E extracta mathematicae Vol. 32, Núm. 2, 125 – 161 (2017) Essential g-Ascent and g-Descent of a Closed Linear Relation in Hilbert Spaces ∗ Zied Garbouj, Häıkel Skhiri Faculté des Sciences de Monastir, Département de Mathématiques, Avenue de l’environnement, 5019 Monastir, Tunisia zied.garbouj.fsm@gmail.com haikel.skhiri@gmail.com haikel.skhiri@fsm.rnu.tn Presented by Jesús M. F. Castillo Received January 26, 2017 Abstract: We define and discuss for a closed linear relation in a Hilbert space the notions of essential g-ascent (resp. g-descent) and g-ascent (resp. g-descent) spectrums. We improve in the Hilbert space case some results given by E. Chafai in a Banach space [Acta Mathematica Sinica, 34 B, 1212-1224, 2014] and several results related to the ascent (resp. essential ascent) spectrum for a bounded linear operator on a Banach space [Studia Math, 187, 59-73, 2008] are extended to closed linear relations on Hilbert spaces. We prove also a decomposition theorem for closed linear relations with finite essential g-ascent or g-descent. Key words: Range subspace, Closed linear relation, Spectrum, Ascent, Essential ascent, Descent, Essential descent, Semi-Fredholm relation. AMS Subject Class. (2010): 47A06, 47A05, 47A10, 47A55 1. Introduction and terminology Let H be a complex Hilbert space. A multivalued linear operator T : H −→ H or simply a linear relation is a mapping from a subspace D(T) ⊆ H, called the domain of T, into the collection of nonempty subsets of H such that T(λx+µy) = λT(x)+µT(y) for all nonzero scalars λ, µ and x, y ∈ D(T). We denote by LR(H) the class of linear relations on H. If T maps the points of its domain to singletons, then T is said to be a single valued linear operator or simply an operator. The graph G(T) of T ∈ LR(H) is defined by : G(T) = {(x, y) ∈ H × H : x ∈ D(T), y ∈ Tx}. We say that T is closed if its graph is a closed subspace of H × H. The class of such linear relations will be denoted by CR(H). A linear relation T ∈ LR(H) is said to be continuous if for each open set Ω ⊆ Im(T), T −1(Ω) is an open ∗This work is supported by the Higher Education And Scientific Research In Tunisia, UR11ES52 : Analyse, Géométrie et Applications 125 126 z. garbouj, h. skhiri set in D(T). Continuous everywhere defined linear relations are referred to as bounded relations. The kernel of a linear relation T is the subspace ker(T) := T −1(0). The subspace Im(T) := T(D(T)) is called the range of T. The nullity and the defect of a linear relation T ∈ LR(H) are defined by α(T) = dim ker(T) and β(T) = dim H/Im(T), respectively. Recall that T ∈ CR(H) is said to be upper semi-Fredholm if T has closed range and α(T) < +∞, and T is said to be lower semi-Fredholm if β(T) < +∞. If T is upper or lower semi-Fredholm we say that T is semi-Fredholm, and we denote by Φ±(H) the class of all semi-Fredholm relations. For T ∈ Φ±(H) we define the index of T by ind(T) = α(T) − β(T). A linear relation is Fredholm if max{α(T), β(T)} < +∞. We denote by Φ(H) (respectively, Φ+(H), Φ−(H)) the class of all Fredholm (respectively, upper semi-Fredholm, lower