Spectral analysis of some classes first-order normal differential operators

In this paper, the general form of all normal differential operators generated by first-order linear singulardifferential expressions in the weighted Hilbert spaces of vector-functions on right semiaxis has been found. Later on,the spectrum set of these type extensions is investigated. Finally, the asymptotical behavior of the singular numbers ofany normal extension is studied.

PDF

___

[1] Allahverdiev BP. Spectral analysis of dissipative Dirac operators with general boundary conditions. Journal of Mathematical Analysis and Applications 2003; 283 (1): 287-303.
[2] Allahverdiev BP. Extensions, dilations and functional models of Dirac operators in limit-circle case. Forum Mathematicum 2005; 17 (4): 591-611.
[3] Allahverdiev BP. Extensions, dilations, and spectral problems of singular Hamiltonian system. Mathematical Methods in the Applied Sciences 2018; 41 (5): 1761-1773.
[4] Coddington EA. Extension Theory of Formally Normal and Symmetric Subspaces. Providence, RI, USA: American Mathematical Society, 1973.
[5] Davis RH. Singular normal differential operators. PhD, University of California, Berkeley, USA, 1955.
[6] Dunford N, Schwartz JT. Linear Operators II. New York, USA: Interscience, 1963.
[7] Gohberg IC, Krein MG. Introduction to the Theory of Linear Non-Self-Adjoint operators. Providence, RI, USA: American Mathematical Society, 1969.
[8] Gorbachuk ML. Selfadjoint boundary value problems for a second order differential equation with an unbounded operator coefficient. Funktsional’nyi Analiz i Ego Prilozheniya 1971; 5 (1): 10-21 (in Russian).
[9] Gorbachuk VI, Gorbachuk ML. Boundary Value Problems for Operator Differential Equations. 1st ed. Dordrecht, the Netherlands: Kluwer Academic Publishers, 1991.
[10] Hörmander L. On the theory of general partial differential operators. Acta Mathematica 1955; 94: 161-248.
[11] Ismailov ZI. Normal boundary value problems of differential equation for second order with bounded operator potential. Differential Equations 1994; 30 (11): 2018-2019 (in Russian).
[12] Ismailov ZI. A three-point normal boundary value problem for an operator-differential equation. Siberian Mathematical Journal 1994; 35 (5): 1058-1061 (in Russian).
[13] Ismailov ZI. Discreteness of the spectrum of first-order normal differential operators. Doklady Mathematics 1998; 358 (2): 162-164 (in Russian).
[14] Ismailov ZI. On the normal solvability and index of the differential operators. Doklady Mathematics 2000; 375 (3): 302-304 (in Russian).
[15] Ismailov ZI. On the normality of first order differential operators. Bulletin of the Polish Academy of Sciences 2003; 51 (2): 39-45.
[16] Ismailov ZI. Compact inverses of first-order normal differential operators. Journal of Mathematical Analysis and Applications 2006; 320 (1): 266-278. doi: 10.1016/j.jmaa.2005.06.090
[17] Ismailov ZI, Erol M. Normal differential operators of third-order. Hacettepe Journal of Mathematics and Statistic 2012; 41 (5): 675-688.
[18] Ismailov ZI, Erol M. Normal differential operators of first-order with smooth coefficients. Rocky Mountain Journal of Mathematics 2012; 42 (2): 1100-1110. doi: 10.1216/RMJ-2012-42-2-633
[19] Ismailov ZI, Karatash H. Some necessary condition for normality of differential operators. Doklady Mathematics 2000; 374 (6): 738-740 (in Russian).
[20] Ismailov ZI, Otkun Çevik E, Unluyol E. Compact inverses of multipoint normal differential operators for first order. Electronic Journal of Differential Equations 2011; 89: 1-11.
[21] Ismailov ZI, Öztürk Mert R. Normal extensions of a singular multipoint differential operator of first order. Electronic Journal of Differential Equations 2012; 36: 1-9.
[22] Ismailov ZI, Öztürk Mert R. Normal extensions of a singular differential operator on the right semi-axis. Eurasian Mathematical Journal 2014; 5 (3): 117-124.
[23] Ismailov ZI, Sertbaş M, Güler BO. Normal extensions a first order differential operators. Filomat 2014; 28 (5): 917-923. doi: 10.2298/FIL1405917I
[24] Kilpi Y. Über lineare normale transformationen in Hilbertschen raum, Annales Academiae Scientiarum Fennicae Mathematica 1953; 154: 1-38 (in German).
[25] Kilpi Y. Über das komplexe momenten problem. Annales Academiae Scientiarum Fennicae Mathematica 1957; 236: 1-32 (in German).
[26] Kilpi Y. Über die anzahl der hypermaximalen normalen fort setzungen normalen transformationen. Annales Universitasis Turkuensis Series Ai 1963; 65: 1-12 (in German).
[27] Kolmogorov AN, Fomin SV. Elements of the Theory of Functions and Functional Analysis. New York, USA: Dover Publications, 1999.
[28] Maksudov F, Ismailov ZI. Normal extensions of second-order differential operators. Differential Equations 1994; 30 (10): 1823-1824 (in Russian).
[29] Maksudov F, Ismailov ZI. Normal boundary value problems for differential equation of first order. Doklady Mathematics 1996; 350 (1): 19-21 (in Russian).
[30] Maksudov F, Ismailov ZI. A necessary condition for the normality of differential operators. Doklady Mathematics 1999; 366 (4): 452-454 (in Russian).
[31] von Neumann J. Allgemeine eigenwerttheories hermitescher funktionaloperatoren. Mathematische Annalen 1929- 1930; 102: 49-131 (in German).