# ON WEIGHTED AND PSEUDO-WEIGHTED SPECTRA OF BOUNDED OPERATORS

• Athmouni, Nassim (Department of Mathematics Faculty of Sciences of Gafsa University of Gafsa) ;
• Baloudi, Hatem (Department of Mathematics Faculty of Sciences of Gafsa University of Gafsa) ;
• Jeribi, Aref (Department of Mathematics Faculty of Sciences of Sfax University of Sfax) ;
• Kacem, Ghazi (Department of Mathematics Faculty of Sciences of Sfax University of Sfax)
• Accepted : 2017.12.21
• Published : 2018.07.31
• 192 11

#### Abstract

In the present paper, we extend the main results of Jeribi in [6] to weighted and pseudo-weighted spectra of operators in a nonseparable Hilbert space ${\mathcal{H}}$. We investigate the characterization, the stability and some properties of these weighted and pseudo-weighted spectra.

#### Keywords

Fredholm operators;${\alpha}-Fredholm$ operator;pseudo-spectrum;weighted spectrum

#### References

1. A. Ammar, B. Boukettaya, and A. Jeribi, A note on the essential pseudospectra and application, Linear Multilinear Algebra 64 (2016), no. 8, 1474-1483. https://doi.org/10.1080/03081087.2015.1091436
2. A. Ammar, H. Daoud, and A. Jeribi, The stability of pseudospectra and essential pseu- dospectra of linear relations, J. Pseudo-Diﬀer. Oper. Appl. 7 (2016), no. 4, 473-491. https://doi.org/10.1007/s11868-016-0150-3
3. S. C. Arora and P. Dharmarha, On weighted Weyl spectrum. II, Bull. Korean Math. Soc. 43 (2006), no. 4, 715-722. https://doi.org/10.4134/BKMS.2006.43.4.715
4. H. Baloudi and A. Jeribi, Left-right Fredholm and Weyl spectra of the sum of two bounded operators and applications, Mediterr. J. Math. 11 (2014), no. 3, 939-953. https://doi.org/10.1007/s00009-013-0372-z
5. L. Burlando, Approximation by semi-Fredholm and $semi-{\alpha}-Fredholm$ operators in Hilbert spaces of arbitrary dimension, Acta Sci. Math. (Szeged) 65 (1999), no. 1-2, 217-275.
6. L. A. Coburn and A. Lebow, Components of invertible elements in quotient algebras of operators, Trans. Amer. Math. Soc. 130 (1968), 359-365. https://doi.org/10.1090/S0002-9947-1968-0223901-2
7. E. B. Davies, Spectral Theory and Diﬀerential Operators, Cambridge Studies in Advanced Mathematics, 42, Cambridge University Press, Cambridge, 1995.
8. S. V. Djordjevic and F. Hernandez-Diaz, ${\alpha}-Fredholm$ spectrum of Hilbert space operators, J. Indian Math. Soc. 83 (2016), no. 3-4, 241-249.
9. S. V. Djordjevic, On ${\alpha}-Weyl$ operators, Advances in Pure Math. 6 (2016), 138-143. https://doi.org/10.4236/apm.2016.63011
10. G. Edgar, J. Ernest, and S. G. Lee, Weighing operator spectra, Indiana Univ. Math. J. 21 (1971/72), 61-80. https://doi.org/10.1512/iumj.1972.21.21005
11. A. Elleuch and A. Jeribi, New description of the structured essential pseudospectra, Indag. Math. (N.S.) 27 (2016), no. 1, 368-382. https://doi.org/10.1016/j.indag.2015.11.002
12. A. Elleuch, On a characterization of the structured Wolf, Schechter and Browder essential pseudospectra, Indag. Math. (N.S.) 27 (2016), no. 1, 212-224. https://doi.org/10.1016/j.indag.2015.10.003
13. I. C. Gohberg, A. S. Markus, and I. A. Feldman, Normally solvable operators and ideals associated with them, Trans. Amer. Math. Soc. 61 (1967), 63-84.
14. K. Gustafson and J. Weidmann, On the essential spectrum, J. Math. Anal. Appl. 25 (1969), 121-127. https://doi.org/10.1016/0022-247X(69)90217-0
15. A. Jeribi, Quelques remarques sur le spectre de Weyl et applications, C. R. Acad. Sci. Paris Ser. I Math. 327 (1998), no. 5, 485-490. https://doi.org/10.1016/S0764-4442(99)80027-5
16. A. Jeribi, Une nouvelle caracterisation du spectre essentiel et application, C. R. Acad. Sci. Paris Ser. I Math. 331 (2000), no. 7, 525-530. https://doi.org/10.1016/S0764-4442(00)01606-2
17. A. Jeribi, Some remarks on the Schechter essential spectrum and applications to transport equations, J. Math. Anal. Appl. 275 (2002), no. 1, 222-237. https://doi.org/10.1016/S0022-247X(02)00323-2
18. A. Jeribi, A characterization of the Schechter essential spectrum on Banach spaces and applications, J. Math. Anal. Appl. 271 (2002), no. 2, 343-358. https://doi.org/10.1016/S0022-247X(02)00115-4
19. A. Jeribi, On the Schechter essential spectrum on Banach spaces and application, Facta Univ. Ser. Math. Inform. No. 17 (2002), 35-55.
20. A. Jeribi, Fredholm operators and essential spectra, Arch. Inequal. Appl. 2 (2004), no. 2-3, 123-140.
21. A. Jeribi, Spectral Theory and Applications of Linear Operators and Block Operator Matrices, Springer, Cham, 2015.
22. E. Luft, The two-sided closed ideals of the algebra of bounded linear operators of a Hilbert space, Czechoslovak Math. J. 18(93) (1968), 595-605.
23. M. Schechter, Principles of Functional Analysis, Academic Press, New York, 1971.
24. L. N. Trefethen, Pseudospectra of matrices, in Numerical analysis 1991 (Dundee, 1991), 234-266, Pitman Res. Notes Math. Ser., 260, Longman Sci. Tech., Harlow, 1991.
25. F. Wolf, On the invariance of the essential spectrum under a change of boundary conditions of partial diﬀerential boundary operators, Nederl. Akad. Wetensch. Proc. Ser. A 62 = Indag. Math. 21 (1959), 142-147.