M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems

Mathematische Nachrichten

Volumn 282, Issue 3, 2009, Pages 314-347

  Brown, B M ^a   Grubb, G ^b   Wood, I G ^c  

^a CARDIFF UNIVERSITY   (United Kingdom)

^b UNIVERSITY OF COPENHAGEN   (Denmark)

^c ABERYSTWYTH UNIVERSITY   (United Kingdom)

Author keywords

Abstract boundary spaces; Boundary triplets; Closed extension; Elliptic PDEs; Essential spectrum; M function; Pseudo differential boundary operators

Indexed keywords

EID: 67449113919     PISSN: 0025584X     EISSN: 15222616     Source Type: Journal    
DOI: 10.1002/mana.200810740     Document Type: Article

References (44)

1. W. O. Amrein and D. B. Pearson, M operators: a generalisation of Weyl-Titchmarsh theory, J. Comput. Appl. Math. 171, 1-26 (2004).
2. F. V. Atkinson, H. Langer, R. Mennicken, and A. A. Shkalikov, The essential spectrum of some matrix operators, Math. Nachr. 167, 5-20 (1994).
3. J. Behrndt and M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal. 243, 536-565 (2007).
4. L. Boutet de Monvel, Comportement d'un opérateur pseudo-différentiel sur une variéte à bord, I-II, J. d'Analyse Fonct. 17, 241-304 (1966).
5. L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126, 11-51 (1971).
6. J. F. Brasche, M. Malamud, and H. Neidhardt, Weyl function and spectral properties of self-adjoint extensions, Integral Equations Operator Theory 43, No. 3, 264-289 (2002).
7. B. M. Brown, M. Marletta, S. Naboko, and I. G. Wood, Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices, J. Lond. Math. Soc., II. Ser. 77, No. 3, 700-718 (2008).
8. J. Brüning, V. Geyler, and K. Pankrashkin, Spectra of self-adjoint extensions and applications to solvable Schrödinger operators, Rev. Math. Phys. 20, 1-70 (2008).
9. V. A. Derkach and M. M. Malamud, On the Weyl function and Hermitian operators with gaps, Soviet Math. Dokl. 35, 393-398 (1987).
10. 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).
11. A. Douglis and L. Nirenberg, Interior estimates for elliptic systems of partial differential equations, Comm. Pure Appl. Math. 8, 503-538 (1955).
12. F. Gesztesy, M. Mitrea, and M. Zinchenko, Variations on a theme of Jost and Pais, J. Funct. Anal. 253, No. 2, 399-448 (2007).
13. G. Geymonat and G. Grubb, The essential spectrum of elliptic systems of mixed order, Math. Ann. 227, No. 3, 247-276 (1977).
14. V. I. Gorbachuk and M. L. Gorbachuk, Boundary Value Problems for Operator Differential Equations (Kluwer, Dordrecht, 1991).
15. G. Grubb, A characterization of the non-local boundary value problems associated with an elliptic operator, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22, 425-513 (1968). Available from www.numdam.org
16. G. Grubb, On coerciveness and semiboundedness of general boundary problems, Israel J. Math. 10, 32-95 (1971).
17. G. Grubb, Properties of normal boundary problems for elliptic even-order systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 1, 1-61 (1974). Available from www.numdam.org
18. G. Grubb, Boundary problems for systems of partial differential operators of mixed order, J. Funct. Anal. 26, 131-165 (1977).
19. G. Grubb, Spectral asymptotics for the "soft" self-adjoint extension of a symmetric elliptic differential operator, J. Operator Theory 10, 9-20 (1983).
20. p spaces, Comm. Partial Differential Equations 15, 289-340 (1990).
21. G. Grubb, Functional Calculus of Pseudo-differential Boundary Problems, Progress in Mathematics Vol. 65, 2nd edition (Birkhäuser, Boston, 1996).
22. G. Grubb, Distributions and Operators, Graduate Text in Mathematics Vol. 252 (Springer-Verlag, New York, 2009).
23. L. Hörmander, Linear Partial Differential Operators, Grundlehren der Mathematischen Wissenschaften Bd. 116 (Springer-Verlag, Berlin, 1963).
24. A. N. Kočubeǐ, Extensions of symmetric operators and symmetric binary relations, Math. Notes 17, No. 1, 25-28 (1975).
25. N. D. Kopachevskiǐ and S. G. Kreǐn, An abstract Green formula for a triple of Hilbert spaces, and abstract boundary value and spectral problems, Ukr. Mat. Visn. 1, No. 1, 69-97 (2004) (in Russian); English transl. in: Ukr. Math. Bull. 1, No. 1, 77-105 (2004).
26. M. G. Kreǐn, Theory of self-adjoint extensions of symmetric semi-bounded operators and applications I, Mat. Sb. (N. S.) l20, No. 62, 431-495 (1947) (in Russian).
27. A. V. Kuzhel, Canonical extensions of Hermitian operators. Dynamical systems 14, J. Math. Sci. (New York) 103, No. 1, 135-138 (2001).
28. A. V. Kuzhel and S. A. Kuzhel, Regular Extensions of Hermitian Operators, translated from the Russian by P. Malyshev and D. Malyshev (VSP, Utrecht, 1998).
29. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes VI, J. Anal. Math. 11, 165-188 (1963).
30. J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications T. 1 (Éditions Dunod, Paris, 1968).
31. V. E. Lyantze and O. G. Storozh, Methods of the Theory of Unbounded Operators (Naukova Dumka, Kiev, 1983) (in Russian).
32. M. M. Malamud and V. I. Mogilevskii, On extensions of dual pairs of operators, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 1, 30-37 (1997).
33. M. M. Malamud and V. I. Mogilevskii, On Weyl functions and Q-function of dual pairs of linear relations, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 4, 32-37 (1999).
34. M. M. Malamud and V. I. Mogilevskii, Kreǐn type formula for canonical resolvents of dual pairs of linear relations, Methods Funct. Anal. Topology 8, No. 4, 72-100 (2002).
35. V. A. Mikhailets and A. V. Sobolev, Common eigenvalue problem and periodic Schrödinger operators, J. Funct. Anal. 165, No. 1, 150-172 (1999).
36. M. A. Naǐmark, Linear Differential Operators. Part II: Linear Differential Operators in Hilbert Space (Frederick Ungar Publishing Co, New York, 1968).
37. A. Posilicano, Self-adjoint extensions of restrictions, Operators and Matrices 2, 483-506 (2008).
38. O. Post, First order operators and boundary triples, Russian J. Math. Phys. 14, 482-492 (2007).
39. V. Ryzhov, A general boundary value problem and its Weyl function, Opusc. Math. 27, No. 2, 305-331 (2007).
40. R. T. Seeley, The resolvent of an elliptic boundary problem, Amer. J. Math. 91, 889-920 (1969).
41. O. G. Storozh, On some analytic and asymptotic properties of the Weyl function of a nonnegative operator, Mat. Metody Fiz.-Mekh. Polya 43, No. 4, 18-23 (2000) (in Ukrainian).
42. L. I. Vainerman, On extensions of closed operators in Hilbert space, Math. Notes 28, 871-875 (1980).
43. V. I. Vishik, On general boundary value problems for elliptic differential operators, Trudy Moskov. Mat. Obshch. 1, 187-246 (1952); Amer. Math. Soc. Transl. Ser. 2 24, 107-172 (1963).
44. L. R. Volevich, On the theory of boundary value problems for general elliptic systems, Dokl. Akad. Nauk SSSR 148, 489-492 (1963); Soviet Math. (Iz. VUZ) 4, 97-100 (1963).