International Journal of Analysis and Applications ISSN 2291-8639 Volume 15, Number 1 (2017), 1-7 http://www.etamaths.com THE ESSENTIAL SPECTRUM OF A SEQUENCE OF LINEAR OPERATORS IN BANACH SPACES AYMEN AMMAR1,∗, NOUI DJAIDJA2 AND AREF JERIBI1 Abstract. In this work we introduce some essential spectra (σei, i = 1, ...,5) of a sequence of closed linear operators (Tn)n∈N on Banach space, we prove that if (Tn)n∈N converges in the generalized sense to a closed linear operator T, then there exists n0 ∈ N such that, for every n ≥ n0, we have σei(λ0 − (Tn + B)) ⊆ σei(λ0 − (T + B)), i = 1, ...,5, where B is a bounded linear operator, and λ0 ∈ C. The same treatment is made when (Tn −T) converges to zero compactly. 1. Introduction Let X and Y be two Banach spaces. We denote by L(X,Y ) (resp., C(X,Y )) the set of all bounded (resp., closed, densely defined) linear operators from X into Y while K(X,Y ) designates the subspace of compact operators from X into Y . If T ∈ C(X,Y ), we write N(T) and R(T) for the null space and range of T, we set α(T)=dimN(T), β(T) = codimR(T). The classes of Fredholm, upper semi- Fredholm and lower semi-Fredholm operators from X into Y are, respectively, the following: Φ(X,Y ) := { T ∈C(X,Y ) : α(T) < ∞ and β(T) < ∞ ,R(T) is closed inY } . Φ+(X,Y ) := { T ∈C(X,Y ) : α(T) < ∞ and R(T) is closed in Y } . Φ−(X,Y ) := { T ∈C(X,Y ) : β(T) < ∞ and R(T) is closed in Y } . The set of semi-Fredholm operators from X into Y is defined by Φ±(X,Y ) := Φ+(X,Y ) ∪ Φ−(X,Y ). The set of Fredholm operators from X into Y is defined by Φ(X,Y ) := Φ+(X,Y ) ∩ Φ−(X,Y ). For T ∈ Φ±(X,Y ), the number i(T) = α(T) −β(T) is called the index of T. Definition 1.1. An operator F ∈L(X,Y ) is called a Fredholm perturbation if T +F ∈ Φ(X,Y ) when- ever T ∈ Φ(X,Y ). F is called an upper (respectively, lower) Fredholm perturbation if T +F ∈ Φ+(X,Y )( respectively, Φ−(X,Y ) ) whenever T ∈ Φ+(X,Y ) ( respectively, Φ−(X,Y ) ) . The sets of Fredholm, up- per semi-Fredholm and lower semi-Fredholm perturbations are denoted by F(X,Y ), F+(X,Y ) and F−(X,Y ), respectively. Let Φb(X,Y ), Φb+(X,Y ) and Φ b −(X,Y ) denote the set Φ(X,Y )∩L(X,Y ), Φ+(X,Y )∩L(X,Y ) and Φ−(X,Y ) ∩L(X,Y ), respectively. Definition 1.2. Let A be a closable linear operator in a Banach space X. The resolvent set and the spectrum of A are, respectively, defined as ρ(A) := { λ ∈ C, such that (λ−A) is injective and (λ−A)−1 ∈L(X) } , σ(A) := C\ρ(A). Received 26th April, 2017; accepted 6th July, 2017; published 1st September, 2017. 2010 Mathematics Subject Classification. 