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Theory of self-adjoint extensions of symmetric operators. entire operators and boundary-value problems

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Abstract

This is a brief survey of M. G. Krein's contribution to the theory of self-adjoint extensions of Hermitian operators and to the theory of boundary-value problems for differential equations. The further development of these results is also considered.

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References

  1. J. von Neumann. “Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren,”Math. Ann.,102, 49–131 (1929).

    Google Scholar 

  2. H. Weyl, “Über gewohnliche lineare Differentialgleichungen mit Singularen Stellen und ihre Eigenfunktionen,”Nachr. Kgl. Ges. Wiss. Göttingen, Math.-Phys. Kl., 37–63 (1909).

  3. H. Weyl, “Über gewohnliche Differentialgleichungen mit Singulariten und die Zugehorigen Entwicklungen willkurlicher Funktionen,”Math. Ann.,68, 220–269 (1910).

    Google Scholar 

  4. M. G. Krein and M. A. Rrasnosel'skii, “Main theorems on extensions of Hermitian operators and their applications to the theory of orthogonal polynomials and moment problem,”Usp. Mat Nauk,3, No. 3, 61–106 (1947).

    Google Scholar 

  5. N. Dunford and J. T. Schwanz,Linear Operators. Pt III. Spectral operators, Wiley-Interscience, New York-London-Sydney-Toronto (1971).

    Google Scholar 

  6. M. G. Krein, “On self-adjoint extensions of bounded and semibounded Hermitian operators,”Dokl Akad. Nauk SSSR,48, No. 5, 323–326 (1945).

    Google Scholar 

  7. M. G. Krein. “Theory of self-adjoint extensions of semibounded Hermitian operators and its applications. I,”Mat. Sb.,20, No. 3, 431–495 (1947).

    Google Scholar 

  8. M. G. Krein, “Theory of self-adjoint extensions of semibounded Hermitian operators and its applications. II,”Mat. Sb.,21, No. 3, 365–404 (1947).

    Google Scholar 

  9. M. I. Vishik, “General boundary-value problems for elliptic differential equations,”Trudy Mosk. Mat. Obshch.,1, 187–246 (1952).

    Google Scholar 

  10. M. Sh. Birman, “To the theory of self-adjoint extensions of positive definite operators,”Mat. Sb.,38, No. 4, 431–450 (1956).

    Google Scholar 

  11. M. A. Naimark, “On the second-kind self-adjoint extensions of a symmetric operator,”Izv. Akad. Nauk SSSR,4, No. 1, 53–104 (1940).

    Google Scholar 

  12. M. A. Naimark, “Spectral functions of a symmetric operator,”Izv. Akad. Nauk SSSR,4, No. 3, 277–318 (1940).

    Google Scholar 

  13. M. G. Krein, “On Hermitian operators whose deficiency indices are equal to one,”Dokl. Akad. Nauk SSSR,43, No. 8, 323–326 (1944).

    Google Scholar 

  14. M. A. Naimark, “On spectral functions of a symmetric operator,”Izv. Akad. Nauk SSSR,7, No. 6, 285–296 (1943).

    Google Scholar 

  15. M. G. Krein, “On Hermitian operators whose deficiency indices are equal to one. II,”Dokl. Akad. Nauk SSSR,44, No. 4, 131–134 (1944).

    Google Scholar 

  16. M. G. Krein, “On a remarkable class of Hermitian operators,”Dokl. Akad. Nauk SSSR,44, No. 5, 191–195 (1944).

    Google Scholar 

  17. M. G. Krein. “On the resolvents of an Hermitian operator with deficiency index (m, m)”Dokl. Akad. Nauk SSSR,52, No. 8, 657–660 (1946).

    Google Scholar 

  18. M. G. Krein. “Foundations of the theory of representations of Hermitian operators with deficiency index (m, m)”Ukr. Mat. Zh.,1, No. 2, 1–65 (1949).

