Advertisement

Integral Equations and Operator Theory

, Volume 85, Issue 4, pp 573–599 | Cite as

The Proper Dissipative Extensions of a Dual Pair

  • Christoph FischbacherEmail author
  • Sergey Naboko
  • Ian Wood
Article

Abstract

Let A and \({(-\widetilde{A})}\) be dissipative operators on a Hilbert space \({\mathcal{H}}\) and let \({(A,\widetilde{A})}\) form a dual pair, i.e. \({A \subset \widetilde{A}^*}\), resp. \({\widetilde{A} \subset A^*}\). We present a method of determining the proper dissipative extensions \({\widehat{A}}\) of this dual pair, i.e. \({A\subset \widehat{A} \subset\widetilde{A}^*}\) provided that \({\mathcal{D}(A)\cap\mathcal{D}(\widetilde{A})}\) is dense in \({\mathcal{H}}\). Applications to symmetric operators, symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators are discussed. Finally, we investigate the stability of the numerical range of the different dissipative extensions.

Keywords

Dissipative operators Operator extensions Dual pairs 

Mathematics Subject Classification

Primary 47B44 47A20 Secondary 47E05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alonso A., Simon B.: The Birman-Kreĭn-Vishik theory of selfadjoint extensions of semibounded operators. J. Oper. Theory 4, 251–270 (1980)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alonso A., Simon B.: Addenda to “The Birman-Kreĭn-Vishik theory of selfadjoint extensions of semibounded operators”. J. Oper. Theory 6, 407 (1981)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ando T., Nishio K.: Positive selfadjoint extensions of positive symmetric operators. Tohóku Math. J. 22, 65–75 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arlinskiĭ Yu.: On proper accretive extension of positive linear relations. Ukr. Math. J. 47(6), 723–730 (1995)Google Scholar
  5. 5.
    Arlinskiĭ Yu.: Boundary triplets and maximal accretive extensions of sectorial operators. In: Hassi, S., de Snoo, H.S.V., Szafraniec, F.H. (eds.) Operator methods for boundary value problems, 1st edn, pp. 35–72. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  6. 6.
    Arlinskiĭ Yu., Kovalev Yu., Tsekanovskiĭ E.: Accretive and sectorial extensions of nonnegative symmetric operators. Complex Anal. Oper. Theory. 6, 677–718 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Arlinskiĭ Yu., Tsekanovskiĭ E.: The von Neumann problem for nonnegative symmetric operators. Integral Equ. Oper. Theory 51, 319–356 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Arlinskiĭ Yu., Tsekanovskiĭ E., Kreĭn’s M.: Research on semi-bounded operators, its contemporary developments and applications. Oper. Theory Adv. Appl. 190, 65–112 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Arsene G., Gheondea A.: Completing matrix contractions. J. Oper. Theory 7, 179–189 (1982)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Birman, M.S.: On the selfadjoint extensions of positive definite operators. Mat. Sbornik 38, 431–450 (1956, Russian)Google Scholar
  11. 11.
    Brown, B.M., Evans, W.D.: Selfadjoint and m sectorial extensions of Sturm–Liouville operators. Integr. Equ. Oper. Theory 85(2), 151–166 (2016)Google Scholar
  12. 12.
    Brown B.M., Grubb G., Wood I.: M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary value problems. Math. Nachr. 3, 314–347 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Brown B.M., Hinchcliffe J., Marletta M., Naboko S., Wood I.: The abstract Titchmarsh-Weyl M-function for adjoint operator pairs and its relation to the spectrum. Integral Equ. Oper. Theory 63, 297–320 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Brown B.M., Marletta M., Naboko S., Wood I.: Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices. J. Lond. Math. Soc. 77(2), 700–718 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Crandall M., Phillips R.: On the extension problem for dissipative operators. J. Funct. Anal. 2, 147–176 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Derkach V., Malamud M.: Generalized resolvents and the boundary value problems for Hermitian operators with gaps. J. Funct. Anal. 95, 1–95 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Edmunds D.E., Evans W.D.: Spectral Theory and Differential Operators. Oxford University Press, Oxford (1987)zbMATHGoogle Scholar
  18. 18.
