Spectral radius of S-essential spectra

  • C. Belabbaci University of Laghouat, Algeria
Keywords: S-essential spectra, measure of noncompactness, spectral radius, Fredholm operators


In this paper, we study the spectral radius of some S-essential spectra of a bounded linear operator defined on a Banach space. More precisely, via the concept of measure of noncompactness,we show that for any two bounded linear operators $T$ and $S$ with $S$ non zero and non compact operator the spectral radius of the S-Gustafson, S-Weidmann, S-Kato and S-Wolf essential spectra are given by the following inequalities
\dfrac{\beta(T)}{\alpha(S)}\leq r_{e, S}(T)\leq \dfrac{\alpha(T)}{\beta(S)},
where $\alpha(.)$ stands for the Kuratowski measure of noncompactness and $\beta(.)$ is defined in [11].
In the particular case when the index of the operator $S$ is equal to zero, we prove the last inequalities for the spectral radius of the S-Schechter essential spectrum. Also, we prove that the spectral radius of the S-Jeribi essential spectrum satisfies inequalities 2) when the Banach space $X$ has no reflexive infinite dimensional subspace and the index of the operator $S$ is equal to zero (the S-Jeribi essential spectrum, introduced in [7]
as a generalisation of the Jeribi essential spectrum).

Author Biography

C. Belabbaci, University of Laghouat, Algeria

Laboratory of Pure and Applied Mathematics Department of mathematics, University of Laghouat, Algeria


B. Abdelmoumen, H. Baklouti, Perturbation results on semi-Fredholm operators and applications, J Inequal Appl. 2009 Article ID 284526, 1-13.

F. Abdmouleh, A. Ammar, A. Jeribi, Stability of the S-essential spectra on a Banach space, Mathematica Slovaca, 63 (2013), 299-320.

R. Akhmerov, M. Kamenski, A. Potapov, A. Rodkina, B. Sadovskii, Measures of Noncompactness and Condensing Operators, Operator Theory: Advances and Applications, 55, Birkhauser Verlag, Basel, Boston, Berlin, 1992.(in Russian).

A. Ammar, B. Boukettaya, A. Jeribi, Stability of the S-left and S-right essential spectra of a linear operator, Acta Math Sci Ser A Chin Ed. 34 (2014), 1922-1934.

A. Ammar, M. Zerai Dhahri, A. Jeribi, A characterization of S-essential spectrum by means of measure of non-strict-singularity and application, Azerb J Math. 5 (2015).

C. Belabbaci, Jeribi essential spectrum, Lib Math(N C). 37 (2018), 65-73.

C. Belabbaci, M. Aissani, M. Terbeche, S-essential spectra and measure of noncompactness, Mathematica Slovaca, 67 (2017), 1203-1212.

M. Benchohra, I. Medjadj, Measure of noncompactness and partial functional differential equations with state-dependent delay, Differ Equ Dyn Syst. (2016), 1-17.

A. Jeribi, Spectral theory and application of linear operators and block operator matrices, springer, 2015.

D. Edmunds, W D. Evans, Spectral Theory and Differential Operators, Oxford Mathematical Monographs, Clarendon Press, 1987.

M. Furi, M. Martelli, A. Vignoli, Contributions to the spectral theory for nonlinear operators in Banach spaces, Annali di Matematica Pura ed Applicata. 118 (1978), 229-294.

A. Jeribi, N. Moalla, S. Yengui, S-essential spectra and application to an example of transport operators, Math Methods Appl Sci. 37 (2014), 2341-2353.

R.D. Nussbaum, The radius of the essential spectrum, Math Duke math J. 37 (1970), 473-478.

E. Zeidler, Nonlinear Functional Analysis and its Applications I, Springer-Verlag, 1986.

How to Cite
Belabbaci C. Spectral radius of S-essential spectra. Mat. Stud. [Internet]. 2020Oct.6 [cited 2021Dec.9];54(1):91-7. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/22