Accretive and nonnegative perturbations of an abstract analogy for the operator of the third boundary problem and corresponding variational problems

Author H. M. Kachurivska, O. G. Storozh
a.kachurivska@mail.ru, storog@ukr.net
ÂÏ ÍÓÁiÏ Óêðà¿íè “Áåðåæàíñüêèé àãðîòåõíi÷íèé iíñòèòóò”, Ëüâiâñüêèé íàöiîíàëüíèé óíiâåðñèòåò iìåíi Iâàíà Ôðàíêà

Abstract In the paper the role of initial object is played by the positively definite operator L0 acting in a Hilbert space H. The main object of the investigation is operator L~ B: It is interpreted as a perturbation of some proper extension of L0. Using methods of the extension theory the criteria of maximal accretivity and maximal nonnegativity for L~ B are established. In the case when L~ B is a positively definite operator, its energetic space is constructed and the solvability of corresponding variational problem is proved. Moreover, the situation when L0 is minimal differential operator generated in the space of infinite-dimensional vector-functions by the Sturm-Liouville differential expression is considered.
Keywords Hilbert space; operator; nonnegative; extension
Reference 1. H.M. Pipa, O.G. Storozh, Accretive perturbations of proper extensions for positively definite operator, Mat. Stud. 25 (2006), ¹2, 181–190. (in Ukrainian)

2. K. Friedrichs, Spektral theorie nalbbeschraanter Operatoren und Anwendungauf die Spektral-zerlegung von Differential operatoren, Math. Ann. 109 (1934), 465–487.

3. M.G. Krein, The theory of self-adjoint extensions of semibounded Hermitian operators and its applications, I, Mat. Sb., 20 (1947), ¹3, 431–495. (in Russian)

4. S.G. Mihlin, Variational Methods in Mathematical Physics, Nauka, Moscow, 1970. (in Russian)

5. A.N. Kochubei, On extensions of positively definite symmetric operator, Dopov. Nats. Acad. Nauk Ukr. 3 (1979), 34–39. (in Ukrainian)

6. V.A. Michailets, Spectra of operators and boundary problems, Math. Inst. of the Acad. of Sc. of Ukr. SSR (1980), 106–131. (in Russian)

7. V.I. Gorbachuk. M.L. Gorbachuk, Boundary Value Problems for Differential-Operator Equations, Naukova Dumka, Kyiv, 1984. (in Russian)

8. Yu.M. Arlinskii, Maximal accretive extensions of sectorial operators.-Manuscript. Thesis for doctor’s degree by speciality 01.01.01 - math. Anal. - Institute of Math., Nats. Acad. Nauk Ukr., Kyiv, 2000. (in Ukrainian)

9. V.A. Derkach, M.M. Malamud, E.R. Tsekanowskii, Sectorial extensions of the positive operator and the chararterystic function, Ukr. Mat. Zh., 41 (1989), ¹2, 151–158. (in Russian)

10. A.W. Straus, On the extensions of semibounded optrator, Dokl Acad. Nauk of the USSR 231 (1973), ¹3, 543–546. (in Russian)

11. M.Sh. Birman, On the selfadjoint extensions of positively definite operators, Mat. Sb. 38 (80) (1956), ¹4, 431–450. (in Russian)

12. M.M. Malamud, On some classes of the extensions of Hermitian operator with the gapes, Ukr. Mat. Zh. 44 (1992), ¹2, 215–233. (in Russian)

13. E.A. Coddington, H.S.V. de Snoo, Positive self-adjoint extensions of positive symmetric subspaces, Math. Z., 159 (1978), ¹3, 203–214.

14. Yu.M. Arlinskii, S. Hassi, Z.Sebestien, H.S.V. de Snoo, On the class of extremal extensions of a nonnegative operator, Oper. Theory Adv. Appl. 127 (2001), 41–81.

15. V.A. Derkach, M.M. Malamud, Weyl function of Hermitian operator and its connection with characteristic function, Preprint 85-9 (104), Donetsk. Fiz.- Tekhn. Inst. Acad. Nauk Ukrainian SSR, Donetsk, 1985. (in Russian)

16. A.N. Kochubei, On extensions of symmetric operators and symmetric binary relations, Mat. Zametki 17 (1975), ¹1, 41–48. (in Russian)

17. H.M. Pipa, O.G. Storozh, On some perturbations chainging the domain of proper extension positively definite operator, Methods of Functional Anal. and Topology 11 (2005), ¹3, 257–269.

18. V.E. Lyantse, O.G. Storozh, Methods of the theory of Unbounded Operators, Naukova Dumka, Kyiv, 1983. (in Russian)

19. O.G. Storozh, On some analytic and asymptotic properties of the Weyl function of a nonnegative operator, Mat. Metody Phys.-Mekh. Polya 43 (2000), ¹4, 18–23. (in Ukrainian) 20. O.G. Storozh, Extensions theory methods and differential-boundary operators, Doctor of Sciences thesis, Lviv, 1995. (in Ukrainian)

21. O.Ya. Mylyo, H.M. Pipa, O.G. Storozh, On the Weyl function and extremal extensions of a semsmooth restriction of positively definite operator, Mat. Metody Phys.-Mekh. Polya 46 (2003), ¹4, 73–80. (in Ukrainian)

22. O.G. Storozh, Second order differential-boundary operator in vector-function, associated with quadratic form, Mat. Stud. 2 (1993), 59–63. (in Ukrainian)

23. O.Ya. Mylyo, O.G. Storozh, Maximal accretivity and maximal nonnegativity conditions for a class of finite-dimensional perturbations of positively definite operator, Math. Stud. 12 (1999), ¹1, 90–100. (in Ukrainian)

24. M.A. Naimark, Linear Differential Operators, Nauka, Moscow, 1969 (in Russian)

25. F.Z. Ziatdinov, On linear differential operators of the second order in Hilbert space of vector-functions taking values in abstract Hilbert space, Izv. Vuzov. Mat. 4 (1960), 89–100. (in Russian)

26. F.S. Rofe-Beketov, Self-adjoint extensions of differential operators in the space of vector-functions, Dokl. of the Acad. of the Sc. USSR, 184 (1969), ¹5, 1034–1037. (in Russian)

27. H.M. Pipa, Nonnegative and accretive perturbations of the operator of the third boundary problem for the Sturm-Liouville differential expression with the bounded operator potentsial,Visn. L’viv Univ. Ser. Mekh.-Math. 68 (2007), 207–214. (in Ukrainian)

28. O.G. Storozh, O.B. Shuvar, On a class of almost bounded perturbations of smooth restrictions of closed operator, Ukr. Mat. Zh. 54 (2002), ¹10, 1396–1402. (in Russian)

Pages 201-212
Volume 36
Issue 2
Year 2011
Journal Matematychni Studii
Full text of paper PDF
Table of content of issue HTML