verallgemeinerte Resolventen,u-Resolventen und ganze Operatoren,”J. Funct. Anal.,30, No. 3, 390–147 (1978).
M. G. Krein, “Foundations of the theory of representations of Hermitian operators with a deficiency index (m, m),”Ukr. Mat. Zh.,1, No. 2, 3–66 (1949).
M. G. Krein and H. Langer, “On some extension problems which are closely connected with the theory of Hermitian operators in a space Π K , III: Indefinite analogue of the Hamburger and Stieltjes moment problems,”Beitr. Analysis,14, 25–40 (1979);15, 27–45 (1981).
M. G. Krein and H. Langer, “Continuous analogs of the orthogonal polynomials on the unit circle with respect to indefinite weight and related continuation problems,”Dokl. Akad. Nauk SSSR,258, No. 3, 537–541 (1981).
M. G. Krein and H. Langer, “On some continuation problems which are closely related to the theory of operators in spaces Π K , IV: Continuous analogues of orthogonal polynomials on the unit circle with respect to an indefinite weight and related continuation problems for some classes of functions,”J. Oper. Theory,13, 299–417 (1985).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 5–17, January–February, 1994.
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Azizov, T.Y., Ginsburg, Y.P. & Langer, H. On M. G. Krein's papers in the theory of spaces with an indefinite metric. Ukr Math J 46, 3–14 (1994). https://doi.org/10.1007/BF01056997
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DOI: https://doi.org/10.1007/BF01056997