International Journal of Analysis and Applications ISSN 2291-8639 Volume 14, Number 1 (2017), 1-8 http://www.etamaths.com A NEW STABILITY OF THE S-ESSENTIAL SPECTRUM OF MULTIVALUED LINEAR OPERATORS AYMEN AMMAR∗, SLIM FAKHFAKH AND AREF JERIBI Abstract. We unfold in this paper two main results. In the first, we give the necessary assumptions for three linear relations A, B and S such that σeap,S(A + B) = σeap,S(A) and σeδ,S(A + B) = σeδ,S(A) is true. In the second, considering the fact that the linear relations A, B and S are not precompact or relatively precompact, we can show that σeap,S(A + B) = σeap,S(A) is true. 1. Introduction Assume that A and S are two bounded operators. Accordingly the map p(λ) := λS −A is a linear bundle. In fact many problems of mathematical physics (for example quantum theory, transport theory,...) are meant to shed light on the essential spectra of λS−A. The spectral theory of Fredholm linear relations is one case worth mentioning given that this type of operators is unstable under the operation closure inverse and conjugate. But this does not hold for the case of multivalued linear operators. On this account, the investigation of the S-essential spectra of multivalued linear operators seems interesting. Historically, in [11] A. Jeribi, N. Moalla, and S. Yengui gave a characterization of the essential spectrum of the operator pencil in order to extend many known results in the literature. In [1] F. Abdmouleh, A. Ammar, and A. Jeribi pursued the study of the S-essential spectra and investigated the S-Browder, the S-upper semi-Browder, and the S-lower semi-Browder essential spectra of bounded linear operators on a Banach space X and they introduced the S-Riesz projection. Moreover, they extended the results of F. Abdmouleh and A. Jeribi [3] to various types of S-essential spectra. In fact, they gave the characterization of the S-essential spectra of the sum of two bounded linear operators. (See for example [10]). In [4] Tereza Alvarez, A. Ammar, and A. Jeribi pursued the study of the S-essential spectra and characterized some S-essential spectra of a closed linear relation in terms of certain linear relations type semi Fredholm. In [6] A. Ammar characterized some essential spectra of a closed linear relation in terms of certain linear relations type α− and β− Atkinson. Throughout this work, let X, Y and Z be three complex normed linear spaces, over K = R or C. A multivalued linear operator (or a linear relation) A from X to Y is a mapping from a subspace of X, D(A) := {x ∈ X : Ax 6= ∅} called the domain of A, into P(Y )\{∅} (collection of non-empty subsets of Y) such that A(αx + βy) = αA(x) + βA(y) for all non-zero scalars α,β ∈ C and x,y ∈D(A). If A maps the points of its domain to singletons, then A is said to be a single valued linear operator (or simply an operator). A linear relation is uniquely determined by its graph, G(A), which is defined by G(A) := {(x,y) ∈ X ×Y : x ∈D(A) and y ∈ Ax}. In this notation, LR(X,Y ) denotes the class of all linear relations on X into Y . If X = Y , we would simply note LR(X,X) := LR(X). The inverse of A is the linear relation A−1 defined by G(A−1) := {(y,x) ∈ Y ×X : (x,y) ∈ G(A)}. Received 23rd November, 2016; accepted 23rd February, 2017; published 2nd May, 2017. 