    Google Scholar 

  19. M. G. Krein and Sh. N. Saakyan, “New results in the theory of resolvents of Hermitian operators,”Dokl. Akad. N auk SSSR,169, No. 6. 1269–1272 (1966).

    Google Scholar 

  20. M. G. Krein and I. E. Ovcharenko, “To the theory of generalized resolvents of nondensely defined Hermitian contractions,”Dokl. Akad. Nauk Ukr. SSR. Ser. A. No. 10. 881–884 (1977).

    Google Scholar 

  21. M. G. Krein and I. E. Ovcharenko, “On theQ-functions and sc-resolvents of nondensely defined Hermitian contractions,”Sib. Mat. Zh.,18, No. 5. 1032–1056 (1987).

    Google Scholar 

  22. M. G. Krein and I. E. Ovcharenko, “On the generalized resolvents and resolvent matrices of positive Hermitian operators,”Dokl. Akad. Nauk SSSR,231, No. 5, 1063–1066 (1977).

    Google Scholar 

  23. M. G. Krein and Sh. N. Saakyan, “A resolvent matrix of an Hermitian operator and the characteristic functions associated with it,”Funkts. Anal. Prilozhen.,4, No. 3, 103–104 (1970).

    Google Scholar 

  24. M. G. Krein, “Analytical problems and results in the theory of linear operators in Hilbert spaces,” in: Proceedings of the Internat. Congress of Mathematicians (Moscow, 1966) [in Russian], Moscow (1968), pp. 189–231.

  25. É. R. Tsekanovskii and Yu. L. Shmul'yan, “Problems of the theory of extensions of unbounded operators in rigged Hilbert spaces,” in:Itogi VINITI, Mathematical Analysis [in Russian], Vol. 14, VINITI, Moscow (1977), pp. 59–100.

    Google Scholar 

  26. F. S. Rofe-Beketov, “Self-adjoint extensions of differential operators in the spaces of vector functions,”Dokl. Akad. Nauk SSSR,184, No. 5, 1034–1037 (1969).

    Google Scholar 

  27. F. S. Rofe-Beketov, “On self-adjoint extensions of differential operators in a space of vector functions,”Teor. Funk., Funkts. Anal. Prilozhen., Issue 8, 3–24, (1969).

    Google Scholar 

  28. M. L. Gorbachuk, “Self-adjoint boundary-value problems for a second-order differential equation with an unbounded operator coefficient,”Funkts. Anal. Prilozhen.,5, No. 1, 10–21 (1971).

    Google Scholar 

  29. V. I. Gorbachuk and M. L. Gorbachuk, “On the spectrum of self-adjoint extensions of the minimal operator generated by the Sturm-Liouville expression with an operator potential,”Ukr. Mat. Zh.,24, No. 6, 726–734 (1972).

    Google Scholar 

  30. V. I. Gorbachuk and M. L. Gorbachuk, “On some classes of boundary-value problems for the Sturm-Liouville equation with an operator potential,”Ukr. Mat. Zh.,24, No. 3, 291–304 (1972).

    Google Scholar 

  31. A. N. Kochubei, “On the extensions of symmetric operators and symmetric binary relations,”Mat. Zametki,17, No. 1, 41–48 (1975).

    Google Scholar 

  32. V. M. Brak, “On a class of boundary-value problems with a spectral parameter in boundary conditions,”Mat. Sb.,100, No. 2, 210–216 (1976).

    Google Scholar 

  33. V. I. Gorbachuk, M. L. Gorbachuk, and A. N. Kochubei, “Theory of extensions of symmetric operators and boundary-value problems for differential equations,”Ukr. Mat. Zh.,41, No. 10, 1299–1313 (1989).

    Google Scholar 

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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 55–62, January–February, 1994.

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Gorbachuk, M.L., Gorbachuk, V.I. Theory of self-adjoint extensions of symmetric operators. entire operators and boundary-value problems. Ukr Math J 46, 54–61 (1994). https://doi.org/10.1007/BF01057000

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