ArticleOriginal scientific text
Title
Resolvent and spectrum of a nonselfadjoint differential operator in a Hilbert space
Authors
Abstract
We consider a second order regular differential operator whose coefficients are nonselfadjoint bounded operators acting in a Hilbert space. An estimate for the resolvent and a bound for the spectrum are established. An operator is said to be stable if its spectrum lies in the right half-plane. By the obtained bounds, stability and instability conditions are established.
Keywords
Abstract differential operator, spectrum, resolvent, stability, instability
Bibliography
- Adiguzelov, E., Karayel, S., A selfadjoint expansion of a symmetric differential operator with operator coefficient, Int. J. Contemp. Math. Sci. 2 (2007), no. 21-24, 1053-1067.
- Amrein, W., Boutet de Monvel-Berthier, A. and Georgescu, V., Hardy type inequalities for abstract differential operators, Mem. Amer. Math. Soc. 70 (1987), no. 375, 119 pp.
- Baksi, O., Sezer, Y. and Karayel, S., The sum of subtraction of the eigenvalues of two selfadjoint differential operators with unbounded operator coefficient, Int. J. Pure Appl. Math. 63 (2010), no. 3, 255-268.
- Gul, E., A regularized trace formula for a differential operator of second order with unbounded operator coefficients given in a finite interval, Int. J. Pure Appl. Math. 32 (2006), no. 2, 225-244.
- Daleckii, Yu L., Krein, M. G., Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs, vol. 43, American Mathematical Society, Providence, R. I., 1974.
- Gil’, M. I., Operator Functions and Localization of Spectra, Lecture Notes in Mathematics, vol. 1830, Springer-Verlag, Berlin, 2003.
- Gil’, M. I., Localization and Perturbation of Zeros of Entire Functions, Lecture Notes in Pure and Applied Mathematics, 258, CRC Press, Boca Raton, FL, 2010.
- Gil’, M. I., Bounds for the spectrum of a matrix differential operator with a damping term, Z. Angew. Math. Phys. 62 (2011), no. 1, 87-97.
- Gohberg, I. C., Krein, M. G., Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs, vol. 18, American Mathematical Society, Providence, R.I., 1969.
- Gohberg, I. C., Krein, M. G., Theory and Applications of Volterra Operators in Hilbert Space, Translations of Mathematical Monographs, vol. 24, American Mathematical Society, R. I., 1970.
- Krein, S. G., Linear Differential Equations in Banach Space, Translations of Mathematical Monographs, vol. 29, American Mathematical Society, Providence, R.I., 1971.
- Kunstmann, P. C., Weis, L., Maximal Lp-regularity for parabolic equations, Fourier multiplier and H1-functional calculus, in: Functional Analytic Methods for Evolution Equations, Lecture Notes in Mathematics, vol. 1855, Springer, Berlin, 2004, 65-311.
- Rofe-Beketov, F. S., Kholkin, A. M., Spectral Analysis of Differential Operators. Interplay between spectral and oscillatory properties, World Scientific Monograph Series in Mathematics, 7, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.
- Yakubov, S., Yakubov, Ya., Differential-Operator Equations. Ordinary and Partial Differential Equations, Chapman and Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 103, Chapman & Hall/CRC, Boca Raton, FL, 2000.
Main language of publication
English
Published
2012
Published online
2016-07-24
Exact and natural sciences