2010 Mathematics Subject Classification. 47A06. Key words and phrases. linear relations; relatively precompact; relatively bounded; S-essential approximate point; S-essential defect. 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, FAKHFAKH AND JERIBI The subspace N(A) := A−1(0) is called the null space of A, and A is called injective if N(A) = {0}; i.e., if A−1 is a single valued linear operator. The range of A is the subspace R(A) := A(D(A)), and A is called surjective if R(A) = Y . When A is injective and surjective, we say that A is bijective. The quantities α(A) := dim(N(A)) and β(A) := codim(R(A)) = dim(Y/R(A)) are called the nullity (or the kernel index) and the deficiency of A, respectively. We also write β(A) := codim(R(A)). The index of A is defined by i(A) := α(A) −β(A) provided that both α(A) and β(A) are not infinite. If α(A) and β(A) are infinite, then A is said to have no index. The set of upper semi-Fredholm linear relations from X into Y is defined by: Φ+(X,Y ) := {T ∈CR(X,Y ) : R(T) is closed, and α(T) < ∞}, the set of lower semi-Fredholm linear relations from X into Y is defined by: Φ−(X,Y ) := {T ∈CR(X,Y ) : R(T) is closed, and β(T) < ∞}. If X = Y , we would simply note Φ+(X,Y ) and Φ−(X,Y ) by respectively Φ+(X) and Φ−(X). Let M be a subspace of X such that M ∩D(A) 6= ∅ and let A ∈LR(X,Y ). Then, the restriction A|M is the linear relation given by: G(A|M ) := {(m,y) ∈ G(A) : m ∈ M} = G(A) ∩ (M ×Y ). For A, B ∈LR(X,Y ) and S ∈LR(Y,Z), the sum A + B and the product SA are the linear relations defined by G(A + B) := {(x,y + z) ∈ X ×Y : (x,y) ∈ G(A) and (x,z) ∈ G(B)}, and G(SA) : {(x,z) ∈ X ×Z : (x,y) ∈ G(A), (y,z) ∈ G(S) for some y ∈ Y} respectively. If λ ∈ K, then λA is defined by: G(λA) := {(x,λy) : (x,y) ∈ G(A)}. If A ∈LR(X) and λ ∈ K, then the linear relation λ−A is given by: G(λ−A) := {(x,y −λx) : (x,y) ∈ G(A)}. We note that ‖Ax‖ and ‖A‖ are not real norms. In fact, a nonzero relation can have a zero norm. A is said to be closed if its graph G(A) is a closed subspace of X ×Y . The closure of A, denoted by A, is defined in terms of its graph G(A) := G(A). We denote by CR(X,Y ) the class of all closed linear relations on X into Y . If X = Y , we would simply note CR(X,X) := CR(X). If A is an extension of A (that is, A|D(A)), we say that A is closable. Let A ∈ LR(X,Y ). We say that A is continuous if for each neighbourhood V in R(A), the inverse image A−1(V ) is a neighbourhood in D(A) equivalently ‖A‖ < ∞; open if A−1 is continuous; bounded if D(A) = X and A is continuous; bounded below if it is injective and open; and compact if QAA(BD(A)) is compact in Y (BD(A) := {x ∈ D(A) : ‖x‖ ≤ 1}). We denote by KR(X,Y ) the class of all compact linear relations on X into Y . If X = Y , we would simply note KR(X,X) := KR(X). We say that A is precompact if QTTBD(T) is totally