    ter Elst A.F.M., Sauter M., Vogt H.: A generalisation of the form method for accretive forms and operators. J. Funct. Anal. 269(3), 705–744 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Evans W.D., Knowles I.: On the extension problem for accretive differential operators. J. Funct. Anal. 63(3), 276–298 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Evans W.D., Knowles I.: On the extension problem for singular accretive differential operators. J. Diff. Equ. 63, 264–288 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gesztesy F., Mitrea M., Zinchenko M.: Variations on a theme by Jost and Pais. J. Funct. Anal. 253, 399–448 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gesztesy F., Mitrea M.: Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains. In: Perspectives in partial differential equations, harmonic analysis and applications, Proc. Sympos. Pure Math., vol. 79, pp. 105–173. Amer. Math. Soc., Providence (2008)Google Scholar
  23. 23.
    Gesztesy F., Mitrea M.: Nonlocal Robin Laplacians and some remarks on a paper by Filonov on eigenvalue inequalities. J. Diff. Equ. 247, 2871–2896 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gesztesy F., Mitrea M.: Robin-to-Robin maps and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains. Oper. Theory Adv. Appl. 191, 81–113 (2009)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Gesztesy F., Mitrea M.: A description of all selfadjoint extensions of the Laplacian and Krein-type resolvent formulas in nonsmooth domains. J. Anal. Math. 113, 53–172 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Grubb G.: Krein resolvent formulas for elliptic boundary problems in nonsmooth domains. Rend. Sem. Mat. Univ. Pol. Torino 66, 13–39 (2008)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Grubb, G.: Spectral asymptotics for nonsmooth singular Green operators. Commun. Partial Diff. Equ. 39(3) (2014).arXiv:1205.0094
  28. 28.
    Grubb, G.: A characterization of the non-local boundary value problems associated with an elliptic operator. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. \({3^{e}}\) série. 22, 425–513 (1968)Google Scholar
  29. 29.
    Hassi S., Malamud M., Mogilevskii V.: Unitary equivalence of proper extensions of a symmetric operator and the Weyl function. Integral Equ. Oper. Theory 77, 449–487 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Hess P., Kato T.: Perturbation of closed operators and their adjoints. Commentarii Mathematici Helvetici 45, 524–529 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kato T.: Perturbation Theory for Linear Operators. Springer-Verlag, New York (1966)CrossRefzbMATHGoogle Scholar
  32. 32.
    Kreĭn, M.G.: The theory of selfadjoint extensions of semibounded Hermitian transformations and its applications, I. Mat. Sbornik 20, 3431495 (1947, Russian)Google Scholar
  33. 33.
    Lyantze, V.E., Storozh, O.G.: Methods of the Theory of Unbounded Operators. Naukova Dumka, Kiev (1983, Russian)Google Scholar
  34. 34.
    Malamud M.: Operator holes and extensions of sectorial operators and dual pairs of contractions. Math. Nach. 279, 625–655 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Malamud M., Mogilevskii V.: On extensions of dual pairs of operators. Dopovidi Nation. Akad. Nauk Ukrainy. 1, 30–37 (1997)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Malamud M., Mogilevskii V.: On Weyl functions and Q-function of dual pairs of linear relations. Dopovidi Nation. Akad. Nauk Ukrainy. 4, 32–37 (1999)MathSciNetGoogle Scholar
  37. 37.
    Malamud M., Mogilevskii V.: Kreĭn type formula for canonical resolvents of dual pairs of linear relations. Methods Funct. Anal. Topol. 8(4), 72–100 (2002)MathSciNetzbMATHGoogle Scholar
  38. 38.
    von Neumann, J.: Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren. Math. Ann. 102, 49–131 (1929, German)Google Scholar
  39. 39.
    Phillips R.: Dissipative operators and hyperbolic systems of partial differential equations. Trans. Am. Math. Soc. 90, 192–254 (1959)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Sz.-Nagy, B., Foias, C., Bercovici, H., Kérchy, L.: Harmonic Analysis of Operators on Hilbert Space. Springer Science & Business Media, New York (2010)CrossRefzbMATHGoogle Scholar
  41. 41.
    Trostorff S.: A characterization of boundary conditions yielding maximal monotone operators. J. Funct. Anal. 267, 2787–2822 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Vishik, M.I.: On general boundary conditions for elliptic differential equations. Tr. Mosk. Mat. Obs. 1, 187–246 (1952, Russian)Google Scholar
  43. 43.
    Vishik M.I.: On general boundary conditions for elliptic differential equations. Am. Math. Soc. Trans. 24, 107–172 (1963)CrossRefzbMATHGoogle Scholar
  44. 44.
    Weidmann J.: Linear Operators in Hilbert Spaces. Springer-Verlag, New York (1980)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Actuarial ScienceUniversity of KentCanterbury, KentUK
  2. 2.Department of Math. Physics, Institute of PhysicsSt. Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations