@ Appl. Gen. Topol. 23, no. 2 (2022), 425-436 doi:10.4995/agt.2022.16940 © AGT, UPV, 2022 Fredholm theory for demicompact linear relations Aymen Ammar, Slim Fakhfakh and Aref Jeribi Department of mathematics, University of Sfax, Faculty of Sciences of Sfax, Soukra Road Km 3.5, B.P 1171, 3000, Sfax, Tunisia (ammar aymen84@yahoo.fr, sfakhfakh@yahoo.fr, Aref.Jeribi@fss.rnu.tn) Communicated by D. Werner Abstract We first attempt to determine conditions on a linear relation T such that µT becomes a demicompact linear relation for each µ ∈ [0,1) (see Theorems 2.4 and 2.5). Second, we display some results on Fredholm and upper semi-Fredholm linear relations involving a demicompact one (see Theorems 3.1 and 3.2). Finally, we provide some results in which a block matrix of linear relations becomes a demicompact block matrix of linear relations (see Theorems 4.2 and 4.3). 2020 MSC: 47A06. Keywords: demicompact linear relations; Fredholm theory; block matrix. 1. Introduction Throughout this work, X, Y and Z are vector spaces over the field K = R or C. A mapping T, whose domain is a linear subspace D(T) := {x ∈ X : Tx 6= ∅} of X, is called a linear relation (or a multivalued linear operator) if for all x,z ∈D(T) and non-zero scalars α; we have Tx + Tz = T(x + z) αTx = T(αx). Evidently, the domain of linear relation is a linear subspace. Received 25 December 2021 – Accepted 10 May 2022 http://dx.doi.org/10.4995/agt.2022.16940 A. Ammar, S. Fakhfakh and A. Jeribi In this notation, LR(X,Y ) denotes the class of all linear relations on X into Y , if X = Y simply denotes LR(X,X) := LR(X). If T maps the points of its domain to singletons, then it is said to be a single valued linear operator (or simply an operator). The simplest naturally occurring example of a multivalued linear operator is the inverse T−1 of a linear map T from X to Y defined by the set of solutions T−1y := {x ∈ X : Tx = y} for equation Tx = y. Each linear relation is identified only by its graph, G(T), which is defined by G(T) := {(x,y) ∈ X ×Y : x ∈D(T) and y ∈ Tx}. The inverse of T is the linear relation, T−1 expressed by G(T−1) := {(y,x) ∈ Y ×X : (x,y) ∈ G(T)}. The subspace N(T) := {x ∈D(T) such that (x, 0) ∈ G(T)} is called the null space of T , and T is called injective if N(T) = {0}, that is, if T−1 is a single valued linear operator. T−1(0) := N(T). The range of T is the subspace R(T) := {y ∈ Y,∃x ∈D(T) : (x,y) ∈ G(T)} and T is called surjective if R(T) = Y . If T is injective and surjective, then we state that T is bijective. The quantities α(T) := dim (N(T)) and β(T) := codim(R(T)) = dim(Y/R(T)) are called the nullity (or the kernel index) and the deficiency of T, respectively. We also write β(T) := codim(R(T)). The index of T is defined by i(T) := α(T) − β(T). If α(T) and β(T) are infinite, then T is said to have no index. Let M be a subspace of X such that M ∩D(T) 6= ∅ and let T ∈ LR(X,Y ); then, the restriction T|M , is the linear relation indicated by G(T|M ) := {(m,y) ∈ G(T) : m ∈ M} = G(T) ∩ (M ×Y ). For S, T ∈LR(X,Y ) and R ∈LR(Y,Z), the sum S + T and the product RS are the linear relations determined by G(T + S) := {(x,y + z) ∈ X ×Y : (x,y) ∈ G(T) and (x,z) ∈ G(S)}, and G(RS) := {(x,z) ∈ X ×Z : (x,y) ∈ G(S), (y,z) ∈ G(R) for some y ∈ Y}, respectively and if λ ∈ K, the λT is computed by G(λT) := {(x,λy) : (x,y) ∈ G(T)}. If T ∈LR(X) and λ ∈ K, then the linear relation λ−T is identified by G(λ−T) := {(x,y −λx) : (x,y) ∈ G(T)}. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 426 Fredholm theory for demicompact linear relations Let T ∈LR(X,Y ). We write QT for the quotient map from Y into Y/T(0). Clearly, QTT is an operator. For all x ∈ D(T), we define ‖Tx‖ := ‖QTTx‖, and the norm of T is defined by ‖T‖ := ‖QTT‖. We note that ‖Tx‖ and ‖T‖ are not real norms. In fact, a non-zero relation can have a zero norm. T is said to be closed if its graph G(T) is a closed subspace of X ×Y . The closure of T denoted by T is defined in terms of its graph G(T) := G(T). We denote by CR(X,Y ) the class of all the closed linear relations on X into Y , if X = Y which simply denotes CR(X,X) := CR(X). If T is an extension to T, we say that T is closable. Let T ∈LR(X,Y ). We say that T is continuous if for each neighbourhood V in R(T), the inverse image T−1(V ) is a neighbourhood in D(T) equivalently if ‖T‖ < ∞; open if T−1 is continuous, bounded if D(T) = X and T is continuous, bounded below if it is injective and open and compact if QTT(BD(T)) is compact in Y (BD(T) := {x ∈D(T) : ‖x‖≤ 1}). We denote by KR(X,Y ) the class of all the compact linear relations on X into Y , if X = Y simply denotes KR(X,X) := KR(X). If X is a normed linear space, then X ′ will denote the dual space of X, i.e., the space of all the continuous linear functionals x′ which 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. Let T ∈LR(X,Y ). The adjoint of T, which is T ′, is defined by G(T ′) = G(−T−1)⊥ ⊂ Y ′ ×X ′ where 〈 (y,x), (y′,x′) 〉 := 〈x,x′〉 + 〈y,y′〉. This means that (y′,x′) ∈ G(T ′) 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 ∈ Tx, x ∈D(T). Hence, x′ ∈ T ′y if, and only if, y′Tx = x′x for all x ∈D(T). Definition 1.1 ([7, Definition, V.1.1]). (i) A linear relation T ∈LR(X,Y ) is said to be upper semi-Fredholm and denoted by T ∈F+(X,Y ), if there exists a finite codimensional subspace M of X for which T|M is injective and open. (ii) A linear relation T is said to be lower semi-Fredholm and denoted by T ∈F−(X,Y ), if its conjugate T ′ is upper semi-Fredholm. If X = Y , this simply denotes F+(X,Y ) and F−(X,Y ) by respectively F+(X) and F−(X). For the case, when X and Y are Banach spaces, we extend the class of closed single valued Fredholm type operators provided earlier to include closed mul- tivalued operators. Note that the definitions of F+(X,Y ) and F−(X,Y ) are, © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 427 A. Ammar, S. Fakhfakh and A. Jeribi respectively, consistent with Φ+(X,Y ) := {T ∈CR(X,Y ) : R(T) is closed, and α(T) < ∞}, Φ−(X,Y ) := {T ∈CR(X,Y ) : R(T) is closed, and β(T) < ∞}. If X = Y , this simply denotes Φ+(X,Y ) and Φ−(X,Y ) by respectively Φ+(X) and Φ−(X). Lemma 1.2 ([1, Lemma 2.1]). Let T : D(T) ⊆ X −→ Y be a closed linear relation. Then, (i) T ∈ Φ+(X,Y ) if, and only if, QTT ∈ Φ+(X,Y/T(0)). (ii) T ∈ Φ−(X,Y ) if, and only if, QTT ∈ Φ−(X,Y/T(0)). Definition 1.3 ([9]). Let X be a Banach space. Let D be a bounded subset of X. We define γ(D), the Kuratowski measure of noncompactness of D, to be inf{d > 0 such that D can be covered by a finite number of sets of a diameter less than or equal to d }. The following Proposition displays some