bounded in Y , and strictly singular if there is no infinite dimensional subspace M of D(A) for which A|M is injective, and open. If X is a normed linear space, then X ′ will denote the dual norm of X, i.e., the space of all continuous linear functionals x′ are defined on X with the norm ‖x′‖ = inf{λ : |x′x| ≤ λ‖x‖ for all x ∈ X}. If K ⊂ X and L ⊂ X ′ , we shall adopt the following notations: K⊥ := {x′ ∈ X ′ : x′ = 0 for all x ∈ K}, L> := {x ∈ X : x′ = 0 for all x′ ∈ L}. Clearly, K⊥ and L> are closed linear subspaces of X ′ and X respectively. The adjoint of T , T ′, is defined by G(A′) = G(−A−1)⊥ ⊂ Y ′ ×X ′ where 〈(y,x), (y′,x′)〉 := 〈x,x′〉 + 〈y,y′〉. This means that STABILITY OF THE S-ESSENTIAL SPECTRUM OF LINEAR RELATIONS 3 (y′,x′) ∈ G(A′) if, and only if, y′y −x′x = 0 for all (x,y) ∈ G(T). Similarly, we have y′y = x′x for all y ∈ Ax, x ∈D(A). Hence x′ ∈ A′y if, and only if, y′Ax = x′x for all x ∈D(A). Let X be a complex Banach space and let A ∈ CR(X,Y ). Suppose that S ∈ LR(X) is A-bounded with A-bounded δ < 1 such that S(0) ⊂ A(0) and D(A) ⊂ D(S). We define the S resolvent set of A by ρS(A) := {λ ∈ C : λS −A is bijective}. In this work, we are concerned with the following S-essential approximate point spectrum of A defined by: σeap,S(A) := ⋂ K∈KA(X) σap,S(A + K). Similarly we are concerned with the following S-essential defect spectrum of A defined by: σeδ,S(A) := ⋂ K∈KA(X) σδ,S(A + K), where KA(X) := {K ∈KR(X) : D(A) ⊂D(K) and K(0) ⊂ A(0)}, σap,S(A) := {λ ∈ C : λS −A is not bounded below}, and σδ,S(A) := {λ ∈ C : λS −A is not surjective}. Note that if S = I, (the identity operator on X), we recover the usual definition of the essential spectra of a bounded linear operator A. The purpose of this paper is to extend the results in [8] mentioned above to the general case of S- essentiel stability in the first place. In the second place, in other hypotheses, we show the stability of S-essential approximate point spectrum. We organize the paper in the following way. Section 2 consists in establishing some preliminary results which will be needed in the sequel. The main results of Section 3 are Lemma 3.1 and Lemma 3.2, which give information concerning the equivalence of norm. In section 4, we investigate the stability of the S- essential approximate point spectrum and the S-essential defect spectrum of closed and closable linear relations under relatively compact and precompact perturbations on a Banach space (see Theorem 4.1), and under different hypotheses we find the stability of the S-essential approximate point spectrum (see Theorem 4.2). 2. Preliminaries The goal of this section consists in establishing some preliminary results which will be needed in the sequel. Definition 2.1. [9, Definition, IV.3.1] Let A ∈LR(X,Y ), and let XA denote the vector space D(A) normed by ‖x‖A := ‖x‖ + ‖Ax‖, for all x ∈D(A). Let GA ∈LR(XA,X) be the identity injection of XA = (D(A),‖.