SPECTRA AND PSEUDOSPECTRA OF THE DIRECT SUM OPERATORS

Abstract

In this paper, the relationships between spectrum and pseudospectrum of the direct sum of linear operators in the direct sum of Hilbert spaces and its coordinate operators have been researched. Also, the analogous relations of some numerical characteristics (spectral and pseudospectral radii) of such operators have been investigated.

Keywords

• Direct sum of operators
• spectrum
• pseudospectrum
• spectral radii

References

1. [1] Arveson W., (1994) C^*-algebras and Numerical Linear Algebra, Journal of Functional Analysis 122, 333-360.
2. [2] Davies E.B., (2002) Non-Self-Adjoint Differential Operators, Bulletin of the London Mathematical Society 34, 513-532.
3. [3] Dencker N., Sjöstrand J. and Zworski M., (2004) Pseudospectra of Semiclassical (Pseudo-) Differential Operators., Communications on Pure and Applied Mathematics 57 (3), 384-415.
4. [4] Bender C.M., Brody D.C and Jones H.F., (2002) Complex Extension of Quantum Mechanics, Physical Review Letters 89 (27), 1-4.
5. [5] Hatano N. and Nelson D.R., (1996) Localization Transitions in Non-Hermitian Quantum Mechanics, Physical Review Letters 77, 570-573.
6. [6] Trefethen L.N. and Chapman S.J., (2004) Wave Packet Pseudomodes of Twisted Toeplitz Matrices, Communications on Pure and Applied Mathematics 57 (9), 1233-1264.
7. [7] Trefethen L.N and Embree M., (2005) Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton, NJ, USA.
8. [8] Hansen A.C., (2011) On the Solvability Complexity Index, the n-Pseudospectrum and Approximations of Spectra of Operators. Journal of the American Mathematical Society 24 (1), 81-124.

9. [9] Brunner H., Iserles A. and Norsett S.P., (2010) The Spectral Problem for a Class of Highly Oscillatory Fredholm Integral Operators. IMA Journal of Numerical Analysis 30: 108-130. [10] Hansen A.C., (2010) Infinite-Dimensional Numerical Linear Algebra: Theory and Applications. Proceedings of the Royal Society London Series A Mathematical, Physical, Engineering Sciences 2010; 466 (2124): 3539-3559.
10. [11] Dunford N. and Schwartz J.T., (1963) Linear Operators II. Interscience, New York, USA.
11. [12] Ismailov Z.I., (2009) Multipoint Normal Differential Operators for First Order, Opuscula Mathematica 29 (4), 399-414.
12. [13] Kochubei A.N., (1979) Symmetric operators and nonclassical spectral problems, Matematicheskie Zametki 25 (3), 425-434.
13. [14] Timoshenko S., (1961) Theory of Elastic Stability. McGraw-Hill Book Co, New York, USA.
14. [15] Zettl A. Sturm-Lioville Theory: Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, USA.
15. [16] Gohberg I.C. and Krein M.G., (1969) Introduction to the Theory of Linear Non-Self-Adjoint Operators. Providence. American Mathematical Society, RI, USA.
16. [17] Pietsch A., (1980) Operators Ideals. North-Holland Publishing Company, Amsterdam, Netherlands.
17. [18] Pietsch A., (1987) Eigenvalues and s-Numbers. Cambridge University Press, Londan, England.
18. [19] Çevik E.O and Ismailov Z.I., (2012) Spectrum of the Direct Sum of Operators, Electronic Journal of Differential Equations 210, 1-8.
19. [20] Ismailov Z.I, Cona L. and Çevik E.O., (2015) Gelfand Numbers of Diagonal Matrices, Hacettepe Journal of Mathematics and Statistics 44 (1), 75-81.
20. [21] Ismailov Z.I., Çevik E.O. and Unluyol E., (2011) Compact Inverses of Multipoint Normal Differential Operators for First Order, Electronic Journal of Differential Equations 89, 1-11.
21. [22] Naimark N.A and Fomin S.V., (1955) Continuous Direct Sums of Hilbert Spaces and Some of Their Applications, Uspekhi Matematicheskikh Nauk 25 (64), 111-434 (in Russian).
22. [23] Cui J., Li C.K. and Poon Y.T., (2014) Pseudospectra of Special Operators and Pseudospectrum Preservers. Journal of Mathematical Analysis and Applications 419: 1261-1273.
23. [24] Hansen A.C. and Nevanlinna O., (2017) Complexity Issues in Computing Spectra, Pseudospectra and Resolvents. Banach Center Publications, Warsaw, Poland.
24. [25] Seidel M, (2012) On (N,ϵ)-Pseudospectra of Operators on Banach Spaces, Journal of Functional Analysis 262, 4916-4927.