semi-Fredholm) relations. The linear relation T ∈ CR(H) is called regular if Im(T) is closed and ker(T) ⊆ Im(T n), for every n ∈ N (see [1]). Recall that the resolvent set of T ∈ LR(H) is defined (see, [4, Chapter VI]) by ϱ(T) = {λ ∈ C : λI − T is injective, open and has dense range} and the spectrum of T is the set σ(T) = C\ϱ(T). It is clear from the closed graph theorem for a linear relation that if T is a closed linear relation then ϱ(T) = {λ ∈ C : λI − T is bijective}. We say that T ∈ LR(H) has a trivial singular chain manifold if Rc(T) = {0} where Rc(T) = [ ∞∪ i=1 ker(T i) ] ∩ [ ∞∪ i=1 T i(0) ] . Let λ ∈ C, by [10, Lemma 7.1], we know that Rc(T) = {0} if and only if Rc(λI − T) = {0}. It is easy to see that Rc(T) = {0} when ϱ(T) ̸= ∅. For two subspaces M and N of H, we recall that dim M/M ∩ N = dim(M + N)/N and (N + W) ∩ M = N ∩ M + W whenever W is a subspace of M. Following [3], the ascent and the descent of T ∈ LR(H) are respectively defined by a(T) = inf{k ∈ N : ker(T k+1) = ker(T k)}, d(T) = inf{k ∈ N : Im(T k+1) = Im(T k)}, essential g-ascent and g-descent of a closed relation 127 whenever these minima exist. If no such numbers exist the ascent and descent of T are defined to be +∞. For T ∈ LR(H) and n ∈ N, we define the following quantities : αn(T) = dim ker(T n+1)/ker(T n), βn(T) = dim Im(T n)/Im(T n+1). Let us recall from [9, Lemma 3.2] and [10, Lemma 4.1], the following properties αn(T) = dim[Im(T n) ∩ ker(T)]/[T n(0) ∩ ker(T)] (1.1) and βn(T) = dim D(T n)/[Im(T) + ker(T n)] ∩ D(T n) = dim[Im(T) + D(T n)]/[Im(T) + ker(T n)]. (1.2) In [6], we show that (βn(T))n≥0 and (αn(T))n≥0 are decreasing sequences. Recall that for T ∈ LR(H), the essential ascent, ae(T), and the essential descent, de(T), are defined by ([3]) ae(T) = inf{n ∈ N : αn(T) < +∞}, de(T) = inf{n ∈ N : βn(T) < +∞}, where the infimum over the empty set is taken to be infinite. For T ∈ LR(H) we consider the two decreasing sequences α̃n(T) = dim Im(T n) ∩ ker(T), β̃n(T) = dim H/[Im(T) + ker(T n)], n ∈ N. Remark 1.1. From the equalities (1.1) and (1.2), we see that αn(T) ≤ α̃n(T) and βn(T) ≤ β̃n(T), for all n ∈ N. Observe that ker(T) ∩ T j(0) ⊆ Rc(T), for all j ∈ N. Thus, by equality (1.1) (resp. (1.2)), it follows that if Rc(T) = {0} (resp. D(T i) + Im(T) = H, for all i ∈ N), so αn(T) = α̃n(T) (resp. βn(T) = β̃n(T)), for all n ∈ N. The above remark leads to the introduction of a new concept of g-ascent (resp. essential g-ascent, g-descent, essential g-descent) for a linear relation. Definition 1.2. Let T ∈ LR(H). (i) The g-ascent, ã(T), of T is defined by ã(T) = inf{n ∈ N : α̃n(T) = 0}. 128 z. garbouj, h. skhiri (ii) The essential g-ascent, ãe(T), of T is defined as ãe(T) = inf{n ∈ N : α̃n(T) < +∞}. (iii) The g-descent, d̃(T), of T is defined by d̃(T) = inf{n ∈ N : β̃n(T) = 0}. (iv) The essential g-descent, d̃e(T), of T is defined by d̃e(T) = inf{n ∈ N : β̃n(T) < +∞}, where as usual the infimum over the empty set is taken to be +∞. It is clear that a(T) ≤ ã(T) and ae(T) ≤ ãe(T) (resp. d(T) ≤ d̃(T) and de(T) ≤ d̃e(T)), and equality holds when Rc(T) = {0} (resp. D(T i)+Im(T) = H, for all i ∈ N). The