Birinci Mertebeden Normal Diferansiyel Operatörlerin Bazı Sınıfları

Yıl 2023, Cilt: 10 Sayı: 2, 356 - 362, 30.11.2023

Rukiye Öztürk Mert

https://doi.org/10.35193/bseufbd.1223029

Öz

Bu çalışmada, sonlu simetrik aralıktaki vektör fonksiyonların Hilbert uzayında, birinci mertebeden lineer diferansiyel-operatör ifadesi tarafından doğrulan minimal ve maksimal operatörleri oluşturulmuştur. Daha sonra, bu minimal operatörün defekt sayıları hesaplanmış ve sınır değer uzayı oluşturulmuştur. Calkin-Gorbachuk yöntemi kullanılarak, formal normal minimal operatörün tüm normal genişlemelerinin sınır değerler dilinde genel formu oluşturulmuştur. Son olarak, bu genişlemelerin spektrum yapısı araştırılmıştır.

Anahtar Kelimeler

Normal Diferansiyel Operatör, Defekt Sayıları, Sınır Değer Uzayı, Spektrum

On Some Class of Normal Differential Operators for First Order

Öz

In this work, we construct the minimal and maximal operators generated by linear differential-operator expression for first order in the Hilbert space of vector-functions on finite symmetric interval. Then, deficiency indices of the minimal operator will be calculated and the space of boundary values of this operator will be constructed. By using of Calkin-Gorbachuk method, the general representation of all normal extensions of the formally normal minimal operator in terms of boundary values will also be established. Moreover we explore the spectrum structure of these extensions.

Anahtar Kelimeler

Normal Differential Operator, Deficiency Indices, Space of Boundary Value, Spectrum