47A10. Key words and phrases. essential spectra; convergence in the generalized sense; convergence to zero compactly. c©2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 1 2 AMMAR, DJAIDJA AND JERIBI Definition 1.3. Let A be a closed linear operator in a Banach space X. We define the sets σe1(A) := { λ ∈ C, such that λ−A 6∈ Φ+(X) } , σe2(A) := { λ ∈ C, such that λ−A 6∈ Φ−(X) } , σe3(A) := { λ ∈ C, such that λ−A 6∈ Φ−(X) ∪ Φ+(X) } , σe4(A) := { λ ∈ C, such that λ−A 6∈ Φ(X) } , σe5(A) := ⋂ k∈K(X) σ(T + K). σe1(.) and σe2(.) are the Gustafson and Weidman’s essential spectra. σe3(.) is the Kato’s essential spectrum. σe4(.) is the Wolf ’s essential spectrum, and σe5(.) is the Schechter’s essential spectrum. Proposition 1.1. [8, Theorem 7.27, p.172] Let T ∈C(X). Then λ /∈ σe5(T) if, and only if, (λ−T) ∈ Φ(X) and i(λ−T) = 0. Definition 1.4. Let X be a Banach space and E,F be closed subspaces of X. Let BE be the unit sphere of E. Let us define δ(E,F) : =   supx∈BE dist(x,F), if E 6= {0},0, otherwise, and δ̂(E,F) := max { δ(E,F),δ(F,E) } . The quantity δ̂(E,F) is called the gap between the subspaces E and F . Remark 1.1. (i) The gap measures the distance between two subspaces and it easily follows, from the definitions, (i1) δ(E,F) = δ(E,F) and δ̂(E,F) = δ̂(E,F). (i2) δ(E,F) = 0 if, and only if, E ⊂ F. (i3) δ̂(E,F) = 0 if, and only if, E = F. (ii) δ̂(·, ·) is a metric on the set V(X) of all linear closed subspaces of X and the convergence En → F in V(X) is obviously defined by δ̂(En,F) → 0. Moreover, (V(X), δ̂) is a complete metric space. Definition 1.5. (i) Let X and Y be two Banach spaces, and let T, S be two closed linear operators acting from X to Y . Let us define δ ( G(T),G(S) ) = sup x ∈D(T) ‖x‖2 + ‖Tx‖2 = 1 [ inf y∈D(S) ( ‖x−y‖2 + ‖Tx−Sy‖2 )1 2 ] . δ̂ ( T,S ) is called the gap between S and T . (ii) Let T and S be two closable operators. We define the gap between T and S by δ(T,S) = δ(T,S) and δ̂(T,S) = δ̂(T,S). Definition 1.6. A sequence (Tn)n∈N of bounded linear operators mapping on X is said to converge to zero compactly if for all x ∈ X, Tnx → 0 and (Tnxn)n is relatively compact for every bounded sequence (xn)n ⊂ X. Remark 1.2. Clearly, Tn converges to 0 implies that Tn converges to zero compactly. Definition 1.7. Let (Tn)n∈N be a sequence of closable linear operators from X into Y and let T be a closable linear operator from X into Y . (Tn)n∈N is said to converge in the generalized sense to T if δ̂(Tn,T) converges to 0 as, n →∞. 