‖A) into X, i.e., D(GA) = XA, GA(x) = x, for all x ∈ XA. Definition 2.2. [9, Definition, VII.2.1] Let A, B ∈ LR(X, Y ). B is said to be A-bounded (or bounded relative to A) if D(A) ⊂D(B) and there exist non-negative constants a, and b, such that ‖Bx‖≤ a‖x‖+ b‖Ax‖ for all x ∈D(A). (2.1) In that case the infimum of all the constant b which satisfies (2.1) is called the A-bound of B. We note that B is A-bounded if, and only if, D(A) ⊂D(B), and BGA is bounded. Definition 2.3. [9, Definition VII.2.1] Let A ∈ LR(X,Y ). A relation B ∈ LR(X,Y ) is said to be A-compact (or compact relative to A) if D(A) ⊂D(B) and BGA is compact. B is called A-precompact (or precompact relative to A) if D(A) ⊂D(B) and BGA is precompact. Lemma 2.1. [7, Lemma, 3.1] Let S, T ∈LR(X,Y ) satisfies S(0) ⊂ T(0) and D(T) ⊂D(S). If S is T -compact, then S is T -bounded. 4 AMMAR, FAKHFAKH AND JERIBI Lemma 2.2. [7, Lemma, 3.6] Let A, B and S ∈LR(X,Y ) satisfy B(0)∪S(0) ⊂ A(0). Suppose that B is A-bounded with A-bound δ1, S is A-bounded with A-bound δ2, and Y is complete. (i) If δ1 + δ2 < 1, and A is closed, then A + B + S is closed. (ii) If δ1 + δ2 < 1 2 , and A + B + S is closed, then A is closed. Lemma 2.3. [7, Lemma, 4.1] Let S ∈ LR(X,Y ) and A ∈ F+(X,Y ) with dimD(A) = ∞. If S is precompact, then S is strictly singular. If additionally S(0) ⊂ A(0), then A + S ∈F+(X,Y ). Proposition 2.1. [5, Theorem 2.17] Let B ∈ LR(X,Y ), A ∈ F+(X,Y ) with G(B) ⊂ G(A) , and dim D(B) = ∞, then B ∈F+(X,Y ). Lemma 2.4. [2, Lemma 2.3] Let X be complete, T ∈CR(X), and K ∈KT (X). (i) If T ∈ Φ+(X), then T + K ∈ Φ+(X) with i(T + K) = i(T). (ii) If T ∈ Φ−(X), then T + K ∈ Φ−(X) with i(T + K) = i(T). Proposition 2.2. [4, Theorem 3.1] Let X be complete, A ∈ CR(X) and λ ∈ C. If S ∈ LR(X) is A-bounded with A-bounded δ < 1 such that S(0) ⊂ A(0) and D(A) ⊂D(S), then (i) λ /∈ σeap,S(A) if, and only if, λS −A ∈ Φ+(X) and i(λS −A) ≤ 0. (ii) λ /∈ σeδ,S(A) if, and only if, λS −A ∈ Φ−(X) and i(λS −A) ≥ 0. To end this section, we present the following Proposition suggested by Cross in [9]. Proposition 2.3. Let A, B ∈LR(X,Y ) (i) [9, Corollary V.2.5] A ∈F+(X,Y ) if, and only if, AGA ∈F+(XA,Y ). (ii) [9, Corollary V.2.3] If A is precompact, then A is continuous. (iii) [9, Proposition III.1.5] Let D(A) ⊂D(B). If B is continuous, then (A + B)′ = A′ + B′. (iv) [9, Proposition V.5.15] Let A ∈CR(X,Y ). A ∈KR(X,Y ) if, and only if, A′ ∈KR(Y ′,X′). (v) [9, Proposition V.7.5] A ∈F+(X,Y ) if, and only if, A′ ∈F−(Y ′ ,X ′ ) and A ′ ∈F+(Y ′ ,X ′ ) if, and only if, A ∈F−(X,Y ). (vi) [9, Proposition V.7.8] If dim B(0) < ∞, then A+B−B ∈F+(X,Y ) if, and only if, A ∈F+(X,Y ). (vii) [9, Proposition V.5.27] If A is closable. Then A ∈F−(X,Y ) if, and only if, AGA ∈F−(XA,Y ). (viii) [9, Proposition V.5.12] Let D(A) ⊂ D(B), and let A ∈ F−(X,Y ). If B is precompact, then A + B ∈F−(X,Y ). 