Ukrainian Mathematical Journal

, Volume 52, Issue 5, pp 694–702 | Cite as

On one generalization of the Berezanskii evolution criterion for the self-adjointness of operators

  • M. L. Gorbachuk
  • V. I. Gorbachuk


We describe all weak solutions of a first-order differential equation in a Banach space on (0, ∞) and investigate their behavior in the neighborhood of zero. We use the results obtained to establish necessary and sufficient conditions for the essential maximal dissipativity of a dissipative operator in a Hilbert space.


Weak Solution Cauchy Problem Vector Function Strong Solution Reflexive Banach Space 
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  1. 1.
    Ph. Clément, H. Heijmans, S. Angenent, C. van Duijn, and B. de Pagter, One-Parameter Semigroups [Russian translation] Mir, Moscow 1992.Google Scholar
  2. 2.
    J. M. Ball, “Strongly continuous semigroups, weak solutions, and the variation constants formula,” Proc. Amer. Math. Soc., 63, No. 2, 370–373 (1977).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York 1983.zbMATHGoogle Scholar
  4. 4.
    A. V. Knyazyuk, “Boundary values of solutions of differential equations in Banach spaces,” Dokl. Akad. Nauk Ukr.SSR, Ser. A, No. 19, 12–14(1984).Google Scholar
  5. 5.
    V. I. Gorbachuk and M. L. Gorbachuk, “Boundary values of solutions of certain classes of differential equations,” Mat. Sb., 102, No. 1, 124–150(1977).MathSciNetGoogle Scholar
  6. 6.
    M. Reed and B. Simon, Methods of Modem Mathematical Physics, Vol. 1, Functional Analysis, Academic Press, New York 1972.Google Scholar
  7. 7.
    K. Hoffman, Banach Spaces of Analytic Functions [Russian translation] Inostrannaya Literatura, Moscow 1963.zbMATHGoogle Scholar
  8. 8.
    V. I. Gorbachuk and A. V. Knyazyuk, “Boundary values of solutions of operator differential equations,” Usp. Mat. Nauk, 44, No. 3, 55–91 (1989).MathSciNetGoogle Scholar
  9. 9.
    S. G. Krein, Linear Differential Equations in Banach Spaces [in Russian], Nauka. Moscow (1967).Google Scholar
  10. 10.
    Yu. M. Berezanskii, Expansions in Eigenfunctions of Self-Adjoint Operators [in Russian] Naukova Dumka, Kiev 1965.Google Scholar
  11. 11.
    Yu. M. Berezanskii, G. F. Us, and Z. G. Sheftel’, Functional Analysis [in Russian] Vyshcha Shkola, Kiev 1990.Google Scholar
  12. 12.
    J. A. Goldstein, Semigroups of Linear Operators and Applications [Russian translation] Vyshcha Shkola, Kiev 1989.Google Scholar
  13. 13.
    A. El Koutri, “Vecteurs a-quasi analytiques et semi-groupes analytiques,” C. R. Acad. Sci., Ser. I, 309, 767–769 (1989).zbMATHGoogle Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 2000

Authors and Affiliations

  • M. L. Gorbachuk
    • 1
  • V. I. Gorbachuk
    • 1
  1. 1.Institute of MathematicsUkranian Academy of SciencesKiev

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