Spectral radius of S-essential spectra - C. Belabbaci (Математичні Студії)
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 \begin{equation} \dfrac{β(T)}{α(S)}≤ r_{e, S}(T)≤ \dfrac{α(T)}{β(S)}, \end{equation} where $α(.)$ stands for the Kuratowski measure of noncompactness and $β(.)$ 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).