2. Preliminaries Theorem 2.1. [2, Theorem 4] Let An be a sequence of bounded linear operators converging to zero compactly and let T be a closed linear operator. If T is a semi-Fredholm operator, there exists n0 ∈ N such that for all n ≥ n0, ESSENTIAL SPECTRUM OF LINEAR OPERATORS 3 (i) (T + An) is semi-Fredholm, (ii) α(T + An) < α(T), (iii) β(T + An) < β(T), and (iv) i(T + An) = i(T). Proposition 2.1. [3, Proposition 7.8.1 ]. Let (Tn)n∈N be a sequence of bounded linear operators and let T ∈L(X) such that Tn −T converges to zero compactly. Then, (i) If Tn ∈Fb(X), then T ∈Fb(X), (ii) If Tn ∈Fb+(X), then T ∈Fb+(X), and (iii) If Tn ∈Fb−(X), then T ∈Fb−(X). Theorem 2.2. [1, theorem 2.1] Let T and S be two closed densely defined linear operators. Then, we have: (i) δ(T,S) = δ(S∗,T∗) and δ̂(T,S) = δ̂(S∗,T∗). (ii) If S and T are one-to-one, then δ(S,T) = δ(S−1,T−1) and δ̂(S,T) = δ̂(S−1,T−1). (iii) Let A ∈L(X,Y ). Then δ̂(A + S,A + T) ≤ 2(1 + ‖A‖2)δ̂(S,T). (iv) Let T be Fredholm operator (respectively semi-Fredholm operator). If δ̂(T,S) < γ(T)(1+[γ(T)]2) −1 2 , then S is Fredholm operator (respectively semi-Fredholm operator ), α(S) ≤ α(T) and β(S) ≤ β(T). Furthermore, there exists b > 0 such that δ̂(T,S) < b, which implies i(S) = i(T). (v) Let T ∈L(X,Y ). If S ∈C(X,Y ) and δ̂(T,S) ≤ [ 1 +‖T‖2 ]−1 2 , then S is bounded operator (so that D(S) is closed). Theorem 2.3. [1, theorem 2.3] Let (Tn)n∈N be a sequence of closable linear operators from X into Y and let T be a closable linear operator from X into Y . (i) The sequence Tn converges in the generalized sense to T if, and only if, Tn + S converges in the generalized sense to T + S, for all S ∈L(X,Y ). (ii) Let T ∈ L(X,Y ). Tn converges in the generalized sense to T if, and only if, Tn ∈ L(X,Y ) for sufficiently larger n and Tn converges to T . (iii) Let Tn converges in the generalized sense to T . Then, T −1 exists and T−1 ∈ L(Y,X), if, and only if, T−1n exists and T −1 n ∈L(Y,X) for sufficiently larger n and T−1n converges to T−1. 3. The main result In this section we investigate the essential spectra (σei, i = 1, . . . , 5) of the sequence of linear operators in a Banach space X. Theorem 3.1. Let (Tn)n∈N be a bounded linear operators mapping on X, and let T and B be two operators in L(X), λ0 ∈ C, and U ⊆ C is open . (a) If ((λ0 −Tn−B)−(λ0 −T −B)) converges to zero compactly, and 0 ∈U, then there exists n0 ∈ N such that, for all n ≥ n0. σei(λ0 −Tn −B) ⊆ σei(λ0 −T −B) + U. And, δ ( σei(λ0 −Tn −B),σei(λ0 −T −B) ) = 0, i=1,. . . ,5 (b) If (λ0 −Tn −B) converges to zero compactly then there exists n0 ∈ N such that for all n ≥ n0 σei((λ0 −T −B) + (λ0 −Tn −B)) ⊆ σei(λ0 −T −B). And, δ ( σei((λ0 −T −B) + (λ0 −Tn −B)),σei(λ0 −T −B) ) = 0, i = 1, . . . , 5. Proof. (a) For i = 1. Assume that the