properties of the Kuratowski mea- sure of noncompactness which are frequently used. Proposition 1.4 ([9]). Let D and D′ be two bounded subsets of X. Then, we have the following properties: (i) γ(D) = 0 if, and only if, D is relatively compact. (ii) if D ⊆ D′, then γ(D) ≤ γ(D′). (iii) γ(D + D′) ≤ γ(D) + γ(D′). (iv) For every α ∈ C, γ(αD) = |α|γ(D). The linear relations, which were introduced into a functional analysis by J. Von Neumann, were motivated by the need to consider adjoints of non- densely defined linear differential operators. These linear relations were widely investigated in a large number of papers (see, for example, [2], [3] and [5]). The notion of demicompactess for linear operators (that is, single valued operators) was introduced into the functional analysis by W.V Petryshyn [10], to discuss fixed points. Since this notion has be come a hot area of research triggering significant scientific concern, several research papers such as [8, 10] invested it in their investigation. In 2012, W. Chaker, A. Jeribi and B. Krichen achieved some results on Fredholm and upper semi-Fredholm operators involv- ing demicompact operators [6]. In what follows, we shall present two definitions set forward by A. Ammar, H. Daoud and A. Jeribi in 2017 [4], who extended the concept of demicompact and k-set-contraction of linear operators on multivalued linear operators and developed some pertinent properties. Definition 1.5 ([4, Definition 3.1]). A linear relation T : D(T) ⊆ X −→ X is said to be demicompact if for every bounded sequence {xn} in D(T), such © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 428 Fredholm theory for demicompact linear relations that QI−T (I − T)xn = QT (I − T)xn → x ∈ X/T(0), there is a convergent subsequence of QTxn. Definition 1.6 ([4, Definition 4.1]). T : D(T) ⊆ X → Y is a linear relation, while δ1 and δ2 are respectively Kuratowski measures of noncompactness in X/D and Y , where D is a closed subspace of N(T). Let k ≥ 0, T is said to be k −D-set-contraction if, for any bounded subset B of D(T), QTT(B) is a bounded subset of Y/T(0) and δ2(QTTB) ≤ kδ1(QDB). If D = {0}, then T is said to be k−{0}-set-contractive linear relation or simply k-set-contractive. According to these definitions and referring to certain notations and some ba- sic concepts of demicompact linear relations, we elaborate the following propo- sitions. Proposition 1.7. Let T : D(T) ⊆ X −→ X be a closed single-valued linear operator. (i) [6, Theorem 4] If T is demicompact, then I − T is an upper single-valued linear operator semi-Fredholm. (ii) [6, Theorem 5] If µT is demicompact for each µ ∈ [0, 1], then I − T is a single-valued linear operator Fredholm and i(I −T) = 0. The basic objective of this paper is to attempt to answer the following ques- tion ”Under which conditions does the linear relation µT for each µ ∈ [0, 1) become a demicompact linear relation ?” Subsequently, we shall exhibit some results on Fredholm linear relations and upper semi-Fredholm demicompact lin- ear relations. Thereafter, we shall display some results about a demicompact block matrix of linear relations. The rest of the current paper is organized as follows. In section 2 which is entitled ”Auxiliary results on demicompact linear relation”, we provide condi- tions so that any linear