On the spectrum of linear operator pencils

F. Bouchelaghem; M. Benharrat

doi:10.30970/ms.52.2.211-220

We consider a linear operator pencil $L (\lambda)=A- \lambda B$, $\lambda \in \mathbb{C}$, where $A$ and $B$ are bounded operators on Hilbert space. The purpose of this paper is to study the conditions under which the spectrum of $L (.)$ to be the whole complex plane or empty. This leads to some criteria for the spectrum to be bounded.

1. R.W. Brockett, Finite Dimensional Linear Systems, John Wiley & Sons, Inc., 1970.

2. J.A. Brooke, P. Busch, B. Pearson, Commutativity up to a factor of bounded operators in complex Hilbert space, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., A, 458 (2002), 109-118.

3. S.L. Campbell, Singular Systems of Differential Equations I, San Francisco, CA, USA: Pitman, 1980.

4. S.L. Campbell, Singular Systems of Differential Equations II, Pitman Advanced Publishing, 1982.

5. S.L. Campbell, C.D. Meyer, N.J. Rose Application of the Drazin inverse to linear systems of differential equations with singular constant coefficients, SIAM J. Appl. Math., 31 (1976), 411-425.

6. R.G. Douglas, On majorization, factorization, and range inclusion of the operators on Hilbert space, Proc. Amer. Math. Soc., 17 (1966), 413-415.

7. P.A. Fillmore, J.P. Williams, On operator ranges, Advances in Math., 7 (1971), 254-281.

8. C.K. Fong, A.R. Sourour, Sums and products of quasi-nilpotent operators, Proc. Roy. Soc. Edinburgh, 99A (1984), 193-200.

9. I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, AMS, 1988.

10. T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1995.

11. T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Anal. Math., 6 (1958), 261.322.

12. J.J. Koliha, A generalized Drazin inverse, Glasgow Math. J., 38 (1996), 367-381.

13. J.J. Koliha, Isolated Spectral Points, Proc. Ame. Math. Soc., 124 (1996), №11, 3417-3424.

14. M. Moller, V. Pivovarchik, Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and their Applications, Birkhauser, 2015.

15. C. Schmoeger, On isolated points of the spectrum of a bounded operators, Proc. Ame. Math. Soc., 117 (1993), 715-719.

16. J.P. Williams, Spectra of products and numerical ranges, J. Math. Anal. Appl., 17 (1967), 214-220.

17. J. Yang, H. Du, A note an commutativity up to a factor of bounded operators, Proc. Amer. Math. Soc., 132 (2004), №4, 1713-1720.

	On the spectrum of linear operator pencils
Author	F. Bouchelaghem¹, M. Benharrat² fairouzbouchelaghem@yahoo.fr¹, benharrat@math.univ-lyon1.fr² 1) Laboratory of Fundamental and Applicable Mathematics of Oran (LMFAO), Department of Mathematics, University of Oran 1 Ahmed Ben Bella, Oran, Algeria; 2) Laboratory of Fundamental and Applicable Mathematics of Oran (LMFAO), Department of Mathematics and informatics, National Polytechnic School of Oran-Maurice Audin (Ex. ENSET of Oran), Oran, Algeria
Abstract	We consider a linear operator pencil $L (\lambda)=A- \lambda B$, $\lambda \in \mathbb{C}$, where $A$ and $B$ are bounded operators on Hilbert space. The purpose of this paper is to study the conditions under which the spectrum of $L (.)$ to be the whole complex plane or empty. This leads to some criteria for the spectrum to be bounded.
Keywords	linear operator pencil; spectral theory; perturbations theory; linear-quadratic optimal control problems
DOI	doi:10.30970/ms.52.2.211-220
Reference	1. R.W. Brockett, Finite Dimensional Linear Systems, John Wiley & Sons, Inc., 1970. 2. J.A. Brooke, P. Busch, B. Pearson, Commutativity up to a factor of bounded operators in complex Hilbert space, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., A, 458 (2002), 109-118. 3. S.L. Campbell, Singular Systems of Differential Equations I, San Francisco, CA, USA: Pitman, 1980. 4. S.L. Campbell, Singular Systems of Differential Equations II, Pitman Advanced Publishing, 1982. 5. S.L. Campbell, C.D. Meyer, N.J. Rose Application of the Drazin inverse to linear systems of differential equations with singular constant coefficients, SIAM J. Appl. Math., 31 (1976), 411-425. 6. R.G. Douglas, On majorization, factorization, and range inclusion of the operators on Hilbert space, Proc. Amer. Math. Soc., 17 (1966), 413-415. 7. P.A. Fillmore, J.P. Williams, On operator ranges, Advances in Math., 7 (1971), 254-281. 8. C.K. Fong, A.R. Sourour, Sums and products of quasi-nilpotent operators, Proc. Roy. Soc. Edinburgh, 99A (1984), 193-200. 9. I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, AMS, 1988. 10. T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1995. 11. T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Anal. Math., 6 (1958), 261.322. 12. J.J. Koliha, A generalized Drazin inverse, Glasgow Math. J., 38 (1996), 367-381. 13. J.J. Koliha, Isolated Spectral Points, Proc. Ame. Math. Soc., 124 (1996), №11, 3417-3424. 14. M. Moller, V. Pivovarchik, Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and their Applications, Birkhauser, 2015. 15. C. Schmoeger, On isolated points of the spectrum of a bounded operators, Proc. Ame. Math. Soc., 117 (1993), 715-719. 16. J.P. Williams, Spectra of products and numerical ranges, J. Math. Anal. Appl., 17 (1967), 214-220. 17. J. Yang, H. Du, A note an commutativity up to a factor of bounded operators, Proc. Amer. Math. Soc., 132 (2004), №4, 1713-1720.
Pages	211-221
Volume	52
Issue	2
Year	2019
Journal	Matematychni Studii
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