The Krein-von Neumann extension and its connection to an abstract buckling problem

Mathematische Nachrichten

Volumn 283, Issue 2, 2010, Pages 165-179

  Ashbaugh, Mark S ^a   Gesztesy, Fritz ^a   Mitrea, Marius ^a   Shterenberg, Roman ^b   Teschl, Gerald ^c,d  

^a UNIVERSITY OF MISSOURI   (United States)

^b UNIVERSITY OF ALABAMA AT BIRMINGHAM   (United States)

^c UNIVERSITY OF VIENNA   (Austria)

^d Erwin Schrodinger International Institute for Mathematical Physics   (Austria)

Author keywords

Buckling problem; Krein von Neumann extension

Indexed keywords

EID: 77952516976     PISSN: 0025584X     EISSN: 15222616     Source Type: Journal    
DOI: 10.1002/mana.200910067     Document Type: Article

References (43)

1. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Volume II (Pitman, Boston, 1981).
2. A. Alonso and B. Simon, The Birman-Krein-Vishik theory of selfadjoint extensions of semibounded operators, J. Operator Theory 4, 251-270 (1980).
3. A. Alonso and B. Simon, The Birman-Krein-Vishik theory of selfadjoint extensions of semibounded operators, Addenda: J. Operator Theory 6, 407 (1981).
4. T. Ando and K. Nishio, Positive selfadjoint extensions of positive symmetric operators, Tohoku Math. J. (2), 22, 65-75 (1970).
5. Yu. M. Arlinskii and E. R. Tsekanovskii, On the theory of non-negative selfadjoint extensions of a non-negative symmetric operator, Rep. Nat. Acad. Sci. Ukraine 2002, No. 11, 30-37.
6. Yu. M. Arlinskiǐ and E. R. Tsekanovskiǐ, On von Neumann's problem in extension theory of nonnegative operators, Proc.Amer.Math.Soc.131, 3143-3154 (2003).
7. Yu. M. Arlinskiǐ and E. R. Tsekanovskiǐ, The von Neumann problem for nonnegative symmetric operators, Integral Equations Operator Theory 51, 319-356 (2005).
8. Yu. Arlinskiǐ and E. Tsekanovskiǐ, M. Kreǐn's research on semibounded operators, its contemporary developments, and applications, in: Modern Analysis and Applications. The Mark Krein Centenary Conference, Vol. 1, edited by V. Adamyan, Y. M. Berezansky, I. Gohberg, M. L. Gorbachuk, V. Gorbachuk, A. N. Kochubei, H. Langer, and G. Popov, Operator Theory: Advances and Applications Vol. 190 (Birkhäuser, Basel, 2009), pp. 65-112.
9. M. S. Ashbaugh, F. Gesztesy, M. Mitrea, and G. Teschl, Spectral theory for perturbed Krein Laplacians in non-smooth domains, Adv. Math., to appear, DOI: 10.1016/j.aim.2009.10.006.
10. M. S. Berger, Nonlinearity and Functional Analysis, Pure and Applied Mathematics Vol. 74 (Academic Press, New York, 1977).
11. M. Sh. Birman, On the theory of self-adjoint extensions of positive definite operators, Mat. Sb. (N.S.) 38, 431-450 (1956) (in Russian)
12. M. Sh. Birman, Perturbations of the continuous spectrum of a singular elliptic operator by varying the boundary and the boundary conditions, Vestnik Leningr. Univ. Mat. Mekh. Astronom. 17, No. 1, 22-55 (1962) (Russian)
13. English transl. in: Spectral Theory of Differential Operators: M. Sh. Birman 80th Anniversary Collection, edited by T. Suslina and D. Yafaev, AMS Translations, Ser. 2, Advances in the Mathematical Sciences Vol. 225 (Amer. Math. Soc., Providence, RI, 2008), pp. 19-53.
14. V. A. Derkach and M. M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95, 1-95 (1991).
15. V. A. Derkach and M. M. Malamud, The extension theory of Hermitian operators and the moment problem, J. Math. Sci. 73, 141-242 (1995).
16. W. G. Faris, Self-Adjoint Operators, Lecture Notes in Mathematics Vol. 433 (Springer, Berlin, 1975).
