Geodesic
Resolvent and spectrum of a nonselfadjoint differential operator in a Hilbert space
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 66 (2012) no. 1.

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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 
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Gil’, Michael. Resolvent and spectrum of a nonselfadjoint differential operator in a Hilbert space. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 66 (2012) no. 1. http://geodesic.mathdoc.fr/item/AUM_2012_66_1_a6/

[1] 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.

[2] 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.

[3] 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.

[4] 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.

[5] 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.

[6] Gil’, M. I., Operator Functions and Localization of Spectra, Lecture Notes in Mathematics, vol. 1830, Springer-Verlag, Berlin, 2003.

[7] 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.

[8] 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.

[9] 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.

[10] 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.

[11] Krein, S. G., Linear Differential Equations in Banach Space, Translations of Mathematical Monographs, vol. 29, American Mathematical Society, Providence, R.I., 1971.

[12] 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.

[13] 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.

[14] 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.