$S$-spectra and $S$-essential pseudospectra of the diagonal block operator matrices

Abstract

In this article, the relationships between the $S$-spectra, the $ S$-spectral radius, the $ \epsilon$-$S$-essential pseudospectra, and the $ \epsilon$-$S$-essential pseudospectral radius of the diagonal block operator matrices in the direct sum of Banach spaces and their block coordinate operators are studied. Then, the results are supported by applications.

Keywords

• $ S$-spectrum
• $ \epsilon$-$S$-essential pseudospectrum
• $ S$-spectral radius
• $ \epsilon$-$S$-essential pseudospectral radius

References

1. [1] F. Abdmouleh, A. Ammar and A. Jeribi, Characterization of the pseudo-browder es- sential spectra of linear operators and application to a transport equation, J. Comput. Theor. Transp. 44 (3), 141-153, 2015.
2. [2] N. Dunford and J.T. Schwartz, Linear operators II, Interscience, 1963.
3. [3] K. Gustafson and J. Weidmann, On the essential spectrum, J. Math. Anal. Appl. 25, 121-127, 1969.
4. [4] A. Jeribi, Spectral Theory and Applications of Linear Operators and Block Operator Matrices, Springer-Verlag, 2015.
5. [5] A. Jeribi, Linear Operators and Their Essential Pseudospectra, Apple Academic Press, 2018.
6. [6] Z.I. Ismailov, Multipoint normal differential operators for first order, Opuscula Math. 29 (4), 399–414, 2009.
7. [7] Z.I. Ismailov, L. Cona and E. Otkun Çevik, Gelfand numbers of diagonal matrices, Hacet. J. Math. Stat. 44 (1), 75-81, 2015.
8. [8] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1980.

9. [9] A.N. Kochubei, Symmetric operators and nonclassical spectral problems, Mat. Za- metki. 25 (3), 425–434, 1979.
10. [10] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Sequence Spaces, Springer-Verlag, 1977.
11. [11] R. Mennicken and A.K. Motovilov, Operator interpretation of resonances arising in spectral problems for 2 × 2 operator matrices, Math. Nachr. 201, 117-181, 1999.
12. [12] E. Otkun Çevik and Z.I. Ismailov, Spectrum of the direct sum of operators, Electron. J. Differential Equations, 210, 1–8, 2012.
13. [13] M. Schechter, On the essential spectrum of an arbitrary operator, I. J. Math. Anal. Appl. 13, 205-215, 1966.
14. [14] S. Timoshenko, Theory of Elastic Stability, McGraw-Hill Book Co, 1961.
15. [15] C. Tretter, Spectral Theory of Block Operator Matrices and Applications, Imperial College Press, 2008.
16. [16] H. Weyl, Uber beschrankte quadratische Formen, deren differenz vollstelig ist, Rend. Circ. Mat. Palermo, 27, 373-392, 1909.
17. [17] A. Zettl, Sturm-Lioville Theory, Mathematical Surveys and Monographs, American Mathematical Society, 2005.