assertion fails. Then by passing to a subsequence, it may be deduced that, for each n, there exists λn ∈ σe1(λ0 −Tn −B) such that λn 6∈ σe1(λ0 −T −B) + U. It is clear that lim n→+∞ λn = λ since (λn) is bounded, this implies that λ 6∈ σe1(λ0 −T −B) + U. Using 4 AMMAR, DJAIDJA AND JERIBI the fact that 0 ∈U, hence we have λ 6∈ σe1(λ0 −T −B), and therefore, (λ− (λ0 −T −B)) ∈ Φb+(X). Let An = λn − λ + (λ0 −T −B) − (λ0 −Tn −B). Since An converges to zero compactly, writing λn−(λ0 −Tn −B) = λ−(λ0 −T −B) +An and according to Theorem 2.1, we infer that, there exists n0 ∈ N such that for all n ≥ n0 we have (λn − (λ0 −Tn −B)) ∈ Φ+(X) and i(λn − (λ0 −Tn −B)) = i(λ− (λ0 −T −B) + An) = i(λ− (λ0 −T −B)). So, λn 6∈ σe1(λ0 −Tn −B), which is a contradiction. Then σe1(λ0 −Tn −B) ⊆ σe1(λ0 −T −B) + U, for all n ≥ n0. Since 0 ∈ U, we obtain σe1(λ0 −Tn −B) ⊆ σe1(λ0 −T −B). Hence by Remark 1.1 (i2), we get δ ( σe1(λ0 −Tn −B),σe1(λ0 −T −B) ) = 0, for all n ≥ n0. For i = 2, 3, 4, by using a similar proof as in (i = 1), by replacing σe1(.), and Φ+(X) by σe2(.), σe3(.), σe4(.), and Φ−(X), Φ−(X) ∪ Φ+(X), Φ(X), respectively, we get If ((λ0 −Tn −B) − (λ0 −T −B)) converges to zero compactly, and 0 ∈ U, then there exists n0 ∈ N such that, for all n ≥ n0. σei(λ0 −Tn −B) ⊆ σei(λ0 −T −B) + U. And δ ( σei(λ0 −Tn −B),σei(λ0 −T −B) ) = 0. For i = 5. Assume that the assertion fails. Then by passing to a subsequence, it may be deduced that, for each n, there exists λn ∈ σe5(λ0 −Tn −B) such that λn 6∈ σe5(λ0 −T −B) + U. It is clear that lim n→+∞ λn = λ since (λn) is bounded, this implies that λ 6∈ σe5(λ0 −T −B) + U. Using the fact that 0 ∈ U, we have λ 6∈ σe5(λ0 −T −B) and therefore, λ − (λ0 −T −B) ∈ Φb(X) and i(λ− (λ0 −T −B)) = 0. Let An = λn −λ + (λ0 −T −B) − (λ0 −Tn −B). Since An converges to zero compactly, writing λn−(λ0 −Tn −B) = λ−(λ0 −T −B)+An and according to Theorem 2.1, we infer that, there exists n0 ∈ N such that for all n ≥ n0 we have λn−(λ0 −Tn−B) ∈ Φ(X) and i(λn− (λ0 −Tn −B)) = i(λ− (λ0 −T −B) + An) = i(λ− (λ0 −T −B)) = 0. So, λn 6∈ σe5(λ0 −Tn −B), which is a contradiction. Then σe5(λ0 −Tn −B) ⊆ σe5(λ0 −T −B) + U, for all n ≥ n0. Since 0 ∈U, we have σe5(λ0 −Tn −B) ⊆ σe5(λ0 −T −B). Hence by Remark 1.1 (i2), we have δ ( σe5(λ0 −Tn −B),σe5(λ0 −T −B) ) = 0, for all n ≥ n0. (b) For i = 1. Let λ 6∈ σe1(λ0 −T −B). Then, (λ− (λ0 −T −B)) ∈ Φb+(X). Since (λ0 −Tn −B) converges to zero compactly and applying [2, Theorem 4] to the operators (λ0−T−B) and (λ0−Tn−B), we prove that, there exists n0 ∈ N such that (λ−(λ0 −T −B)+(λ0 −Tn −B)) ∈ Φ+(X) for all n ≥ n0. Hence λ 6∈ σe1((λ0 −T −B) + (λ0 −Tn −B)). We conclude that σe1(λ0 −Tn −B) ⊆ σe1(λ0 −T −B). Now applying Remark 1.1 (i2) we obtain δ ( σe1((λ0 −T −B) + (λ0 −Tn −B)),σe1(λ0 −T −B) ) = 0, for all n ≥ n0. For i = 2, 3, 4, by using a similar proof as in (i = 1), by replacing σe1(.), and Φ+(X) by σe2(.), σe3(.), σe4(.), and Φ−(X), Φ−(X) ∪ Φ+(X), Φ(X), respectively, we get If (λ0 −Tn −B) converges to zero compactly then there exists n0 ∈ N such that for all n ≥ n0. σei((λ0 −T + B) + (λ0 −Tn −B)) ⊆ σei(λ0 −T −B). And, δ ( σei((λ0 −T −B) + (λ0 −Tn −B)),σei(λ0 −T −B) ) = 0, for all n ≥ n0. For i = 5. Let λ 6∈ σe5(λ0 −T −B). Then, (λ− (λ0 −T −B)) ∈ Φb(X) and i(λ − (λ0 −T −B)) = 0. Since (λ0 −Tn −B) converges to zero compactly and by applying the [2, Theorem 4] to the operators (λ0 −T −B) and (λ0 −Tn −B), we prove that, there exists n0 ∈ N such that (λ− (λ0 −T −B) + (λ0 −Tn −B)) ∈ Φ(X) for all n ≥ n0. Hence λ 6∈ σe5((λ0 −T −B) + (λ0 −Tn −B)). We conclude that σe5(λ0 −Tn −B) ⊆ σe5(λ0 −T −B) ESSENTIAL SPECTRUM OF LINEAR OPERATORS 5 Now applying Remark 1.1 (i2) we have δ ( σe5((λ0 −T −B) + (λ0 −Tn −B)),σe5(λ0 −T −B) ) = 0, for all n ≥ n0. � Theorem 3.2. Let (Tn)n∈N be a sequence of closed linear operators mapping on Banach spaces X and let T ∈ C(X),and let B and L be two operators in L(X), λ0 ∈ C such that Tn converges in the generalized sense to T , and λ0 ∈ ρ(T + B), U ⊆ C is open. (a) If 0 ∈U, then there exists n0 ∈ N such that, for every n ≥ n0, we have σei(λ0 −Tn −B) ⊆ σei(λ0 −T −B) + U. (3.1) And, δ ( σei(λ0 −Tn −B),σei(λ0 −T −B) ) = 0, i = 1, . . . , 5. (b) There exist ε > 0 and n ∈ N such that, for all ‖L‖ < ε, we have σei(λ0 −Tn −B + L) ⊆ σei(λ0 −T −B) + U, for all n ≥ n0. And, δ ( σei(λ0−Tn−B +L),σei(λ0−T −B) ) = δ ( σei(λ0−T −B +L),σei(λ0−T −B) ) , i = 1, . . . , 5. ♦ Proof. (a) For i = 1, since (B−λ0) be a bounded operator and λ0 ∈ ρ(T +B). According to Theorem 2.3 (i) and (iii)the sequence (λ0 −Tn −B) converges in the generalized sense to (λ0 −T −B), and λ0 ∈ ρ(Tn + B) for a sufficiently large n and (λ0 −Tn −B) −1 converges to (λ0 −T −B) −1 . Now to prove such that the inclusion (3.1)holds it suffices to prove there exist n0 ∈ N, such that for all n ≥ n0, we have σe1(λ0 −Tn −B)−1 ⊆ σe1(λ0 −T −B)−1 + U. (3.2) In first step by an indirect proof, we suppose that the (3.2) does not hold, and for each n ∈ N there exists λn ∈ σe1(λ0 −Tn −B)−1 such that λn 6∈ σe1(λ0 −T −B)−1 + U. It is clear that lim n→+∞ λn = λ since (λn) is bounded, this implies that λ 6∈ σe1(λ0 −T −B)−1 + U. Using the fact that 0 ∈U hence we have λ 6∈ σe1(λ0 −T −B)−1. Therefore (λ−(λ0 −T −B)−1) ∈ Φb+(X) and applying Theorem 2.3 (ii), we conclude that δ̂(λn − (λ0 −Tn −B)−1,λ− (λ0 −T −B)−1) → 0, as n →∞. Let γ(λ − (λ0 −T −B)−1) = δ > 0. Then there exists N ∈ N such that, for all n ≥ N we have δ̂(λn − (λ0 −Tn −B)−1,λ − (λ0 −T −B)−1) ≤ δ√1+δ2 . According Theorem 2.2 (iv) we infer (λn − (λ0 −Tn −B)−1) ∈ Φb+(X). Then we obtain λn /∈ σe1((λ0 −Tn −B)−1), which this is a contradicts our assumption. Hence (3.2) holds. Now, if λ ∈ σe1(λ0 −Tn −B) then 1λ ∈ σe1((λ0 −Tn −B) −1). According then (3.1) we conclude that 1 λ ∈ σe1((λ0 −T −B)−1) + U. (3.3) Since 0 ∈U, then (3.3) implies that 1 λ ∈ σe1((λ0 −T −B)−1). We have to prove λ ∈ σe1(λ0 −T −B) + U. (3.4) We will proceed by contradiction, we suppose that λ 6∈ σe1(λ0 −T −B) + U. The fact that 0 ∈ U implies that λ 6∈ σe1(λ0 −T −B) and so, 1λ 6∈ σe1((λ0 −T −B) −1) which this is a contradicts our assumption. So λ ∈ σe1(λ0 −T −B) +U. Therefore (3.1) holds. Since U is an arbitrary neighborhood of 0 and by using the relation (3.1) we have σe1(λ0 −Tn −B) ⊆ σe1(T + B −λ0), for all n ≥ n0. Hence by Remark 1.1 (i2) δ ( σe1(λ0 −Tn −B),σe1(λ0 −T −B) ) = δ ( σe1(λ0 −Tn −B),σe1(λ0 −T + B) ) = 0 for all n ≥ n0.This ends the proof (i=1). For i = 2, 3, 4, by using a similar proof as in ((a) for i = 1), by replacing σe1(.), and Φ+(X) by σe2(.), σe3(.), σe4(.), and Φ−(X), Φ−(X) ∪ Φ+(X), Φ(X), respectively, we get σei(λ0 −Tn −B) ⊆ σei(λ0 −T −B) + U. And, δ ( σei(λ0 −Tn −B),σei(λ0 −T −B) ) = 0, for all n ≥ n0. 6 AMMAR, DJAIDJA AND JERIBI For i = 5, since (λ0 − B) be a bounded operator and λ0 ∈ ρ(T + B), according to Theorem 2.3 (i) and (iii) the sequence (λ0 − Tn − B) converges in the generalized sense to (λ0 −T −B), and λ0 ∈ ρ(Tn + B) for a sufficiently large n and (λ0 −Tn −B) −1 converges to (λ0 −T −B) −1 . Now to prove that (3.1)holds it suffices to prove there exist n0 ∈ N, such that for all n ≥ n0, we have σe5(λ0 −Tn −B)−1 ⊆ σe5(λ0 −T −B)−1 + U. (3.5) In first step by an indirect proof, we suppose that the inclusion (3.5) does not hold, and for each n ∈ N there exists λn ∈ σe5(λ0 −Tn −B)−1 such that λn 6∈ σe5(λ0 −T −B)−1 + U. It is clear that lim n→+∞ λn = λ since (λn) is bounded, this implies that λ 6∈ σe5(λ0 −T −B)−1 + U. Using the fact that 0 ∈ U, hence we have λ 6∈ σe5(λ0 −T −B)−1. Therefore (λ − (λ0 −T −B)−1) ∈ Φb(X) and i(λ− (λ0 −T −B)−1)=0, and applying Theorem 2.3 (ii), we conclude that δ̂(λn − (λ0 −Tn −B)−1,λ− (λ0 −T −B)−1) → 0 as n →∞. Let γ(λ− (λ0 −T −B)−1) = δ > 0. Then there exists N ∈ N such that, for all n ≥ N we have δ̂(λn − (λ0 −Tn −B)−1,λ− (λ0 −T −B)−1) ≤ δ √ 1 + δ2 . According to Theorem 2.2 (iv) we infer (λn − (λ0 −Tn −B)−1) ∈ Φb(X) and i(λn − (λ0 −Tn −B)−1) = i(λ− (λ0 −T −B)−1)=0. Then we obtain λn /∈ σe5((λ0 −Tn −B)−1), which this is a contradicts our assumption. Hence (3.1) holds. Now, if λ ∈ σe5(λ0 − Tn − B) then 1 λ ∈ σe5((λ0 −Tn −B)−1). According then (3.1) we conclude that 1 λ ∈ σe5((λ0 −T −B)−1) + U. (3.6) Since 0 ∈U, then (3.6) implies that 1 λ ∈ σe5(λ0 −T −B)−1. We have to prove λ ∈ σe5(λ0 −T −B) + U. (3.7) We will proceed by contradiction , we suppose that λ 6∈ σe5(λ0 − T − B) + U. The fact that 0 ∈ U implies that λ 6∈ σe5(λ0 − T − B) and so, 1λ 6∈ σe5((λ0 − T − B) −1) which this is a contradicts our assumption. So λ ∈ σe5(λ0 −T −B) + U. Therefore (3.1)holds. Since U is an arbitrary neighborhood of 0 and by using (3.1) we have σe5(λ0 −Tn −B) ⊆ σe5(λ0 −T −B) for all n ≥ n0. Hence by Remark 1.1 (i2) δ ( σe5(λ0 −Tn −B),σe5(λ0 −T −B) ) = δ ( σe5(λ0 −Tn −B),σe5(λ0 −T −B) ) = 0 for all n ≥ n0.This ends the proof of, (a) . (b) For i = 1, since λ0 ∈ ρ(T +B), then (T +B−λ0)−1 exists and bounded. We put 1‖(λ0−T−B)−1‖ = ε1. Let L ∈L(X) such that ‖L‖ < ε1 this implies ‖L (λ0 −T −B) −1 ‖ < 1. By according Theorem 2.3 (i) the squence (λ0−Tn−B +L) converges in the generalized sense to (λ0− T−B+L), and the Neumann series ∑∞ k=0(−L (λ0 −T −B) −1 )k converges to (I+L (λ0 −T −B) −1 )−1 and ‖(I + L (λ0 −T −B) −1 )−1‖ < 1 1 −‖L‖‖(λ0 −T −B) −1 ‖ Since (λ0 − T − B + L)−1 = (λ0 −T −B) −1 )(I + L (λ0 −T −B) −1 ))−1, then λ0 ∈ ρ(T + B + L). Now applying ((a)for i = 1), we deduce that there exists n0 ∈ N such that σe1(λ0 −Tn −B + L) ⊆ σe1(λ0 −T −B + L) +U, for all n ≥ n0. Let λ 6∈ σe1(λ0−T −B). Then (λ−(λ0−T −B)) ∈ Φ+(X). By applying [8, Theorem 7.9] there exists ε2 > 0 such that for ‖L‖ < ε2, one has (λ−(λ0−T−B)−L) ∈ Φ+(X) and, this implies that λ 6∈ σe1(λ0 − T − B + L). From what has been mentioned and if we take ε = min(ε1,ε2) then for all ‖L‖ < ε, there exists n0 ∈ N such that σe1(λ0 −Tn −B + L) ⊆ σe1(λ0 −T −B) + U, for all n ≥ n0. Since 0 ∈U then we have δ ( σe1(λ0 −Tn −B + L),σe1(λ0 −T −B) ) = 0 and δ ( σe1(λ0 −T −B + L),σe1(λ0 −T −B) ) = 0. ESSENTIAL SPECTRUM OF LINEAR OPERATORS 7 Therefore, (i = 1) holds. For i = 2, 3, 4. By using a similar proof as in ((b)for i = 1), by replacing σe1(.), and Φ+(X) by σe2(.), σe3(.), σe4(.), and Φ−(X), Φ−(X) ∪ Φ+(X), Φ(X), respectively, we get There exist ε > 0 and n ∈ N such that, for all ‖L‖ < ε, we have σei(λ0 −Tn −B + L) ⊆ σei(λ0 −T −B) + U, for all n ≥ n0. And, δ ( σei(λ0 −Tn −B + L),σei(λ0 −T −B) ) = δ ( σei(λ0 −T −B + L),σei(λ0 −T −B) ) . For i = 5, since λ0 ∈ ρ(T + B), then (λ0 −T −B)−1 exists and bounded. We put 1‖(λ0−T−B)−1‖ = ε1. Let L ∈L(X) such that ‖L‖ < ε1 this implies ‖L (λ0 −T −B) −1 ‖ < 1. By according theorem 2.3 (i) we have (λ0−Tn−B +L) converges in the generalized sense to (λ0−T − B + L) , and the Neumann series ∑∞ k=0(−L (λ0 −T −B) −1 )k converges to (I + L (λ0 −T −B) −1 )−1 and ‖(I + L (λ0 −T −B) −1 )−1‖ < 1 1 −‖L‖‖(λ0 −T −B) −1 ‖ . Since (λ0 − T − B + L)−1 = (λ0 −T −B) −1 )(I + L (λ0 −T −B) −1 ))−1, then λ0 ∈ ρ(T + B + L). Now applying ((a)for i = 5), we deduce that there exists n0 ∈ N such that σe5(λ0 −Tn −B + L) ⊆ σe5(λ0−T −B +L) +U, for all n ≥ n0. Let λ 6∈ σe5(λ0−T −B). Then (λ−(λ0−T −B)) ∈ Φ(X). By applying [8, Theorem 7.9] there exists ε2 > 0 such that for ‖L‖ < ε2, one has (λ−(λ0 −T −B)−L) ∈ Φ(X) and i(λ−(λ0−T −B−L)) = i(λ−(λ0−T −B) = 0. This implies that λ 6∈ σe5(λ0−T −B +L). From what has been mentioned and if we take ε = min(ε1,ε2) then for all ‖L‖ < ε, there exists n0 ∈ N such that σe5(λ0 −Tn −B + L) ⊆ σe5(λ0 −T −B) + U, for all n ≥ n0. Since 0 ∈U then we have δ ( σe5(λ0 −Tn −B + L),σe5(λ0 −T −B) ) = 0 = δ ( σe5(λ0 −T −B + L),σe5(λ0 −T −B) ) . Therefore, (i = 5) holds. � References [1] A. Ammar and A. Jeribi, The weyl essential spectrum of a sequence of linear operators in Banach spaces, Indag. Math., New Ser. 27 (1) (2016), 282-295. [2] S. Goldberg, Perturbations of semi-Fredholm operators by operators converging to zero compactly, Proc. Amer. Math. Soc. 45 (1974), 93-98 . [3] A. Jeribi, Spectral Theory and Applications of Linear Operators and Block Operator Matrices. Springer-Verlag, New York, 2015. [4] A. Jeribi and N. Moalla, A characterisation of some subsets of Schechter’s essential spectrum and applications to singular transport equation, J. Math. Anal. Appl. 358 (2) (2009), 434-444. [5] A.Jeribi, Une nouvelle caractrisation du spectre essentiel et application, Comp Rend.Acad.Paris serie I, 331 (2000),525-530. [6] T. Kato, Perturbation theory for linear operators, Second edition. Grundlehren der Mathematischen Wissenschaften, Band 132. Springer-Verlag, Berlin-New York, 1966. [7] K. Latrach, A. Jeribi, Some results on Fredholm operators, essential spectra, and application, J. Math. Anal. Appl. 225 (1998), 461-485. [8] M. Schechter, Principles of Functional Analysis, Grad. Stud. Math. vol. 36, Amer. Math. Soc., Providence, 2002. [9] M. Schechter, Riesz operators and Fredholm perturbations, Bull. Amer. Math. Soc. 74 (1968), 1139-1144. 1Department of Mathematics, University of Sfax, Faculty of Sciences of Sfax, Tunisia 2Department of Mathematics, University of Mohamed Boudiaf, M’sila, Algeria ∗Corresponding author: ammar aymen84@yahoo.fr 1. Introduction 2. Preliminaries 3. The main result References