� EXTRACTA MATHEMATICAE Volumen 33, Número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura EXTRACTA MATHEMATICAE Vol. 36, Num. 1 (2021), 63 – 80 doi:10.17398/2605-5686.36.1.63 Available online May 5, 2021 Stability of some essential B-spectra of pencil operators and application A. Ben Ali, M. Boudhief, N. Moalla Department of Mathematics, Science Faculty of Sfax University of Sfax, Tunisia boudhiafmon198@gmail.com , Nedra.moalla@ipeis.rnu.tn Received January 1, 2021 Presented by Pietro Aiena Accepted April 6, 2021 Abstract: In this paper, we give some results on the essential B-spectra of a linear operator pencil, which are used to determine the essential B-spectra of an integro-differential operator with abstract boundary conditions in the Banach space Lp([−a,a] × [−1, 1]), p ≥ 1 and a > 0. Key words: Operator pencil, finite-rank and power finite-rank perturbations, essential B-spectra, transport operator. MSC (2020): 47A53, 47A55. 1. Introduction Let X be a Banach space. We will denote by C(X) (resp. L(X)) the set of all closed linear (resp. the algebra of all bounded) linear operators from X into X. For T ∈ C(X), we write D(T) ⊂ X for the domain, N(T) ⊂ X for the null space and R(T) ⊂ X for the range of T. We denote by α(T) the dimension of N(T) and β(T) the codimension of R(T) in X. For T ∈ C(X) and M ∈ L(X), we define the resolvent of the linear operator pencil λM −T, where λ ∈ C, or the M-resolvent of T by ρM (T) := {λ ∈ C : λM −T has a bounded inverse} , and its spectrum by σ(M,T) = C\ρM (T) . For T ∈C(X), we define the set 4(T) = { n ∈ N : ∀m ∈ N, m ≥ n ⇒ R(Tn) ∩N(T) ⊂ R(Tm) ∩N(T) } . The degree of stable iteration of T is defined as dis(T) = inf 4(T), where dis(T) = ∞ if 4(T) = ∅. ISSN: 0213-8743 (print), 2605-5686 (online) c©The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0) https://doi.org/10.17398/2605-5686.36.1.63 mailto:boudhiafmon198@gmail.com mailto:Nedra.moalla@ipeis.rnu.tn https://publicaciones.unex.es/index.php/EM https://creativecommons.org/licenses/by-nc/3.0/ 64 a. ben ali, m. boudhief, n. moalla We define the set of upper semi-Fredholm operators by Φ+(X) = { T ∈C(X) : α(T) < ∞ and R(T) is closed in X } , and the set of lower semi-Fredholm operators by Φ−(X) = { T ∈C(X) : β(T) < ∞ and R(T) is closed in X } . Φ(X) := Φ+(X) ∩ Φ−(X) will denote the set of Fredholm operators from X into X. The index of a Fredholm operator T is defined by i(T) = α(T)−β(T). According to [17], an operator T ∈C(X) is called quasi-Fredholm of degree d ∈ N if the following three conditions are satisfied: (i) dis(T) = d ; (ii) R(Td) ∩N(T) is a closed and complemented subspace of X; (iii) R(T) + N(Td) is a closed and complemented subspace of X. This set of operators will be denoted by QF(d). Following [10, Definition 2.4], an operator T ∈ C(X) is called upper semi B-Fredholm (resp. lower semi B-Fredholm) if there exists an integer d ∈ N such that T ∈ QF(d) and such that N(T)∩R(Td) is of finite dimension (resp. R(T) + N(Td) is of finite codimension). These sets are denoted respectively by Φ+B(X) and Φ − B(X). We denote by ΦB(X) := Φ + B(X) ∩ Φ − B(X), the set of B-Fredholm operators from X into X. In this case, the index of T is defined as the integer: ind(T) = dim(N(T) ∩ R(Td)) − codim(R(T) + N(Td)). An operator T ∈ C(X) is called B-Weyl if it is a B-Fredholm operator of index zero. We will denote this set by BW(X). For T ∈C(X), we define the ascent a(T) of T by a(T) = inf { n ∈ N : N ( Tn ) = N ( Tn+1 )} , and the descent d(T) of T by d(T) = inf { n ∈ N : R ( Tn ) = R ( Tn+1 )} . We define respectively the set of Drazin invertible operators, left Drazin invertible operators and right Drazin invertible operators as follows: DR(X) := {T ∈C(X) : a(T) and d(T) are both finite} , LD(X) := { T ∈C(X) : a(T) is finite and R ( Ta(T)+1 ) is closed } , RD(X) := { T ∈C(X) : d(T) is finite and R ( Td(T) ) is closed } . An operator T ∈ C(X) is of Kato type if there exist an integer d ∈ N and a pair of two closed subspaces (N1,N2) of X such that: stability of some essential b-spectra 65 (i) X = N1 ⊕N2; (ii) T(N1) ⊂ N1 and T/N1 is semi-regular; (iii) T(N2) ⊂ N2 and (T/N2 ) d = 0, i.e., T/N2 is nilpotent. Note that, in the case of Hilbert spaces, Labrousse in [17, Theorem 3.2.2] has shown that, the set of Quasi-Fredholm operators coincides with the Kato type operators for the class of closed operators. According to [17, p. 206, Remark], this equivalence is also true in the case of Banach spaces. For T ∈C(X) and M ∈ L(X), we define the B-Fredholm spectrum, Drazin spectrum, the upper semi B-Fredholm spectrum, the lower semi B-Fredholm spectrum, the left Drazin spectrum, the right Drazin spectrum, the B-Weyl spectrum, the closed-range spectrum and the Kato spectrum of the linear operator pencil λM −T, where λ ∈ C, or the pair (M,T) as follows: σBF (M,T) = { λ ∈ C : λM −T /∈ ΦB(X) } , σD(M,T) = { λ ∈ C : λM −T /∈ DR(X) } , σBF+ (M,T) = { λ ∈ C : λM −T /∈ Φ+B(X) } , σBF−(M,T) = { λ ∈ C : λM −T /∈ Φ−B(X) } , σLD(M,T) = { λ ∈ C : λM −T /∈ LD(X) } , σRD(M,T) = { λ ∈ C : λM −T /∈ RD(X) } , σBW (M,T) = { λ ∈ C : λM −T /∈ BW(X) } , σec(M,T) = { λ ∈ C : R(λM −T) is not closed } , σek(M,T) = { λ ∈ C : λM −T is not Kato } . The point spectrum, the residual spectrum and the continuous spectrum of the pair (M,T), when T ∈C(X) and M ∈ L(X), are defined respectively by: σp(M,T) = { λ ∈ C : λM −T is not injective } , σr(M,T) = { λ ∈ C : N(λM −T) = {0} and R(λM −T) X } , σc(M,T) = { λ ∈ C : N(λM −T) = {0}, R(λM −T) = X and R(λM −T) 6= X } , where, R(λM −T) is the closure of R(λM − T). The collection {σp(M,T), σr(M,T),σc(M,T)} forms a partition of the spectrum σ(M,T), which means that they are pairwise disjoint and σ(M,T) = σp(M,T)∪σr(M,T)∪σc(M,T). 66 a. ben ali, m. boudhief, n. moalla For T ∈ C(X) and M ∈ L(X), the upper semi B-Fredholm, the lower semi B-Fredholm and the B-Fredholm resolvent of the linear operator pencil λM −T, where λ ∈ C, are defined respectively by ρBF+ (M,T) = C\σBF+ (M,T) , ρBF−(M,T) = C\σBF−(M,T) , ρBF (M,T) = ρBF+ (M,T) ∩ρBF−(M,T) . The present paper is a generalization of the results obtained by A. Jeribi et al. in [15] and some of stability results obtained by M. Berkani et al. in [6] for the usual essential B-spectra. It generalizes also the works obtained by A. Jeribi in [13, 14] about the invariance of the S-essential spectra un- der weakly compact or strictly singular perturbations, which are not applied in the B-Fredholm theory. So that, we can use, by adding some hypothe- sis, the perturbations of the B-Fredholm spectra under finite-rank and power finite-rank commuting operators. More precisely, let T1,T2 ∈ C(X) be two commuting closed linear operators such that the bounded linear operator M commutes in the resolvent sense with T1 and T2 (see Definition 2.2) and satisfy- ing (λM −T1)−1 −(λM −T2)−1 ∈F(X) (resp. (λM −T1)−1 −(λM −T2)−1 ∈ Fp(X) or nilpotent) for some λ ∈ ρM (T1) ∩ ρM (T2). Then, we prove that σ∗(M,T1) = σ∗(M,T2), where σ∗(M,.) ∈ { σBF (M,.),σBF+ (M,.),σBF−(M,.), σBW (M,.),σLD(M,.),σRD(M,.),σD(M,.) } . These perturbation results are needed to extend the results obtained in [15], on the S-essential spectra of closed densely defined linear operators to essential B-spectra of operator pencil λM −T, when T ∈C(X), M ∈ L(X) and λ ∈ C. Moreover, under the additional hypothesis M(C(T)) = C(T) (see Definition 2.1), we give the relationship between the closed-range spectrum and the Kato spectrum of the linear operator pencil λM −T, when T ∈ C(X), M ∈ L(X) and λ ∈ C (see Proposition 2.4), which generalizes a result obtained in [9, Proposition 3.2] in the case of bounded operators and [8, Corollary 3] for closed densely defined linear operators for the usual spectrum. We establish also, the equality between the closed-range spectrum and some essential B-spectra of operator pencil acting on a Banach space (Theorem 2.5). The obtained results, are finally used to describe the essential B-spectra of the operator pencil of the following integro-differential operator with abstract boundary conditions in the Banach space Xp := Lp ( [−a,a] × [−1, 1],dxdy ) , a > 0, 1 ≤ p < ∞, AH = TH + K , stability of some essential b-spectra 67 where TH, K and M are defined by  TH : D(TH) ⊆ Xp −→ Xp ψ 7−→ −ξ ∂ψ ∂x (x,ξ) −σ(ξ)ψ(x,ξ) , D(TH) = { ψ ∈Wp : ψi = Hψo } , where Wp := { ψ ∈ Xp : ξ∂ψ∂x ∈ Xp } ,  K : Xp −→ Xp u 7−→ ∫ 1 −1 k(x,ξ,ν)u(x,ν)dν , and { M : Xp −→ Xp ϕ 7−→ M(ϕ)(x,ξ) = η(ξ)ϕ(x,ξ) , where σ(.) and η(.) are in L∞(−1, 1), k(., ., .) is a measurable function, and H is the boundary operator connecting the outgoing and the incoming fluxes. The outline of this work is organized in the following way: in Section 2, we give some stability results of some essential B-spectra of linear operator pencil. The main results of this section are Theorem 2.3 and Theorem 2.5. In Section 3, we apply the results developed in Section 2 to characterize the B-essential spectra of a transport operator with abstract boundary conditions on Lp-spaces, 1 ≤ p < ∞. 2. Stability of some essential B-spectra of pencil operators We are interested, in this section, in some of stability results of the essential B-spectra of an operator pencil λM − T , where M ∈ L(X), T ∈ C(X) and λ ∈ C. Since, the Kato decomposition theorem, remains true in the case of Banach spaces as shown in [17, p. 206], we can directly use the following proposition inspired from [10], when necessary, in the case of Banach spaces without proof. Proposition 2.1. Let T ∈ C(X). If T is a semi B-Fredholm operator, then there exist two closed subspaces X0 and X1 of X such that (i) X = X0 ⊕X1, (ii) T(X0) ⊂ X0 and T0 = T/X0 is a semi-Fredholm operator, (iii) T(X1) ⊂ X1 and T1 = T/X1 is a nilpotent operator. 68 a. ben ali, m. boudhief, n. moalla First we recall the following subspace, introduced by P. Saphar in [20], and it was defined by P. Aiena in [1] in purely algebraic terms. Definition 2.1. The algebraic core C(T) of a linear operator T is defined to be the greatest subspace N ⊂ D(T) for which T(N) = N. For more details for the algebraic core C(T), we can refer to [1]. Theorem 2.1. Let T ∈C(X) and M ∈ L(X) such that M(C(T)) = C(T) and ρM (T) 6= ∅. If T is a semi B-Fredholm operator, then there exists ε > 0 such that T − µM is a semi-Fredholm operator, for each µ ∈ D(0,ε)\{0}. Moreover, we have α(T −µM) and β(T −µM) are constants on D(0,ε)\{0}. Proof. If M = I, then we obtain the result established in [7]. If M 6= I, then the fact that T is a semi B-Fredholm operator, this implies from Proposition 2.1, the existence of two T-invariant closed subspaces X0 and X1 such that • X = X0 ⊕X1, • T(X0) ⊂ X0 and T0 = T/X0 is a semi-Fredholm operator, • T(X1) ⊂ X1 and T1 = T/X1 is nilpotent. Since M(C(T)) = C(T) and T(C(T)) = C(T), then we can conclude, by using the definition of C(T), that X0 and X1 are invariants under the operator M. So, we can consider M0 = M/X0 and M1 = M/X1 such that M = M0⊕M1. Case 1: If X0 = {0}, then we get M0 = 0, T0 = 0, M = M1 and T = T1 is nilpotent. Since, the operator T1 is nilpotent then we have T1 − µI1 is invertible, for µ 6= 0, where I1 = I/X1 . We have T1 −µM1 = (T1 −µI1)[I1 −µ(T1 −µI1)−1(M1 − I1)] is invertible for µ such that 0 < |µ| < 1 ‖(T1 −µI1)−1(M1 − I1)‖ = γ . Hence, T − µM = T1 − µM1 is a semi-Fredholm operator, for µ such that 0 < |µ| < γ. Moreover, we have α(T − µM) = α(T1 − µM1) = 0 and β(T −µM) = β(T1 −µM1) = 0 on D(0,γ)\{0}. Case 2: Suppose X0 6= {0}. • If M0 = 0, then by using Case 1, we obtain that T−µM = T0⊕T1−µM1 is a semi-Fredholm operator for µ such that 0 < |µ| < γ. • If M0 6= 0, then T − µM = T0 − µM0 ⊕ T1 − µM1. It follows from [19, Theorem 7.9], that T0 − µM0 is a semi-Fredholm for µ such that stability of some essential b-spectra 69 µ ∈ D ( 0, ε ′ ‖M0‖ ) \{0}, for some ε′ > 0. Set ε = min ( ε′ ‖M0‖ ,γ ) . Therefore, the operator T − µM is semi-Fredholm for µ such that µ ∈ D(0,ε)\{0}. Again, from [19, Theorem 7.9], we get α(T − µM) = α(T0 − µM0) and β(T −µM) = β(T0 −µM0) are constants on D(0,ε)\{0}. Using Theorem 2.1, we can deduce the following result which is a general- ization of [15, Proposition 2.1] Corollary 2.1. Let T ∈ C(X) and M ∈ L(X) such that M(C(T)) = C(T) and ρM (T) 6= ∅. Then, (i) ρBF+ (M,T), ρBF−(M,T) and ρBF (M,T) are open subsets of C; (ii) ind(λM−T) is constant on any component of ρBF+ (M,T), ρBF−(M,T) and ρBF (M,T). Proof. (i) Let λ0 ∈ ρBF+ (M,T), then from Theorem 2.1 there exists an ε > 0 such that T − µM is an upper semi-Fredholm operator, for each µ ∈ D(λ0,ε)\{0}. This implies that ρBF+ (M,T) is an open subset of C. The same proof is used to show that, ρBF−(M,T) and ρBF (M,T) are open subsets of C. (ii) Let Ω be a component of ρBF+ (M,T) (resp. ρBF−(M,T)), λ0 ∈ Ω be a fixed point and λ1 ∈ Ω be an arbitrary point that are connected by a polygonal line Γ contained in Ω. It follows from the assertion (i) of this corollary that, for each µ ∈ Γ, there exists an open disc D(µ,ε), such that ind(µM − T) = ind(λM − T), for each λ ∈ D(µ,ε). By the Heine-Borel theorem, there exist a finite number of open discs that cover Γ. This allows us to deduce that ind(λ0M −T) = ind(λ1M −T). Now, we recall the following definition considered in [12] for bounded linear operators and it remains also true in the general case of closed linear operators: Definition 2.2. Let M ∈ L(X) and T ∈C(X) such that ρM (T) 6= ∅. We say that M and T commute in the sense of resolvent if for all λ ∈ ρM (T), M(T −λM)−1 = (T −λM)−1M . Remarks 2.1. (a) If M and T commute in the sense of the resolvent, the assumption M(C(T)) ⊂ C(T) is verified. Indeed, let x ∈ C(T). Then, from [1, Theorem 1.8], there exists a sequence (un) ⊂ D(T) such that x = u0 and Tun+1 = un, for every n ∈ Z+. Set yn = Mun. The commutativity 70 a. ben ali, m. boudhief, n. moalla of the resolvent of the operators M and T, permits us to deduce that M = (λM − T)−1M(λM − T) on D(T) and TM = MT on D(T), which entails that Mun ∈ D(T). Thus, we get y0 = Mu0 = Mx and Tyn+1 = TMun+1 = MTun+1 = Mun = yn, for every n ∈ Z+. This, implies that Mx ∈ M(C(T)) and finally we obtain that M(C(T)) ⊂ C(T). (b) If M and T commute in the sense of the resolvent and M is invertible, then we get M−1(C(T)) ⊂ C(T) and finally we can conclude that C(T) ⊂ M(C(T)). Proposition 2.2. Let M ∈ L(X) be an invertible operator and T ∈C(X) such that 0 ∈ ρM (T). If MT−1 = T−1M, then (i) for λ 6= 0 and n ≥ 1: (T −λM)n(T−1)n = (−λ)nMn ( T−1 −λ−1M−1 )n ; (ii) for all n ≥ 1: R ( (T −λM)n ) = R ( (T−1 −λ−1M−1)n ) . Proof. (i) For n = 1, the equality is obvious. Let n ≥ 1 and assume that (T −λM)n(T−1)n = (−λ)nMn(T−1 −λ−1M−1)n, then (T −λM)n+1(T−1)n+1 = (T −λM) [ (T −λM)n ( T−1 )n] T−1 = (T −λM) [ (−λ)nMn ( T−1 −λ−1M−1 )n] T−1 = (T −λM)T−1 [ (−λ)nMn ( T−1 −λ−1M−1 )n] = (−λ)n+1Mn+1 ( T−1 −λ−1M−1 )n+1 . (ii) It follows from (i), that R [ Mn(T−1 −λ−1M−1)n ] = R [ (T−1 −λ−1M−1)n ] ⊆ R [ (T −λM)n ] . Conversely, if y ∈ R((T − λM)n), then there exists x ∈ D(Tn) such that y = (T − λM)nx. The fact that (T−1)n(X) = D(Tn), this enable us the existence of t ∈ X such that x = (T−1)n(t). Hence, y = (T −λM)n ( T−1 )n (t) and finally y = (−λ)nMn ( T−1 −λ−1M−1 )n (t) ∈ R [ (T−1 −λ−1M−1)n ] . Proposition 2.3. Let M ∈ L(X) be an invertible operator and T ∈C(X) such that 0 ∈ ρM (T). If MT−1 = T−1M, then (i) for λ 6= 0 and n ≥ 1: (T−1)n(T −λM)n = (−λ)nMn ( T−1 −λ−1M−1 )n and Tn(T−1 −λ−1M−1)n = (−λ−1)n(M−1)n(T −λM)n on D(Tn); (ii) for all n ≥ 1: N((T −λM)n) = N ( (T−1 −λ−1M−1)n ) . stability of some essential b-spectra 71 Proof. (i) Since T is invertible and D(T −λM) = D(T), then R ( (T−1)n ) = D(Tn) and R(Tn) = X. Therefore, the operators (T−1)n(T − λM)n and Tn ( T−1 − λ−1M−1 )n are well defined. Then, we verify directly that (T−1)n(T −λM)n = (−λ)nMn ( T−1 −λ−1M−1 )n and Tn ( T−1 −λ−1M−1 )n = (−λ−1)n(M−1)n(T −λM)n on D(Tn). (ii) It is a direct consequence of (i). Remark 2.2. If M = I, we recover the results obtained in [10]. Now, we state the main result of this section. Theorem 2.2. Let M ∈ L(X) be an invertible operator and T ∈ C(X) such that 0 ∈ ρM (T). If MT−1 = T−1M, then σ∗(M,T) = { λ−1 : λ ∈ σ∗ ( M−1,T−1 ) \{0} } , where σ∗(M,T) ∈ { σBF+ (M,T),σBF−(M,T),σBF (M,T), σD(M,T),σBW (M,T),σLD(M,T),σRD(M,T) } . Proof. Let λ 6= 0. By using Proposition 2.2 and Proposition 2.3, we get that λM−1 − T−1 is a B-Fredholm operator if and only if λ−1M − T is a B-Fredholm one. The same arguments are used to prove the other B-spectra. In order to give, the stability result of some essential B-spectra of operator pencil by means of some class of commuting perturbations which generalizes some results established in [6], we shall define the following class of operators: We say that a linear operator is of finite-rank if its range is of finite dimension. If there exists an integer p ∈ N∗ such that dim R(Tp) < ∞, then it is called a power finite-rank operator. We will denote by F(X) (resp. Fp(X)) the set of all finite-rank linear bounded (resp. power finite-rank) operators. In many applications (see Section 3) the perturbed operator is not of finite rank but we have some information about the difference of the resolvent, so the usual result. Theorem 2.3. Let M ∈ L(X) be an invertible operator and T1,T2 ∈ C(X) such that M commutes with T1 and T2 in the sense of resolvent. If for some λ ∈ ρM (T1)∩ρM (T2), the operator (T1−λM)−1−(T2−λM)−1 ∈F(X), 72 a. ben ali, m. boudhief, n. moalla then σ∗(M,T1) = σ∗(M,T2) , where, σ∗(M,.) ∈{σBF (M,.),σBF+ (M,.),σBF−(M,.),σBW (M,.)}. Proof. Without loss of generality, we can assume that λ = 0, then T−11 − T −1 2 ∈ F(X). Let µ 6= 0. The use of Theorem 2.2 shows that, µM − T1 is a B-Fredholm operator if and only if µ−1M−1 − T−11 is a B- Fredholm one. Since T−11 −T −1 2 ∈F(X), then by using [3, Corollary 3.10], we get µ−1M−1 −T−12 is a B-Fredholm operator. Again by Theorem 2.2, this is equivalent to µM−T2 is also a B-Fredholm operator, which finish the proof of the B-Fredholm spectrum equality. For the upper semi B-Fredholm spectrum, the lower semi B-Fredholm spectrum and the B-Weyl spectrum, we use the same technique as above and [11, Proposition 2.7]. Definition 2.3. ([18]) Let X be a Banach space, A : D(A) ⊂ X → X and T : D(T) ⊂ X → X two linear operators. We say that A commutes with T , and we denote AT = TA, if (i) D(A) ⊂ D(T); (ii) Tx ∈ D(A) whenever x ∈ D(A); (iii) AT = TA on {x ∈ D(A) : Ax ∈ D(T)}. Under the additional hypothesis of commutativity of operators, we get a stronger version of Theorem 2.3: Theorem 2.4. Let M ∈ L(X) be an invertible operator and T1,T2 ∈ C(X) such that M commutes with T1 and T2 in the resolvent sense and T1T2 = T2T1. If for some λ ∈ ρM (T1) ∩ ρM (T2), the operator (T1 − λM)−1 − (T2 − λM)−1 ∈Fp(X), then σ∗(M,T1) = σ∗(M,T2) , where σ∗(M,T) ∈ { σBF (M,.),σBF+ (M,.),σBF−(M,.), σBW (M,.),σLD(M,.),σRD(M,.),σD(M,.) } . Proof. Without loss of generality, we can assume that λ = 0, then T−11 − T−12 ∈ Fp(X). Let µ 6= 0. It follows from Theorem 2.2, that µM − T1 is stability of some essential b-spectra 73 a B-Fredholm operator if and only if µ−1M−1 − T−11 is a B-Fredholm one. Since, T−11 −T −1 2 ∈Fp(X), then from [16], we obtain that µ −1M−1 −T−12 is also a B-Fredholm operator, which is equivalent to µM −T1 is a B-Fredholm operator by Theorem 2.2. This, shows that σBF (M,T1) = σBF (M,T2). For the other equalities, we use the same technique as above. Corollary 2.2. Let M ∈ L(X) be an invertible operator and T1,T2 ∈ C(X) such that M commutes with T1 and T2 in the resolvent sense and T1T2 = T2T1. If for some λ ∈ ρM (T1) ∩ ρM (T2), the operator (T1 − λM)−1 − (T2 − λM)−1 is nilpotent, then σ∗(M,T1) = σ∗(M,T2) , where σ∗() ∈ { σBF (M,T),σBF+ (M,T),σBF−(M,T), σBW (M,T),σLD(M,T),σRD(M,T),σD(M,T) } . Corollary 2.3. Let T ∈C(X), M ∈ L(X) be an invertible operator and Q ∈ L(X) a nilpotent operator such that ρM (T) 6= ∅. If TQ = QT on D(T), MQ = QM and M,T commute in the resolvent sense, then σ∗(M,T) = σ∗(M,T + Q) , where σ∗() ∈ { σBF (M,T),σBF+ (M,T),σBF−(M,T), σBW (M,T),σLD(M,T),σRD(M,T),σD(M,T) } . Proof. Since, TQ = QT and Q is a nilpotent operator, then (λM−T)−1Q is nilpotent, for all λ ∈ ρM (T). Thus, its spectral radius is equal to zero, which implies that λ ∈ ρM (T + Q) and that (λM −T −Q)−1 = (λM −T)−1(M − (λI −T)−1Q)−1 = (λM −T)−1 n∑ k=0 ((λM −T)−1Q)k = (λM −T)−1 + (λM −T)−1Q n−1∑ k=1 ((λM −T)−1)kQk−1 74 a. ben ali, m. boudhief, n. moalla with n is the nilpotent-index of (λM−T)−1Q. Hence, (λM−T−Q)−1−(λM− T)−1 is nilpotent. So, we deduce from Corollary 2.2, that σ∗(M,T + Q) = σ∗(M,T). The following proposition is proved in [12] for bounded linear operators and it holds also true in the general case of closed densely-defined linear operators. Proposition 2.4. Let T ∈ C(X) be densely-defined linear operator and M ∈ L(X) such that M(C(T)) = C(T). If λ ∈ σec(M,T) is non-isolated, then λ ∈ σek(M,T). Remark 2.3. Proposition 2.4 is also true if we replace σek(M,T) by σqf (M,T), where σqf (M,T) = {λ ∈ C : λM − T s not a Quazi-Fredholm operator}. The following theorem, shows the equality between the closed-range spectrum and some essential B-spectra of operator pencil acting on the Banach space. Theorem 2.5. Let T ∈C(X) be densely-defined linear operator and M ∈ L(X) such that M(C(T)) = C(T). If σec(M,T) = σ(M,T) and every λ ∈ σec(M,T) is non-isolated. Then σ(M,T) = σBF (M,T) = σBW (M,T) = σBF+ (M,T) = σBF−(M,T) = σD(M,T) . Proof. Since σBF (M,T) ⊂ σ(M,T), it suffices to show that σ(M,T) ⊂ σBF (M,T). Let λ ∈ σ(M,T), then from Proposition 2.4, we have λ ∈ σek(M,T). Since, a B-Fredholm operator is a Quasi-Fredholm one, this shows that σek(M,T) ⊂ σBF (M,T). The same arguments are used for the upper semi B-Fredholm, the lower semi B-Fredholm and the B-Weyl spectrum. The fact that, a Drazin invertible operator is a B-Fredholm one, then by using the same arguments as above we can prove that, σ(M,T) = σD(M,T). Fi- nally, we conclude that σ(M,T) = σec(M,T) = σBF (M,T) = σBW (M,T) = σBF+ (M,T) = σBF−(M,T) = σD(M,T). 3. Application In this section, we will use the previous results to treat the essential B- spectra of a transport operator with abstract boundary conditions. Let Xp := Lp ( (−a,a) × (−1, 1),dxdξ ) , a > 0 , 1 ≤ p < ∞ . stability of some essential b-spectra 75 We consider the following integro-differential operator with abstract boundary conditions: AH = TH + K , where TH is defined by  TH : D(TH) ⊆ Xp −→ Xp ψ 7−→ THψ(x,ξ) = −ξ ∂ψ ∂x (x,ξ) −σ(ξ)ψ(x,ξ) , D(TH) = { ψ ∈Wp : ψi = Hψo } , where Wp := { ϕ ∈ Xp : ξ∂ϕ∂x ∈ Xp } and σ(.) ∈ L∞(−1, 1); ψo,ψi are, respectively, the outgoing and the incoming fluxes related by the boundary operator H (“o” for the outgoing and “i” for the incoming), and given by  ψi(ξ) = ψ(−a,ξ) , ξ ∈ (0, 1) , ψi(ξ) = ψ(a,ξ) , ξ ∈ (−1, 0) , ψo(ξ) = ψ(−a,ξ) , ξ ∈ (−1, 0) , ψo(ξ) = ψ(a,ξ) , ξ ∈ (0, 1) . The bounded collision operator K is defined by  K : Xp −→ Xp u 7−→ K(u)(x,ξ) = ∫ 1 −1 k(x,ξ,ν)u(x,ν) dν , where the kernel k : (−a,a) × (−1, 1) × (−1, 1) −→ R is assumed to be measurable. The following boundary spaces denoted by Xop and X i p are defined as fol- lows: Xop := Lp [ {−a}× (−1, 0); |ξ|dξ ] ×Lp [ {a}× (0, 1); |ξ|dξ ] := Xo1,p ×X o 2,p equipped with the norm ∥∥u◦,X◦p∥∥ := (∥∥u◦1,X◦1,p∥∥p + ∥∥u◦2,X◦2,p∥∥p)1p = [∫ 0 −1 |u(−a,v)|p |v|dv + ∫ 1 0 |u(a,v)|p |v|dv ]1 p , 76 a. ben ali, m. boudhief, n. moalla and Xip := Lp [ {−a}× (0, 1); |ξ|dξ ] ×Lp [ {a}× (−1, 0); |ξ|dξ ] := Xi1,p ×X i 2,p equipped with the norm ∥∥ui,Xip∥∥ := (∥∥ui1,Xi1,p∥∥p + ∥∥ui2,Xi2,p∥∥p)1p = [∫ 1 0 |u(−a,v)|p |v|dv + ∫ 0 −1 |u(a,v)|p |v|dv ]1 p . In this part, we will determine the essential B-spectra of the pair (M,AH), where M is the operator defined by{ M : Xp −→ Xp ϕ 7−→ M(ϕ)(x,ξ) = η(ξ)ϕ(x,ξ) , where η(.) ∈ L∞(−1, 1). To clarify our subsequent analysis, we define the following bounded oper- ators introduced in [15]:  Mλ : X i p −→ Xop , Mλu := ( M+λ u,M − λ u ) , with (M+λ u)(−a,ξ) := u(−a,ξ) e −2a |ξ| (λη(ξ)+σ(ξ)) , 0 < ξ < 1 , (M−λ u)(a,ξ) := u(a,ξ) e −2a |ξ| (λη(ξ)+σ(ξ)) , −1 < ξ < 0 ,   Bλ : X i p −→ Xp , Bλu := χ(−1,0)(ξ)B − λ u + χ(0,1)(ξ)B + λ u, with (B−λ u)(x,ξ) := u(a,ξ) e −(λη(ξ)+σ(ξ))|ξ| |a−x| , −1 < ξ < 0 , (B+λ u)(x,ξ) := u(−a,ξ) e −(λη(ξ)+σ(ξ))|ξ| |a+x| , 0 < ξ < 1 ,   Gλ : Xp −→ Xop , Gλϕ := ( G+λ ϕ,G − λ ϕ ) , with G−λ ϕ(a,ξ) := 1 |ξ| ∫ a −a e −(λη(ξ)+σ(ξ)) |ξ| |a+x|ϕ(x,ξ) dx, −1 < ξ < 0 , G+λ ϕ(−a,ξ) := 1 |ξ| ∫ a −a e −(λη(ξ)+σ(ξ)) |ξ| |a−x|ϕ(x,ξ) dx, 0 < ξ < 1 , stability of some essential b-spectra 77 and finally  Cλ : Xp −→ Xp , Cλϕ = χ(−1,0)C − λ ϕ + χ(0,1)C + λ ϕ, with C−λ ϕ(x,ξ) := 1 |ξ| ∫ a x e −(λη(ξ)+σ(ξ)) |ξ| |x−x ′| ϕ(x′,ξ) dx′ , −1 < ξ < 0 , C+λ ϕ(x,ξ) := 1 |ξ| ∫ x −a e −(λη(ξ)+σ(ξ)) |ξ| |x−x ′| ϕ(x′,ξ) dx′ , 0 < ξ < 1 , where χ(0,1)(.) and χ(−1,0)(.) denote, respectively, the characteristic functions of the intervals (0, 1) and (−1, 0). Note that, from [15, Proposition 3.1], the operators Mλ, Bλ, Gλ and Cλ are bounded respectively by exp(−2aµ∗ Re λ), (pµ∗ Re λ)−1/p, (µ∗ Re λ)−1/q and (µ∗ Re λ)−1 where q is the conjugate of p. In what follows, we will show that the M-spectrum of T0 (i.e., TH with H = 0) is the continuous spectrum σc(M,T0) of the pair (M,T0). Lemma 3.1. (i) The point spectrum of the pair (M,T0) is empty. (ii) The residual spectrum of the pair (M,T0) is empty. Proof. (i) We consider for λ ∈ C such that Re(λ) ≤ 0, the eigenvalue problem (λM − T0)ψ = 0, where the unknown ψ must be in D(T0). His solution is formally given by ψ(x,ξ) = K(ξ)e −1 |ξ| [λη(ξ)+σ(ξ)]x. Moreover, since ψ ∈ D(T0), then we get ψi = 0. So we obtain K(ξ) = 0 on (−1, 1). Consequently, ψ = 0. (ii) To prove that the residual spectrum σr(M,T0) is also empty, we shall determine the point spectrum of the adjoint operator pencil densely defined λM −T0, where λ ∈ C. The adjoint operators T∗0 and M ∗ are, respectively, given by:  T∗0 : D(T ∗ 0 ) ⊆ Xq −→ Xq ψ 7−→ T∗0 ψ(x,ξ) = ξ ∂ψ ∂x (x,ξ) −σ(ξ)ψ(x,ξ) , D(T∗0 ) = { ψ ∈Wq : ψo = 0 } , where q is the conjugate of p ( 1 p + 1 q = 1 ) ,{ M∗ : Xq −→ Xq ϕ −→ M∗(ϕ)(x,v) = η(v)ϕ(x,v) . 78 a. ben ali, m. boudhief, n. moalla Let us consider the eigenvalue problem (λM∗ −T∗0 )ψ = 0. (3.1) In view of the boundary conditions, a straightforward estimation shows that, the problem (3.1) admits only the trivial solution. Then, we obtain σp(M ∗,T∗0 ) = ∅. Since, σr(M,T0) ⊆ σp(M ∗,T∗0 ), then we can easily obtain the desired result. Now, by using the previous results, we can deduce the following theorem: Theorem 3.1. Let M ∈ L(X). Then, σ(M,T0) = σc(M,T0) = σec(M,T0) = { λ ∈ C : Re(λ) ≤ 0 } . Proof. Since, σ(M,T0) = σp(M,T0)∪σr(M,T0)∪σc(M,T0), then by using Lemma 3.1, we deduce σ(M,T0) = σc(M,T0). On the other hand, it follows from [15, Theorem 3.1], that σ(M,T0) = {λ ∈ C : Re(λ) ≤ 0}. Combining these two results, we can obtain the assertion of the present theorem. Now, we are able to express the B-spectra of the pair (M,TH): Theorem 3.2. If the boundary operator H is of finite rank and commutes with M, and every point λ in σec(M,T0) is non-isolated, then σ∗(M,TH) = σ∗(M,T0) = { λ ∈ C : Re(λ) ≤ 0 } , where σ∗(M,.) ∈ { σBF (M,.),σBF+ (M,.),σBF−(M,.),σBW (M,.) } . Proof. According to [15], we have (λM −TH)−1 = H ∑ n≥0 Bλ(MλH) nGλ + Cλ, (3.2) where Cλ = (λM −T0)−1. Since (λM −TH)−1 − (λM −T0)−1 ∈F(X), this implies by Theorem 2.3 that σ∗(M,TH) = σ∗(M,T0). The fact that M and TH commute in the sense of the resolvent and M is invertible this allows us, by the use of Remarks 2.1. Theorem 2.5 and Theorem 3.1 to conclude that σ∗(M,T0) = { λ ∈ C : Re(λ) ≤ 0 } , where σ∗(M,.) ∈ { σBF (M,.),σBF+ (M,.),σBF−(M,.),σBW (M,.) } stability of some essential b-spectra 79 Theorem 3.3. Suppose that the boundary operator H and the collision operator K are of finite rank. 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Helv. 43 (1968), 87 – 97. [19] M. Schechter, “ Principles of Functional Analysis ”, Academic Press, New York-London, 1971. [20] P. Saphar, Contribution à l’étude des applications linéaires dans un espace de Banach, Bull. Soc. Math. France 92 (1964), 363 – 384. Introduction Stability of some essential B-spectra of pencil operators Application