notion of ascent (resp. descent, essential ascent, essential descent) of a linear operator was studied in several papers (see for examples [2, 5]). In recent years some work has been devoted to extend these concepts to the case of linear relations, (see [3, 6, 10]). In [3], many basic results related to the ascent (resp. descent, essential ascent, essential descent) spectrum of linear operators have been extended to linear relations (usually with additional conditions). In this context, we prove that the results in [3] related to the spectral mapping theorem of ascent and essential ascent (resp. descent and essential descent) spectrums of a closed linear relation everywhere defined such that ϱ(T) ̸= ∅ (resp. dim T(0) < +∞) remain valid when T ∈ Υ(H) (see page 142) (resp. T ∈ LR(H)) and without the assumption that ϱ(T) ̸= ∅ (resp. dim T(0) < +∞) and D(T) = H. In [2], the ascent spectrum and the essential ascent spectrum of a bounded operator acting in Banach spaces are introduced and studied. In this paper, we extend in the Hilbert space case these notions to multivalued linear operators. However, the techniques used in this work are different from those used in [2, 3, 5]. Our approach here is based in the concept of range subspaces of Hilbert spaces (see, [7]). The paper is organized as follows. In the next section, we first established some algebraic lemmas that will be used throughout this work. In Sections 3, 4 and 5, we are interested in the spectral theory of closed linear relations in Hilbert spaces having a finite essential g-ascent or finite essential g-descent. For example, in Theorem 3.4, we show that a closed linear relation with finite essential g-ascent and g-descent of a closed relation 129 essential g-ascent is stable under perturbations of the form λI, where λ ∈ C. In Theorem 3.8, we study the spectrum boundary points of a closed linear relation with finite essential g-ascent. In Theorem 4.12 and Theorem 5.4, we prove, under some conditions, that the essential g-ascent spectrum and essential g-descent spectrum satisfy the polynomial version of the spectral mapping theorem for a closed linear relation. Finally, in Section 6, we prove that if T ∈ CR(H), then there exists n ∈ N such that ãe(T) ≤ n and Im(T) + ker(T n) is closed (resp. d̃e(T) ≤ n and Im(T n) ∩ ker(T) is closed) if and only if there exist d ∈ N and two closed subspaces M and N such that : (i) H = M u N; (ii) Im(T d) ⊆ M, T(M) ⊆ M, N ⊆ ker(T d) and, if d > 0, N ̸⊂ ker(T d−1); (iii) G(T) = [G(T) ∩ (M × M)] u [G(T) ∩ (N × N)]; (iv) the restriction of T to M is both upper semi-Fredholm (resp. lower semi-Fredholm) and regular relation; (v) if A ∈ LR(N) such that its graph is the subspace G(T) ∩ (N × N), then A is a bounded operator everywhere defined and G(Ad) = N × {0}. 2. Algebraic preliminaries Throughout this paper the symbol u denotes the topological direct sum of closed subspaces in H, i.e., X0 = X1 u X2 if the linear space X0 = X1 + X2 is closed and X1 ∩ X2 = {0}. Next we give an example of quantities introduced below. Example 2.1. (i) Let M be a subspace of H and define the linear relation T in H by : D(T) = M and T(x) = M, ∀ x ∈ M. Clearly we have ker(T n) = Im(T n) = M, ∀ n ≥ 1. (1) • Case 1 : if dim M = +∞, from (1), we have a(T) = ae(T) = 1 < ã(T) = ãe(T) = +∞. 