Discreteness of Spectrum of Normal Differential Operators for First Order

In this work under the condition A−1 ∈ C∞(H), we investigate the discreteness of spectrum of normalextensions in detail. Later on, the asymptotical behavior of eigenvalues of any normal extension has been examined.

PDF

___

[1] Coddington, E.A., Extension theory of formally normal and symmetric subspaces , Mem. Amer. Math. Soc., 134 (1973), 1 80.
[2] Davis, R. H, Singular Normal Differential Operators, Tech. Rep., Dep. Math., California Univ., 1955.
[3] Dunford, N., Schwartz, J. T., Linear Operators I, II, Second ed., Interscience, New York, 1958; 1963.
[4] Gohberg, I.C., Krein, M.G., Introduction to the Theory of Linear Non-Self-Adjoint Operators, Amer. Math. Soc., Providence, RI, 1969.
[5] Gorbachuk, M.L., Self-Adjoint Boundary Value Problems for the Differential Equations for Second Order with the Unbounded Operator Coefficient, Funktsional. Anal. i Prilozhen. 5 (1971), 10-21 (in Russian).
[6] Gorbachuk, V.I., Gorbachuk, M.L., Boundary Value Problems for Operator Differential Equations, Kluwer Academic, Dordrecht, 1991.
[7] Hormander, L., ¨ On the theory of general partial differential operators, Acta Mathematica, 94 (1955), 161-248.
[8] Ipek Al, P., Yılmaz, B., Ismailov, Z.I., The general form of normal quasi-differential operators for first order and their spectrum, Turkish Journal of Mathematics and Computer Science, 8 (2018), 22-28.
[9] Ismailov, Z. I., Compact inverses of first-order normal differential operators, J. Math., Anal. Appl. USA, 320,1(2006),266-278.
[10] Kilpi, Y., Uber lineare normale transformationen in Hilbertschen raum ¨ , Ann. Acad. Sci. Fenn. Math. Ser. AI 154 (1953).
[11] Kilpi, Y., Uber die anzahl der hypermaximalen normalen fort setzungen normalen transformationen ¨ , Ann. Univ. Turkuensis. Ser. AI 65 (1963).
[12] Kolmogorov, A.N., Fomin, S.V., Elements of the Theory of Functions and Functional Analysis, Dover Books on Mathematics, 1999.
[13] Zettl, A., Sun, J., Survey article: Self-adjoint ordinary differential operators and their spectrum, Roky Mountain Journal of Mathematics, 45, (2015), 763-886.