Filomat 27:6 (2013), 1147–1155 DOI DOI: 10.2298/FIL1306147A Published by Faculty of Sciences and Mathematics, University of Niˇ s, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Fredholm Perturbations and Seminorms Related to Upper Semi-Fredholm Perturbations Boulbeba Abdelmoumen a , Hamadi Baklouti b a D´ epartement de math´ ematiques, IPEIS, Sfax University b D´ epartement de math´ ematiques, FSS, Sfax University Abstract. In this paper, we investigate the stability in the set of Fredholm perturbations under composition with bounded operators. Moreover, we introduce the concept of a measure of non-Fredholm perturbations, which allows us to give a general approach to the question of obtaining perturbation theorems for semi- Fredholm operators. Finally, we prove some localization results about the Wolf, the Schechter and the Browder essential spectrum of bounded operators on a Banach space X. 1. Introduction Let X and Y be two Banach spaces. We denote by L(X, Y) (respectively K (X, Y)) the space of all bounded (respectively compact) linear operators from X into Y. If T ∈L(X, Y), we write N (T) ⊆ X and Ran(T) ⊆ Y for the null space and the range of T. We set α(T):= dim N (T) and β(T):= codim Ran(T). The sets of upper and lower semi-Fredholm operators in L(X, Y) are denoted by Φ + (X, Y) and Φ − (X, Y). We use Φ ± (X, Y):=Φ + (X, Y) ∪ Φ − (X, Y) for the set of semi-Fredholm operators, and Φ(X, Y):=Φ + (X, Y) ∩ Φ − (X, Y) for the set of Fredholm operators. If T ∈ Φ ± (X, Y), then i(T):= α(T) − β(T) is called the index of T. If X = Y, we simply write L(X), K (X), Φ + (X), Φ − (X), Φ ± (X) and Φ(X). Set N ∞ (T) = ⊔ n N (T n ), R ∞ (T) = ∩ n Ran(T n ), and denote by a(T) respectively δ(T), the ascent and the descent of T ∈L(X). The sets of upper and lower semi-Browder operators are denoted by B + (X) and B − (X). The set of Browder operators on X is B(X) = B + (X) ∩B − (X). In this paper we are concerned with the following essential spectra: Wolf essential spectrum: σ e (T):= {λ ∈ C such that λ − T < Φ(X)}, Schechter essential spectrum: σ ess (T):= C\{λ − T ∈ Φ(X) such that i(λ − T) = 0}, Browder essential spectrum : σ b = {λ ∈ C; λ − T < B(X)}. An operator T ∈L(X, Y) is said to be left Atkinson if T ∈ Φ + (X, Y) and Ran(T) is complemented. The operator T ∈L(X, Y) is said to be right Atkinson if T ∈ Φ − (X, Y) and N (T) is complemented. The class of left Atkinson operators and right Atkinson operators will be denoted by Φ l (X, Y) and Φ r (X, Y), respectively. 2010 Mathematics Subject Classification. Primary 47A53; Secondary 47A55, 47A68 Keywords. Measure of noncompactness, Fredholm operators, Fredholm perturbations, Essential spectra. Received: 05 December 2012; Accepted: 15 March 2013 Communicated by Dragana Cvetkovic-Ilic Email addresses: Boulbeba.Abdelmoumen@ipeis.rnu.tn (Boulbeba Abdelmoumen), h.baklouti@gmail.com (Hamadi Baklouti)