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

Michael Gil

Annales UMCS, Mathematica (2012)

  • Volume: 66, Issue: 1, page 25-39
  • ISSN: 2083-7402

Abstract

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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.

How to cite

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Michael Gil. "Resolvent and spectrum of a nonselfadjoint differential operator in a Hilbert space." Annales UMCS, Mathematica 66.1 (2012): 25-39. <http://eudml.org/doc/268249>.

@article{MichaelGil2012,
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.},
author = {Michael Gil},
journal = {Annales UMCS, Mathematica},
keywords = {Abstract differential operator; spectrum; resolvent; stability; instability; abstract differential operator},
language = {eng},
number = {1},
pages = {25-39},
title = {Resolvent and spectrum of a nonselfadjoint differential operator in a Hilbert space},
url = {http://eudml.org/doc/268249},
volume = {66},
year = {2012},
}

TY - JOUR
AU - Michael Gil
TI - Resolvent and spectrum of a nonselfadjoint differential operator in a Hilbert space
JO - Annales UMCS, Mathematica
PY - 2012
VL - 66
IS - 1
SP - 25
EP - 39
AB - 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.
LA - eng
KW - Abstract differential operator; spectrum; resolvent; stability; instability; abstract differential operator
UR - http://eudml.org/doc/268249
ER -

References

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  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. Zbl0633.35009
  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. Zbl1220.47004
  4. Gűl, 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. Zbl1134.47032
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  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.[WoS] Zbl1230.34072
  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. Zbl0181.13504
  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. Zbl0194.43804
  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 H ∞-functional calculus, in: Functional Analytic Methods for Evolution Equations, Lecture Notes in Mathematics, vol. 1855, Springer, Berlin, 2004, 65-311. Zbl1097.47041
  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. Zbl1090.47030
  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. Zbl0936.35002

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