130 z. garbouj, h. skhiri • Case 2 : if 0 < dim M < +∞, by (1), we deduce that ae(T) = ãe(T) = 0 < a(T) = 1 < ã(T) = +∞. • Case 3 : if 0 < dim H/M < +∞, from (1), we obtain d̃e(T) = de(T) = 0 < d(T) = 1 < d̃(T) = +∞. • Case 4 : if dim H/M = +∞, it follows from (1) that d(T) = de(T) = 1 < d̃e(T) = d̃(T) = +∞. (ii) Let M and N be a pair of closed subspaces of H such that H = M u N and {0} N H. Let P be the linear projection with domain H, range N and kernel M and let L = P −1, then D(L) = N and L(x) = x + M, ∀ x ∈ N. Since (I − L)x = M, with x ∈ N, it follows that ker[(I − L)n] = N and Im[(I − L)n] = M, for all n ≥ 1. Thus ãe(I − L) = ae(I − L) ≤ a(I − L) = ã(I − L) = 1, d̃e(I − L) = de(I − L) ≤ d(I − L) = d̃(I − L) = 1. In particular, if dim N < +∞, then ãe(I −L) = ae(I −L) = d̃e(I −L) = de(I − L) = 0. In this section, we prove some algebraic results of the theory of linear relations which are used to prove the main results in this work. For T ∈ LR(H), we consider the sequence Sn(T) = dim[Im(T n) ∩ ker(T)]/[Im(T n+1) ∩ ker(T)], n ∈ N. From [10, Lemma 4.2], we get Sn(T) = dim[Im(T) + ker(T n+1)]/[Im(T) + ker(T n)], ∀ n ∈ N. The degree of stable iteration, p(T), of T is defined by p(T) = inf{n ∈ N : Sm(T) = 0, ∀ m ≥ n}, where the infimum over the empty set is taken to be infinite. The following lemma helps to characterize the relationship between the degree of stable iteration and both the finite essential ascent and essential g-ascent of linear relations. essential g-ascent and g-descent of a closed relation 131 Lemma 2.2. Let T ∈ LR(H). (i) If ae(T) < +∞, then p(T) ≤ inf{n ∈ N : αn(T) = αi(T), ∀ i ≥ n} < +∞. (ii) If ãe(T) < +∞ (resp. ã(T) < +∞), then p(T) = inf{n ∈ N : α̃n(T) = α̃m(T), ∀ m ≥ n} < +∞ and ãe(T) ≤ p(T) (resp. ã(T) = p(T)). Proof. (i) If ae(T) < +∞, then αn(T) < +∞, for every n ≥ ae(T). So, there exists m ≥ ae(T) such that αn(T) = αm(T), for all n ≥ m. Now by [6, equality (16)], we get Sn(T) = 0, for all n ≥ m, and this proves that p(T) ≤ m. (ii) By [6, equality (18)], we can prove this assertion similarly as in (i), which completes the proof. Remark 2.3. Let T ∈ LR(H) such that ae(T) < +∞ (resp. a(T) < +∞). We note that ae(T) ≤ p(T) (resp. a(T) = p(T)) in general is not true. Indeed, let T be defined as in Case 1 of Example 2.1, then p(T) = 0 < ae(T) = a(T) = 1. The next lemma exhibits some useful entirely algebraic properties of the degree of stable iteration of linear relations. Lemma 2.4. Let T ∈ LR(H). (i) If de(T) < +∞, then p(T) ≤ m = inf{n ∈ N : βn(T) = βi(T), ∀ i ≥ n} < +∞. (ii) If d̃e(T) < +∞ (resp. d̃(T) < +∞), then p(T) = inf{n ∈ N : β̃n(T) = β̃m(T), ∀ m ≥ n} < +∞ and d̃e(T) ≤ p(T) (resp. d̃(T) = p(T)). 