relation becomes a demicompact linear relation and we present the results deriving from these demicompact relations (see Theorems 2.4 and 2.5). In section 3 which is entitled ”Fredholm and upper semi-Fredholm linear relations”, we investigate Fredholm linear relations as well as upper semi- Fredholm demicompact linear relations (see Theorems 3.1 and 3.2). Finally we exhibit some results in which a block matrix of linear relations becomes a demicompact block matrix of linear relations (see Theorems 4.2 and 4.3). 2. Auxiliary results on demicompact linear relations In this Section, we try to answer the following question ”Under which con- ditions does the linear relation µT for each µ ∈ [0, 1) become a demicompact linear relation ?” We then present some fundamental results about demicom- pact linear relations. Lemma 2.1. Let T : D(T) ⊆ X −→ X be a linear relation. If I − QT is compact, then T is demicompact if, and only if, QTT is demicompact. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 429 A. Ammar, S. Fakhfakh and A. Jeribi Proof. We suppose that T is demicompact. Let {xn} be a bounded sequence of D(T) such that xn −QTTxn → y. We have xn −QTTxn = (I −QT )xn + QTxn −QTTxn.(2.1) Based upon Eq. (2.1) and considering that I−QT is compact and {xn−QTTxn} is a convergent sequence, {QTxn − QTTxn} has a convergent subsequence. When the latter is added to demicompact T , we get {QTxn}, as a convergent subsequence. On the other side, we have xn = xn −QTxn + QTxn = (I −QT )xn + QTxn. Since I −QT is compact and {QTxn} has a convergent subsequence, {xn} has a convergent subsequence. Conversely, we suppose that QTT is demicompact. Let {xn} be a bounded sequence of D(T) such that QTxn −QTTxn → y. We have QTxn −QTTxn = −(I −QT )xn + xn −QTTxn.(2.2) According to Eq. (2.2), and considering the fact that I −QT is compact and {QTxn −QTTxn} is a convergent sequence, we infer that {xn −QTTxn} has a convergent subsequence. Bearing in mind the fact that QTT is demicom- pact and {xn − QTTxn} has a convergent subsequence, we obtain {xn} as a convergent subsequence. On the other side, we have QTxn = QTxn −xn + xn = −(I −QT )xn + xn. Besides, we have I − QT which is compact and {xn} which has a convergent subsequence. Thus, {QTxn} has a convergent subsequence. � Proposition 2.2. Let T : D(T) ⊆ X −→ X be a continuous linear relation. If T is a k −T(0)-set-contraction, then µT is demicompact for each µk < 1. Proof. Let {xn} be a bounded sequence of D(T) such that QµTxn−QµTµTxn → y. We have QµTxn = QµT (xn −µTxn) + QµTµTxn.(2.3) Suppose that γ({QµTxn}) 6= 0. Therefore, using Eq. (2.3) and Proposition 1.4, we obtain γ({QµTxn}) ≤ γ({QµT (xn −µTxn)}) + γ({QµTµTxn}) ≤ µkγ({QµTxn}) < γ({QµTxn}). However, the result is not accurate. It follows that γ({QµTxn}) = 0. Hence, {QµTxn} is relatively compact. � An immediate consequence of Proposition 2.2 is the following Corollary: Corollary 2.3. Let k ≥ 0 and T : D(T) ⊆ X −→ X be a continuous linear relation. If T is a k −T(0)-set-contraction, then 1 1+k T is demicompact. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 430 Fredholm theory for demicompact linear relations Theorem 2.4. Let T : D(T) ⊆ X −→ X be a linear relation. If m ∈ N∗, (QTµT) m is compact for each µ ∈ [0, 1) and I − QµT is compact, then µT is demicompact for each µ ∈ [0, 1). Proof. Let {xn} be a bounded sequence of D(T) such that yn = QµTxn −QµTµTxn → y. Let’s consider the various cases for m: Case 1: For m = 1. Using Lemma 2.1, we notice that, µT is demicompact for each µ ∈ [0, 1). Case 2: For m ∈ N∗ \{1}, we have m−1∑ k=0 (QµTµT) kQµTxn = m−1∑ k=0 (QµTµT) kyn + m−1∑ k=0 (QµTµT) k+1xn QµTxn + m−1∑ k=1 (QµTµT) kQµTxn = m−1∑ k=0 (QµTµT) kyn + m−2∑ k=0 (QµTµT) k+1xn +(QµTµT) mxn QµTxn + m−2∑ k=0 (QµTµT) k+1QµTxn = m−1∑ k=0 (QµTµT) kyn + m−2∑ k=0 (QµTµT) k+1xn +(QµTµT) mxn. Since QTTQT and (QTT) nQT are single-valued linear operators for all n ≥ 1, we get m−2∑ k=0 (QµTµT) k+1QµTxn which is single-valued. Therefore, QµTxn = m−1∑ k=0 (QµTµT) kyn + m−2∑ k=0 (QµTµT) k+1(I −QµT )xn +(QµTµT) mxn. As a matter of fact, γ({QµTxn}) ≤ m−1∑ k=0 γ((QµTµT) k)γ({yn}) + γ((QµTµT)m)γ({xn}) + m−2∑ k=0 γ((QµTµT) k+1)γ(I −QµT )γ({xn}) = 0. We conclude that γ({QµTxn}) = 0. Hence, {QµTxn} is relatively compact. Thus, there is a convergent subsequence of {QµTxn}. � Theorem 2.5. Let T : D(T) ⊆ X −→ X be a linear relation and k ≥ 0. (i) If m ∈ N∗, (QTT)m and I −QµT are compact, then 11+kT is demicompact. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 431 A. Ammar, S. Fakhfakh and A. Jeribi (ii) If (QTµT) m is compact for each µ ∈ [0, 1) and m > 0, then µT is demi- compact for each µ ∈ [0, 1). (iii) If m ∈ N∗, γ((QTµT)m) ≤ k and I −QµT is compact, then µT is demi- compact for each 0 ≤ µmk < 1. (iv) If m ∈ N∗, γ(Tm) ≤ k and I−Q 1 1+k T is compact, then 1 1+k T is demicom- pact. Proof. (i) An immediate consequence of Theorem 2.4 for µ = 1 1+k . (ii) Likewise, based on the preceding proof of Theorem 2.4, we obtain QµTxn = m−1∑ k=0 (QµTµT) kyn + m−2∑ k=0 (QµTµT) k+1(I −QµT )xn +(QµTµT) mxn. As a matter of fact, γ({QµTxn}) ≤ m−1∑ k=0 γ((QµTµT) k)γ({yn}) + γ((QµTµT)m)γ({xn}) + m−2∑ k=0 γ((QµTµT) k+1)γ(I −QµT )γ({xn}) = 0. We conclude that γ({QµTxn}) = 0. Hence, {QµTxn} is relatively compact. Thus, there is a convergent subsequence of {QµTxn}. (iii) Let {xn} be a bounded sequence of D(T) such that yn = QµTxn − QµTµTxn → y. Suppose that γ({QµTxn}) 6= 0. We have γ({QµTxn}) ≤ m−1∑ k=0 γ((QµTµT) k)γ({yn}) + γ((QµTµT)m)γ({xn}) + m−1∑ k=1 γ((QµTµT) k)γ(I −QµT )γ({xn}) ≤ µmγ((QµTT)m)γ({xn}) ≤ µmγ((QTT)m)γ({xn}) ≤ µmkγ({xn}) < γ({xn}). However, this result is not accurate. It follows that γ({QµTxn}) = 0. Hence, {QµTxn} is relatively compact. (iv) An immediate consequence of (iii) for µ = 1 1+k resides in the fact that, we have k (1+k)m < 1 for each k ≥ 0. � © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 432 Fredholm theory for demicompact linear relations 3. Fredholm and upper semi-Fredholm demicompact linear relations In this Section, we set forward some results on Fredholm and upper semi- Fredholm linear relations involving demicompact linear relations. In particular, in both Theorems stated below, we extend Proposition 1.7, to linear relations: Theorem 3.1. Let T : D(T) ⊆ X −→ X be a closed linear relation. If T is demicompact and I − QT is compact, then I − T is an upper semi-Fredholm relation. Proof. Let T be a demicompact and I−QT be a compact linear relation. Using Lemma 2.1, we infer that QTT is demicompact. Based on the latter and using Proposition 1.7 (i), we obtain I−QTT which is an upper semi-Fredholm single valued linear operator. On