3. Main results In [9], book1 claims that ‖A‖ − ‖B‖ ≤ ‖A − B‖ is not in general true. He gives an example (see [9, Exercise, II.1.12]). In the first Lemma in this section, we give a necessary and sufficient condition for two linear relations A and B so that the equality of ‖A‖−‖B‖≤‖A−B‖ become justified. Lemma 3.1. Let A, B ∈LR(X,Y ). If B(0) ⊂ A(0) and D(A) ⊂D(B), then (i) ‖Ax‖−‖Bx‖≤‖Ax−Bx‖, for x ∈D(A). (ii) ‖Ax‖−‖Bx‖≤‖Ax + Bx‖, for x ∈D(A). Proof We have for x ∈D(A), by Lemma [4, Lemma 2.2 (iii)], we get (A−B + B)x = Ax, then ‖(A−B + B)x‖ = ‖Ax‖, (3.1) (i) Using [9, Proposition, II.1.5] and from Eqs (3.1), we obtain ‖Ax‖ ≤ ‖(A − B)x‖ + ‖Bx‖. So ‖Ax‖−‖Bx‖≤‖Ax−Bx‖. STABILITY OF THE S-ESSENTIAL SPECTRUM OF LINEAR RELATIONS 5 (ii) Using [9, Proposition, II.1.5] and from Eqs (3.1), we obtain ‖Ax‖‖ ≤ ‖(A + B)x‖ + ‖Bx‖. So ‖Ax‖−‖Bx‖≤‖Ax + Bx‖. Lemma 3.2. Let A, B, and S ∈LR(X,Y ) verifying B(0) ⊂ A(0) and λ ∈ C. If S is A-bounded with A-bound δ1 and B is A-bounded with A-bound δ2 such that δ2 + |λ|δ1 < 1, then ‖.‖A and ‖.‖λS−(A+B) are equivalent. In particular, ‖.‖A and ‖.‖λS−A are equivalent. Proof Since S is A-bounded with bound δ1 and B is A-bounded with bound δ2, there exist non- negative constants a, b, a1 and b1 such that, for x ∈ D(A), ‖Sx‖ ≤ a‖x‖ + b‖Ax‖ and ‖Bx‖ ≤ a1‖x‖+b1‖Ax‖. So we have −‖Bx‖≥−a1‖x‖−b1‖Ax‖, thus ‖Ax‖−‖Bx‖≥−a1‖x‖+ (1−b1)‖Ax‖. Using Lemma 3.1 (ii), we get ‖Ax + Bx‖≥−a1‖x‖ + (1 − b1)‖Ax‖. On the other hand, ‖x‖λS−(A+B) = ‖x‖ + ‖(λS − (A + B))x‖, ≥ ‖x‖ + ‖(A + B))x‖−|λ|‖Sx‖, ≥ ‖x‖−a1‖x‖ + (1 − b1)‖Ax‖−|λ|‖Sx‖, ≥ ‖x‖−a1‖x‖ + (1 − b1)‖Ax‖−|λ|a‖x‖− b|λ|‖Ax‖, ≥ (1 −a1 −|λ|a)‖x‖ + (1 − b1 −|λ|b)|‖Ax‖, ≥ min(1 −a1 −|λ|a, 1 − b1 −|λ|b)|(‖x‖ + ‖Ax‖). Therefore, ‖x‖λS−(A+B) ≥ K|‖x‖A, with K = min(1 −a1 −|λ|a, 1 − b1 −|λ|b). On the other hand, we obtain ‖x‖λS−(A+B) = ‖x‖ + ‖(λS − (A + B))x‖, ≤ ‖x‖ + ‖Ax‖ + ‖Bx‖ + |λ|‖Sx‖, ≤ ‖x‖ + a1‖x‖ + b1‖Ax‖ + ‖Ax‖ + |λ|a‖x‖ + b|λ|‖Ax‖, ≤ (1 + a1 + |λ|a)‖x‖ + (1 + b1 + |λ|b)|‖Ax‖, ≤ max(1 + a1 + |λ|a, 1 + b1 + |λ|b)|(‖x‖ + ‖Ax‖). Therefore, ‖x‖λS−(A+B) ≤ H|‖x‖A, with H = max(1 + a1 + |λ|a, 1 + b1 + |λ|b). We deduce that ‖.‖A and ‖.‖λS−(A+B) are equivalent. Lemma 3.3. Let A, B, and S ∈LR(X) and let λ ∈ C. (i) R((λS −A)GB) = R(λS −A). (ii) N((λS −A)GB) = N(λS −A). Proof (i) Using the fact that GBx = (GB) −1x = x, R(A) = AD(A) and D(AB) = B−1D(A). R((λS −A)GB) = (λS −A)GBD((λS −A)GB), = (λS −A)D((λS −A)GB), = (λS −A)GBD(λS −A), = (λS −A)D(λS −A), = R(λS −A). (ii)N((λS −A)GB) = {x ∈D((λS −A)GB), (λS −A)GB(x) = (λS −A)GB(0)}, = {x ∈D(λS −A), (λS −A)(x) = (λS −A)(0)}, = N(λS −A). Proposition 3.1. Let X be complete, let A, B, S ∈LR(X) satisfy B(0) ⊂ A(0) and let λ ∈ C. If B is A-precompact, then i(λS −A) = i(λS − (A + B)). 6 AMMAR, FAKHFAKH AND JERIBI Proof Since B is A-precompact, then BGA is precompact, and X is complete. By Remark [9, Note V.1 p 134] BGA is compact. i(λS −A) = i((λS −A)GA), by Lemma 3.3, = i((λS −A)GA + BGA), by Lemma 2.4 (BGA is compact), = i((λS − (A + B))GA), = i(λS − (A + B)), by Lemma 3.3. Lemma 3.4. Let A, B ∈LR(X,Y ) such that G(A) G(B). We have (i) α(A) ≤ α(B). (ii) β(B) ≤ β(A). (iii) i(A) ≤ i(B). Proof (i) We have α(A) := dim(N(A)). Then N(A) := {x ∈D(A) : (x, 0) ∈ G(A)}, {x ∈D(A) : (x, 0) ∈ G(B)}, = N(B|D(A)), ⊂ N(B). So, α(A) ≤ α(B). (ii) We have β(A) := codim(R(A)) = dim(Y/R(A)). Let y ∈ R(A). Then, y ∈ Ax for all x ∈ D(A). We get by G(A) G(B), y ∈ Bx for all x ∈D(A). So, y ∈R(B|D(A)). Thus y ∈R(B). We infer that Y/R(B) ⊂ Y/R(A). Then β(B) ≤ β(A). (iii) i(A) := α(A) −β(A) ≤ α(B) −β(B) = i(B). Lemma 3.5. Let A, B ∈LR(X,Y ). If G(A) G(B), then G(A) G(A + B). Proof G(A) := {(x,y) ∈ X ×Y : x ∈D(A) D(B) and y ∈ Ax Bx}, {(x,y) ∈ X ×Y : x ∈D(A) ∩D(B) and y ∈ Ax + Bx}, := {(x,y) ∈ X ×Y : x ∈D(A + B) and y ∈ (A + B)x}, := G(A + B). 