The Glazman-Krein-Naimark Theorem for Ordinary Differential Operators

  • W. N. Everitt
  • L. Markus
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 98)


The original form of symmetric boundary conditions for Lagrange symmetric (formally self-adjoint) ordinary linear differential expressions, was obtained by I.M. Glazman in his seminal paper of 1950. This result described all self-adjoint differential operators, in the underlying Hilbert function space, generated by real even-order differential expressions.


Linear Manifold Linear Differential Operator Chapter Versus Ordinary Differential Operator Deficiency Index 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    N.I. Akhiezer and I.M. Glazman. Theory of linear operators in Hilbert space; I and II. (Pitman and Scottish Academic Press, London and Edinburgh; 1980: translated, from the material prepared for the third Russian edition of 1978, by E.R. Dawson and edited by W.N. Everitt.)Google Scholar
  2. [2]
    N. Dunford and J.T. Schwartz. Linear operators II: Spectral theory. (Wiley, New York; 1963.)Google Scholar
  3. [3]
    W.N. Everitt. ‘On the deficiency index problem for ordinary differential operators.’ Lecture notes for the 1977 International Conference: Differential Equations, Uppsala, Sweden; pages 62 to 81. (University of Uppsala, Sweden, 1977; distributed by Almquist and Wiskell International, Stockholm, Sweden.)Google Scholar
  4. [4]
    W.N. Everitt. ‘Linear ordinary quasi-differential expressions’. Lecture notes for the Fourth International Symposium on Differential Equations and Differential Geometry, Beijing, Peoples’ Republic of China 1983; pages 1 to 28. (Department of Mathematics, University of Beijing, Peoples’ Republic of China; 1986.)Google Scholar
  5. [5]
    W.N. Everitt and L. Markus. ‘Controllability of [r]-matrix quasi-differential equations. J. of Diff. Equations. 89 (1991), 95–109.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    W.N. Everitt and L. Markus. Boundary value problems and symplectic geometry for ordinary differential and quasi-differential equations. (In manuscript; to be submitted for publication in 1996.)Google Scholar
  7. [7]
    W.N. Everitt and D. Race. ‘Some remarks on linear ordinary quasi-differential expressions.’ Proc. London Math. Soc. (3) 54 (1987), 300–320.MathSciNetCrossRefGoogle Scholar
  8. [8]
    W.N. Everitt and A. Zettl. ‘Generalized symmetric ordinary differential expressions I: the general theory.’ Nieuw Arch. Wish (3) 27 (1979), 363–397.MathSciNetGoogle Scholar
  9. [9]
    W.N. Everitt and A. Zettl. ‘Differential operators generated by a countable number of quasi-differential expressions on the real line.’ Proc. London Math. Soc. (3) 64 (1991), 524–544.MathSciNetGoogle Scholar
  10. [10]
    I.M. Glazman. ‘On the theory of singular differential operators.’ Uspehi Math. Nauk. 40 (1950), 102–135. (English translation in Amer. Math. Soc. Translations (1) 4 (1962), 331-372.)MathSciNetGoogle Scholar
  11. [11]
    I. Halperin, ‘Closures and adjoints of linear differential operators.’ Ann. of Math. 38 (1937), 880–919.MathSciNetCrossRefGoogle Scholar
  12. [12]
    T. Kimura and M. Takahasi. ‘Sur les operateurs differentiels ordinaires lineaires formellement autoadjoint.’ Funkcial. Ekuac. 7 (1965), 35–90.MathSciNetzbMATHGoogle Scholar
  13. [13]
    K. Kodaira. ‘On ordinary differential equations of any even order, and the corresponding eigenfunction expansions.’ Amer. J. of Math. 72 (1950), 502–544.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    M.A. Naimark. Linear differential operators; II. (Ungar, New York; 1968: translated from the second Russian edition of 1966 by E.R. Dawson, and edited by W.N. Everitt.)Google Scholar
  15. [15]
    D. Shin. ‘Existence theorems for quasi-differential equations of order n.’ Doklad. Akad. Nauk SSSR. 18 (1938), 515–518.Google Scholar
  16. [16]
    D. Shin. ‘On quasi-differential operators in Hilbert space.’ Doklad. Akad. Nauk. SSSR. 18 (1938), 523–526.Google Scholar
  17. [17]
    D. Shin. ‘On the solutions of a linear quasi-differential equation of order n.’ Mat. Sb. 7 (1940), 479–532.Google Scholar
  18. [18]
    J. Weidmann. Spectral theory of ordinary differential operators. (Lecture Notes in Mathematics, 1259; Springer-Verlag, Berlin and Heidelberg; 1987.)Google Scholar
  19. [19]
    A. Zettl. ‘Formally self-adjoint quasi-differential operators.’ Rocky Mountain J. Math. 5 (1975), 453–474.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1997

Authors and Affiliations

  • W. N. Everitt
    • 1
  • L. Markus
    • 2
  1. 1.School of Mathematics and StatisticsUniversity of BirminghamEdgbaston BirminghamEngland, UK
  2. 2.School of MathematicsUniversity of MinnesotaMinnesotaUSA

Personalised recommendations