17. H. Freudenthal, Über die Friedrichsche Fortsetzung halbbeschränkter Hermitescher Operatoren, Kon. Akad. Wetensch., Amsterdam, Proc. 39, 832-833 (1936).
18. K. Friedrichs, Spektraltheorie halbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differential- operatoren I, II, Math. Ann. 109, 465-487, 685-713 (1934)
19. corrs. in Math. Ann. 110, 777-779 (1935).
20. M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes (de Gruyter, Berlin, 1994).
21. F. Gesztesy and M. Mitrea, Self-adjoint extensions of the Laplacian and Krein-type resolvent formulas in rough domains, preprint, 2009, arXiv:0907.1750.
22. G. Grubb, A characterization of the non-local boundary value problems associated with an elliptic operator, Ann. Scuola Norm. Sup. Pisa (3), 22, 425-513 (1968).
23. G. Grubb, Les problèmes aux limites généraux d'un opérateur elliptique, provenant de le théorie variationnelle, Bull. Sci. Math. (2), 94, 113-157 (1970).
24. G. Grubb, On coerciveness and semiboundedness of general boundary problems, Israel J. Math. 10, 32-95 (1971).
25. G. Grubb, Spectral asymptotics for the "soft" selfadjoint extension of a symmetric elliptic differential operator, J. Operator Theory 10, 9-20 (1983).
26. G. Grubb, Distributions and Operators, Graduate Texts in Mathematics Vol. 252 (Springer, New York, 2009).
27. S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, enlarged 2nd printing (Springer, Berlin, 2006).
28. S. Hassi, M. Malamud, and H. de Snoo, On Kreǐn's extension theory of nonnegative operators, Math. Nachr. 274-275, 40-73 (2004).
29. H. Heuser, Funktionalanalysis (Teubner, Stuttgart, 1986).
30. V. A. Kozlov, Estimation of the remainder in a formula for the asymptotic behavior of the spectrum of nonsemibounded elliptic systems, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1979, No. 4, 112-113, 125 (in Russian).
31. V. A. Kozlov, Estimates of the remainder in formulas for the asymptotic behavior of the spectrum for linear operator bundles, Funktsional. Anal. i Prilozhen 17, No. 2, 80-81 (1983)
32. Engl. transl. in: Funct. Anal. Appl. 17, No. 2, 147-149 (1983).
33. V. A. Kozlov, Remainder estimates in spectral asymptotic formulas for linear operator pencils, in: Linear and Nonlinear Partial Differential Equations. Spectral Asymptotic Behavior, Probl. Mat. Anal. 9 (Leningrad Univ., Leningrad, 1984), pp. 34-56
34. Engl. transl. in: J. Sov. Math. 35, 2180-2193 (1986).
35. M. G. Krein, The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. I, Mat. Sb. (N.S.) 20, 431-495 (1947) (in Russian).
36. M. G. Krein, The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. II, Mat. Sb. (N.S.) 21, 365-404 (1947) (in Russian).
37. M. M. Malamud, Certain classes of extensions of a lacunary Hermitian operator, Ukrainian Math. J. 44, No. 2, 190-204 (1992).
38. W. McLean, Strongly Elliptic Systems and Boundary Integral Equations (Cambridge University Press, Cambridge, 2000).
39. A. V. Štraus, On extensions of a semibounded operator, Sov. Math. Dokl. 14, 1075-1079 (1973).
40. M. L. Vǐsik, On general boundary problems for elliptic differential equations, Trudy Moskov. Mat. Obsc. 1, 187-246 (1952) (in Russian)
41. Engl. transl. in: Amer. Math. Soc. Transl. (2), 24, 107-172 (1963).
42. J. von Neumann, Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Math. Ann. 102, 49-131 (1929-30).
43. J. Weidmann, Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics Vol. 68 (Springer, New York, 1980).