4. Stability of σeap,S(.) and σeδ,S(.) In this section, on one level, we study the stability of the S-essential approximate point spectrum and the S-essential defect spectrum of closed and closable linear relations under relatively precom- pact perturbations on a Banach space. On another level, we study the stability of the S-essential approximate point spectrum but under assumptions different from those adopted above. Theorem 4.1. Let X be complete, A ∈ CR(X), B, S ∈ LR(X) satisfy B(0) ⊂ S(0) ⊂ A(0) and dimD(B) = ∞, and λ ∈ C. If S is A-bounded with A-bound δ1 and B is A-precompact with A-bound δ2 such that δ2 + |λ|δ1 < 1, then σeap,S(A + B) = σeap,S(A), and σeδ,S(A + B) = σeδ,S(A). Proof Let B be A-precompact, then BGA is precompact, and X and XA are complete. By Remark [9, Note V.1 p 134], we get BGA is compact. By Lemma 2.1, we get BGA is bounded, then B is A-bounded with A-bound δ2. Using the fact that S is A-bounded with A-bound δ1 and δ2 + |λ|δ1 < 1 and by applying Lemma 2.2 (i), we obtain λS − (A + B) is closed. Suppose that λ /∈ σeap,S(A), then by Proposition 2.2, λS − A ∈ Φ+(X). By Proposition 2.3 (i), we get (λS − A)GλS−A ∈ Φ+(XA), which gives (λS − A)GA ∈ Φ+(XA) by referring to Lemma 3.2. Since BGA is compact and dimD(B) = dimD(BGA) = ∞, then using Lemma 2.3 it follows that (λS−(A+B))GA ∈ Φ+(XA). By Lemma 3.2, we obtain (λS−(A+B))GλS−(A+B) ∈ Φ+(XA). Using STABILITY OF THE S-ESSENTIAL SPECTRUM OF LINEAR RELATIONS 7 Proposition 2.3 (i), we get λS − (A + B) ∈ Φ+(X) and we have i(λS − A) = i(λS − (A + B)) by Proposition 3.1, that is λ /∈ σeap,S(A+B) by Proposition 2.2. So σeap,S(A+B) ⊆ σeap,S(A). Conversely, let λ /∈ σeap,S(A+B). Then by proposition 2.2, we have λS−(A+B) ∈ Φ+(X). Using Proposition 2.3 (i), we get (λS−(A+B))GλS−(A+B) ∈ Φ+(XA), which gives (λS−(A+B))GA ∈ Φ+(XA) by referring to Lemma 3.2. Since BGA is compact, then by Lemma 2.3, it follows that (λS−A)GA ∈ Φ+(XA). By Lemma 3.2, we obtain (λS−A)GλS−A ∈ Φ+(XA). Using Proposition 2.3 (i), we get λS−A ∈ Φ+(X). We have i(λS −A) = i(λS − (A + B)) by Proposition 3.1, that is λ /∈ σeap,S(A) by Proposition 2.2. We infer that σeap,S(A + B) = σeap,S(A). Now suppose that λ /∈ σeδ,S(A), then by Proposition 2.2, we have λS − A ∈ Φ−(X). Applying Proposition 2.3 (vii), we obtain (λS −A)GλS−A ∈ Φ−(XA). Using Lemma 3.2, we get (λS −A)GA ∈ Φ−(XA). Since BGA is precompact, then by Proposition 2.3 (viii), we obtain (λS − (A + B))GA ∈ Φ−(XA). Resorting to Lemma 3.2, we get (λS − (A + B))GλS−(A+B) ∈ Φ−(XA). So applying Proposition 2.3 (vii), we get (λS − (A + B)) ∈ Φ−(X). We have i(λS − A) = i(λS − (A + B)) by Proposition 3.1, that is λ /∈ σeδ,S(A + B) by Proposition 2.2. Then σeδ,S(A + B) ⊂ σeδ,S(A). Conversely, let λ /∈ σeδ,S(A + B), then by