Resolvent and spectrum of a nonselfadjoint differential operator in a Hilbert space

Michael Gil’

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

Full Text:

PDF

References


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.




DOI: http://dx.doi.org/10.2478/v10062-012-0004-2
Date of publication: 2016-07-24 20:22:24
Date of submission: 2016-07-24 15:49:17


Statistics


Total abstract view - 667
Downloads (from 2020-06-17) - PDF - 521

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2012 Michael Gil’