the other side, QI−T (I −T) = QT (I −T) = QTI −QTT + I − I = −(I −QT )I + I −QTT. Since I−QT is compact and I−QTT is an upper single valued linear operator semi-Fredholm, we notice that QI−T (I − T) is an upper single valued linear operator semi-Fredholm. Using Lemma 1.2, we obtain I−T which is an upper semi-Fredholm relation. � Theorem 3.2. Let T : D(T) ⊆ X −→ X be a closed linear relation. If µT is demicompact and I −QT is compact, then I −T is a Fredholm relation and i(I −T) = 0. Proof. Let T be a demicompact linear relation and I−QT be a compact opera- tor. Applying Lemma 2.1, we get QTT which is a demicompact linear relation. Using Proposition 1.7 (ii) and demicompact QTT , we obtain I −QTT , which is a single valued linear operator Fredholm and i(I −QTT) = 0. On the other side, QI−T (I −T) = QT (I −T) = QTI −QTT + I − I = −(I −QT )I + I −QTT. Moreover, we have I − QT which is compact and I − QTT which is a single valued linear operator Fredholm and i(I −QTT) = 0. Therefore, QI−T (I −T) is a Fredholm operator of index zero. Using Lemma 1.2, we notice that I −T is a Fredholm relation and i(I −T) = 0. � Proposition 3.3. Let T : D(T) ⊆ X −→ X be a continuous linear relation. If T is demicompact, k −T(0) is a set-contraction and I −QT is compact, then I −T is a Fredholm relation and i(I −T) = 0. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 433 A. Ammar, S. Fakhfakh and A. Jeribi Proof. Since T is demicompact, k − T(0) is a set-contraction and I − QT is compact, grounded on Corollary 2.3, we deduce that 1 1+k T is demicompact. Using Theorem 3.2, we obtain I−T which is Fredholm relation and i(I−T) = 0. � 4. Demicompact block matrix of linear relations In this section, a block matrix of linear relations L is identified. Afterwards, some results, where this block matrix of linear relations L becomes a demicom- pact block matrix of linear relations, are displayed. In the Banach space X ⊕Y , we consider the linear relation L provided by the block matrix of linear relations (4.1) L = ( A B C D ) , where A : D(A) ⊆ X −→ X, B : D(B) ⊆ Y −→ X, C : D(C) ⊆ X −→ Y and D : D(D) ⊆ Y −→ Y are linear relations with their natural domain D(L) := ( D(A) ∩D(C) ) ⊕ ( D(B) ∩D(D) ) . The graph of L is defined by G(L) := {( (x1,x2),(y1,y2) ) : (x1,x2) ∈ D(L), y1 ∈ Ax1 + Bx2 and y2 ∈ Cx1 + Dx2 } . Lemma 4.1 ([5, Remark 2.3]). Let L = ( A B C D ) be a block matrix of linear relations where A : D(A) ⊆ X −→ X, B : D(B) ⊆ Y −→ X, C : D(C) ⊆ X −→ Y and D : D(D) ⊆ Y −→ Y . If B(0) ⊂ A(0) and C(0) ⊂ D(0), then QLL = ( QAA QAB QDC QDD ) . Theorem 4.2. Let A : D(A) ⊆ X −→ X and D : D(D) ⊆ Y −→ Y be two demi- compact linear relations. Then, M = ( A 0 0 D ) is a demicompact linear relation. Proof. Let {tn} = ( xn yn ) be a bounded sequence of D(M) such that QMtn − QMMtn is convergent. We have QMtn −QMMtn = ( QA 0 0 QD ) ( xn yn ) − ( QAA 0 0 QDD ) ( xn yn ) = ( QAxn −QAAxn QDyn −QDDyn ) . Since {xn} is a bounded sequence of D(A), QAxn−QAAxn are convergent and A is a demicompact linear relation; then {QAxn} has a convergent subsequence. Similarly, we get {QDyn} which has a convergent subsequence. Hence, {QMtn} has a convergent subsequence. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 434 Fredholm theory for demicompact linear relations � Theorem 4.3. Let A : D(A) ⊆ X −→ X and D : D(D) ⊆ Y −→ Y be two demicompact linear relations and let B : D(B) ⊆ Y −→ X and C : D(C) ⊆ X −→ Y be two linear relations. If QA(I −B) and QD(I −C) are compact and B(0) ⊆ A(0) and C(0) ⊆ D(0), then L = ( A B C D ) is a demicompact linear relation. Proof. Let {tn} = ( xn yn ) be a bounded sequence of D(L) such that QLtn −QLLtn is convergent. Using Lemma 4.1, we obtain QLtn −QLLtn = ( QA QA QD QD ) ( xn yn ) − ( QAA QAB QDC QDD ) ( xn yn ) = ( QAxn + QAyn −QAAxn −QAByn QDxn + QDyn −QDCxn −QDDyn ) = ( QAxn −QAAxn + QA(I −B)yn QDyn −QDDyn + QD(I −C)xn ) . We have {xn} which is a bounded sequence and QA(I −B) which is compact. Then, {QA(I − B)yn} is bounded. Since {QAxn − QAAxn + QA(I − B)yn} is convergent and {QA(I−B)yn} is bounded, {QAxn−QAAxn} is convergent. Subsequently, using the fact that A is a demicompact linear relation, {QAxn}, therefore, has a convergent subsequence. Similarly, we get {QDyn} which has a convergent subsequence. As a matter of fact, {QLtn} has a convergent subsequence. � An immediate consequence of Theorem 4.3 is the following Corollary: Corollary 4.4. Let A : D(A) ⊆ X −→ X and D : D(D) ⊆ Y −→ Y be two demicompact linear relations and let B : D(B) ⊆ Y −→ X and C : D(C) ⊆ X −→ Y be two linear relations. Thus, the block matrix L = ( A B C D ) is a demicompact linear relation, if one of the following conditions holds: a.: QA(I−B) and QC(I−D) are compact and B(0) ⊆ A(0) and D(0) ⊆ C(0). b.: QB(I−A) and QD(I−C) are compact and A(0) ⊆ B(0) and C(0) ⊆ D(0). c.: QB(I−A) and QC(I−D) are compact and A(0) ⊆ B(0) and D(0) ⊆ C(0). Acknowledgements. The authors wish to express their gratitude to the referee for his valuable comments which helped to improve the quality of this paper. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 435 A. Ammar, S. Fakhfakh and A. Jeribi References [1] F. Abdmouleh, T. Álvarez, A. Ammar and A. Jeribi, Spectral mapping theorem for Rakocević and Schmoeger essential spectra of a multivalued linear operator, Mediterr. J. Math. 12, no. 3 (2015), 1019–1031. [2] A. Ammar, A characterization of some subsets of essential spectra of a multivalued linear operator, Complex Anal. Oper. Theory 11, no. 1 (2017), 175–196. [3] A. Ammar, Some results on semi-Fredholm perturbations of multivalued linear opera- tors, Linear Multilinear Algebra 66, no. 7 (2018), 1311–1332. [4] A. Ammar, H. Daoud and A. Jeribi, Demicompact and K-D-setcontractive multivalued linear operators, Mediterr. J. Math. 15, no. 2 (2018): 41. [5] A. Ammar, S. Fakhfakh and A. Jeribi, Stability of the essential spectrum of the diago- nally and off-diagonally dominant block matrix linear relations, J. Pseudo-Differ. Oper. Appl. 7, no. 4 (2016), 493–509. [6] W. Chaker, A. Jeribi and B. Krichen, Demicompact linear operators, essential spectrum and some perturbation results, Math. Nachr. 288, no. 13 (2015), 1476–1486. [7] R. W. Cross, Multivalued Linear Operators, Marcel Dekker, (1998). [8] A. Jeribi, Spectral Theory and Applications of Linear Operator and Block Operator Matrices, Springer-Verlag, New York, 2015. [9] K. Kuratowski, Sur les espaces complets, Fund. Math. 15 (1930), 301–309. [10] W. V. Petryshyn, Remarks on condensing and k-set-contractive mappings, J. Math. Appl. 39 (1972),3 717–741. © AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 436