Proposition 2.2, we obtain λS − (A + B) ∈ Φ−(X). Using Proposition 2.3 (vii), we get (λS − (A + B))GλS−(A+B) ∈ Φ−(XA). Applying Lemma 3.2, we get (λS − (A + B))GA ∈ Φ−(XA). The latter holds if, and only if, ((λS−(A + B))GA) ′ ∈ Φ+(X ′ A) by Proposition 2.3 (v). Subsequently, using Proposition 2.3 (ii) and (iii), we get ((λS − A)GA) ′ + (BGA) ′ ∈ Φ+(X ′ A). Since BGA is precompact, then by Proposition 2.3 (iv) we have (BGA) ′ is precompact. Applying Lemma 2.4, we have ((λS−A)GA) ′ ∈ Φ+(X ′ A). Besides using Proposition 2.3 (v), we get (λS−A)GA ∈ Φ−(XA). So by Proposition 2.3 (vii), ((λS −A) ∈ Φ−(X). We have i(λS −A) = i(λS − (A + B)) by Proposition 3.1. That is λ /∈ σeδ,S(A) by Proposition 2.2. We conclude that σeδ,S(A + B) = σeδ,S(A). Theorem 4.2. Let A ∈ CR(X), B, S ∈ LR(X) and let λ ∈ C. Suppose that S is A-bounded with A-bound δ1 and B is A-bounded with A-bound δ2 such that δ2 + |λ|δ1 < 1. If G(B) G(λS) G(A) and dimD(B) = ∞, then (i) σeap,S(A + B) ⊂ σeap,S(A). (ii) If dim B(0) < ∞, then σeap,S(A + B) = σeap,S(A). Proof Since S is A-bounded with A-bound δ1 and B is A-bounded with A-bound δ2 such that δ2 + |λ|δ1 < 1, then applying Lemma 2.2, we obtain λS − (A + B) is closed. (i) Suppose that λ /∈ σeap,S(A), then by Proposition 2.2, λS −A ∈ Φ+(X) and i(λS −A) ≤ 0. Since G(B) G(λS) and G(λS) G(A), then applying Lemma 3.5, we get G(λS) G(λS −A), and then G(B) G(λS −A). On the one hand, we have G(λS − (A + B)) := {(x,y) ∈ X ×X : (x,y1) ∈ G(λS −A) and (x,y2) ∈ G(B) G(λS −A), where y = y1 + y2}, G(λS −A). On the other hand, dimD(λS − (A + B)) = dim(D(λS − A) ∩D(B)) = dimD(B) = ∞. Then by Proposition 2.1, we obtain λS − (A + B) ∈ Φ+(X). We have G(λS − (A + B)) G(λS −A), using Lemma 3.4, we get i(λS − (A + B)) ≤ (λS −A) ≤ 0. So by Proposition 2.2, we obtain λ /∈ σeap,S(A + B). Then σeap,S(A + B) ⊂ σeap,S(A). 8 AMMAR, FAKHFAKH AND JERIBI (ii) Since G(B) G(λS) and G(λS) G(A), then applying Lemma 3.5, we get G(λS) G(λS −A) and G(B) G(λS −A). By Lemma 3.5, we obtain G(B) G(λS − (A + B)). On the one hand, G(λS −A−B + B) := {(x,y + z) ∈ X ×Y : (x,y) ∈ G(λS − (A + B)) and (x,z) ∈ G(B)} {(x,y + z) ∈ X ×Y : (x,y) ∈ G(λS − (A + B)) and (x,z) ∈ G(S) G(λS − (A + B))} := G(λS − (A + B)). On the other hand, dimD(λS − A − B + B) = dim(D(λS) ∩D(A) ∩D(B)) = dimD(B) = ∞. Let λ /∈ σeap,S(A + B). Then by proposition 2.2, we have λS − (A + B) ∈ Φ+(X). Since λS −A is closed and dim B(0) < 0, then λS−A−B + B is closed and we have G(λS−A−B + B) G(λS−(A + B)), dimD(λS−A−B+B) = ∞, then by Proposition 2.1, we get λS−A−B+B ∈ Φ+(X). Thus by Propo- sition 2.3 (vi), we obtain λS−A ∈ Φ+(X). Using Lemma 3.4, we get i(λS−A) ≤ i(λS−A−B +B) ≤ (λS−(A + B)) ≤ 0. So by Proposition2.2, we obtain λ /∈ σeap,S(A). Thus, σeap,S(A) ⊂ σeap,S(A + B). We infer that σeap,S(A + B) = σeap,S(A). 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Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, P.O.Box 1171, 3000 Sfax, Tunisia ∗Corresponding author: ammar aymen84@yahoo.fr 1. Introduction 2. Preliminaries 3. Main results 4. Stability of eap,S(.) and e,S(.) References