Kaynakça

• Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., & Holden, H. (2005). Solvable Models in Quantum Mechanics. AMS Chelsea Publishing, Providence, RI, USA, 488.
• Zettl, A. (2005). Sturm-Liouville Theory. American Mathematical Society, Mathematical Surveys and Monographs, 121, Providence, RI, USA, 328.
• Von Neumann, J. (1929-1930). Allgemeine eigenwerttheories hermitescher funktionaloperatoren. Mathematische Annalen, 102, 49-131 (in German).
• Stone, M. H. (1932). Linear transformations in Hilbert space and their applications in analysis. Amer. Math. Soc. Collag., 15, 49-131.
• Glazman, I. M. (1950). On the theory of singular differential operators. Uspehi Math. Nauk, 40, 102-135.
• Naimark, M. A. (1968). Linear Differential Operators, II. Ungar, NewYork, 352.
• Gorbachuk V. I. & Gorbachuk, M. L. (1991). Boundary Value Problems for Operator Differential Equations, Kluwer Academic Publishers, Dordrecht, the Netherlands, 347.
• Rofe-Beketov, F. S. & Kholkin, A. M. (2005). Spectral analysis of differential operators, World Scientific Monograph Series in Mathematics, 7, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 438.
• El-Gebeily, M. A., O'Regan, D. & Agarwal, R. (2011). Characterization of self-adjoint ordinary differential operators. Mathematical and Computer Modelling, 54, 659-672.
• Everitt, W. N. & Markus, L. (1997). The Glazman-Krein-Naimark Theorem for ordinary differential operators. Operator Theory, Advances and Applications, 98, 118-130.
• Everitt, W. N. & Poulkou, A. (2003). Some observations and remarks on differential operators generated by first order boundary value problems. Journal of Computational and Applied Mathematics, 153, 201-211.
• Zettl, A. & Sun, J. (2015). Survey Article: Self-adjoint ordinary differential operators and their spectrum. Rosky Mountain Journal of Mathematics, 45 (1), 763-886.
• Coddington, E. A. (1973). Extension Theory of Formally Normal and Symmetric Subspaces, American Mathematical Society, Providence, RI, USA, 80.
• Kilpi, Y. (1953). Über lineare normale transformationen in Hilbertschen raum. Annales Academiae Scientiarum Fennicae Mathematica, 154, 1-38 (in German).
• Kilpi, Y. (1957). Über das komplexe momenten problem. Annales Academiae Scientiarum Fennicae Mathematica, 236, 1-32 (in German).
• Kilpi, Y. (1963). Über die anzahl der hypermaximalen normalen fort setzungen normalen transformationen. Annales Universitasis Turkuensis Series Ai, 65, 1-12 (in German).
• Davis, R. H. (1955). Singular normal differential operators, PhD, University of California, Berkeley, USA, 96.
• Ismailov, Z. I. (2006). Compact inverses of first-order normal differential operators. Journal of Mathematical Analysis and Applications, 320 (1), 266-278.
• Ismailov, Z. I. & Erol, M. (2012). Normal differential operators of first-order with smooth coefficients. Rocky Mountain Journal of Mathematics, 42 (2), 1100-1110.
• Ismailov, Z. I. & Erol, M. (2012). Normal differential operators of third-order. Hacettepe Journal of Mathematics and Statistic, 41 (5), 675-688.
• Ismailov, Z. I. & Öztürk Mert, R. (2012). Normal extensions of a singular multipoint differential operator of first order. Electronic Journal of Differential Equations, 36, 1-9.
• Ismailov, Z. I. & Öztürk Mert, R. (2014). Normal extensions of a singular differential operator on the right semi-axis. Eurasian Mathematical Journal, 5 (3), 117-124.
• Ismailov, Z. I., Sertbaş, M., & Güler, B. O. (2014). Normal extensions a first order differential operator. Filomat, 28 (5), 917-923.
• Ismailov, Z. I., Güler, B. Ö., & Ipek, P. (2015). Solvability of first order functional differential operators. Journal of Mathematical Chemistry, 53 (9), 2065-2077.
• Ismailov, Z. I., Güler, B. Ö., & Ipek, P. (2016). Solvable time-delay differential operators for first order and their spectrums. Hacettepe Journal of Mathematics and Statistics, 3 (45), 755-764.
• Ismailov, Z. I. & Ipek, P. (2014). Spectrums of solvable pantograph differential-operators for first order. Abstract and Applied Analysis, 2014, 1-8.
• Ismailov, Z. I. & Ipek, P. (2015). Solvability of multipoint differential operators of first order. Electronic Journal of Differential Equations, 36, 1-10.
• Ismailov, Z. I. & Ipek Al, P. (2019). Boundedly solvable neutral type delay differential operators of the first order. Eurasian Mathematical Journal, 10 (3), 20-27.
• Ismailov, Z. I., Yılmaz, B., & Ipek, P. (2017). Delay differential operators and some solvable models in life sciences. Communications Faculty of Science University of Ankara Series A1 Mathematics and Statistics, 66 (2), 91-99.
• Ipek Al, P. & Akbaba, Ü. (2020). On the compactly solvable differential operators for first order. Lobachevskii Journal of Mathematics, 41 (6), 1078-1086.
• Ipek Al, P. & Ismailov, Z. I. (2021). First order selfadjoint differential operators with involution. Lobachevskii Journal of Mathematics, 42 (3), 496-501.
• Ipek Al, P. & Akbaba, Ü. (2021). Maximally dissipative differential operators of first order in the weight Hilbert space. Lobachevskii Journal of Mathematics, 42 (3), 490-495.
• Ipek Al, P. & Ismailov, Z.I. (2018). Maximal accretive singular quasi-differential operators. Hacettepe Journal of Mathematics and Statistics, 47, 1120-1127.
• Akbaba, Ü. & Ipek Al, P. (2021). Maximally accretive differential operators of first order in the weight Hilbert space. Lobachevskii Journal of Mathematics, 42 (12), 2707-2713.
• Hörmander, L. (1955). On the theory of general partial differential operators. Acta Mathematica, 94, 161-248.