132 z. garbouj, h. skhiri Proof. Since ker(T n) + Im(T) ⊆ ker(T n+1) + Im(T) ⊆ D(T n+1) + Im(T) ⊆ D(T n) + Im(T), we deduce that βn(T) = dim[D(T n) + Im(T)]/[D(T n+1) + Im(T)] + βn+1(T) + Sn(T). (1) But, since Im(T) + ker(T n) ⊆ Im(T) + ker(T n+1) ⊆ H, it follows that β̃n(T) = Sn(T) + β̃n+1(T). (2) Finally, the assertions (i) and (ii) follow from (1) and (2). The proof is there- fore complete. Remark 2.5. Let T ∈ LR(H) such that de(T) < +∞ (resp. d(T) < +∞). We note that de(T) ≤ p(T) (resp. d(T) = p(T)) in general is not true. Indeed, let T be defined as in Case 4 of Example 2.1, then p(T) = 0 < de(T) = d(T) = 1. 3. Essential g-ascent spectrum and g-ascent spectrum of a closed relation This section contains the main results of this work, in which we generalize some results of [2, Section 2] and our results of [5, Section 3] to the case of a closed linear relation in a Hilbert space H. Throughout the remainder of the paper, for T ∈ LR(H) and λ ∈ C, we denote by Tλ the relation λI − T. The ascent resolvent set of T ∈ CR(H) is the set ϱasc(T) = {λ ∈ C : a(Tλ) < +∞ and Im(Tλ) + ker[(Tλ)a(Tλ)] is closed} and its ascent spectrum σasc(T) = C\ϱasc(T). The essential ascent resolvent and the essential ascent spectrum of T ∈ CR(H) are defined respectively by ϱeasc(T) = {λ ∈ C : ae(Tλ) < +∞ and Im(Tλ) + ker[(Tλ) ae(Tλ)] is closed} essential g-ascent and g-descent of a closed relation 133 and σeasc(T) = C\ϱ e asc(T). The g-ascent resolvent set of T ∈ CR(H) is the set ϱgasc(T) = {λ ∈ C : ã(Tλ) < +∞ and Im(Tλ) + ker[(Tλ) ã(Tλ)] is closed} and its g-ascent spectrum σgasc(T) = C\ϱ g asc(T). The essential g-ascent resolvent and the essential g-ascent spectrum of T ∈ CR(H) are defined respectively by ϱe, gasc(T) = {λ ∈ C : ãe(Tλ) < +∞ and Im(Tλ) + ker[(Tλ) ãe(Tλ)] is closed} and σe, gasc(T) = C\ϱ e, g asc(T). From [6, Lemma 2.9], we deduce easily the following ϱ(T) ⊆ ϱgasc(T) ⊆ ϱasc(T) ⊆ ϱ e asc(T), ϱ(T) ⊆ ϱgasc(T) ⊆ ϱ e, g asc(T) ⊆ ϱ e asc(T). Let us recall the following definition. Definition 3.1. ([7], Definition 3.3, Definition 4.1) (i) A subspace M of a Hilbert space H is said to be a range subspace of H if there exist a Hilbert space N and a bounded operator T from N to H such that M = Im(T). In particular, a closed subspace of a Hilbert space H is a range subspace of H. (ii) An operator or a relation T ∈ LR(H) is said to be a range space operator or range space relation if its graph G(T) is a range subspace of H × H. It is clear that a closed relation in a Hilbert space H is a range space relation in H. Our approach here is based in the concept of range subspaces of Hilbert spaces (see, [7]). We have the following lemma, which will be needed in the sequel. 134 z. garbouj, h. skhiri Lemma 3.2. Let T ∈ CR(H). (i) If ae(T) < +∞, T k(0) and Im(T) + ker(T ae(T)) are closed for all k ∈ N, then a) Im(T n)+ker(T m) is closed, for all m+n ≥ p(T). In particular Im(T n) is closed for all n ≥ p(T). b) Im(T n) + ker(T m) is closed, for all n ∈ N and m ≥ ae(T). (ii) If ãe(T) < +∞ and Im(T) + ker(T ãe(T)) is closed, then a) T n(0) is closed, for all n ∈ N, and so from [6, Lemma 2.9] and (i), Im(T n) + ker(T m) is closed, for all m + n ≥ p(T) (resp. n ∈ N and m ≥ ae(T)), b) ker(T n) and Im(T ae(T)+n) are closed, for all n ∈ N. Proof. (i) From [6, equality (17)], we have Im(T n) + ker(T m) is closed for all n, m ∈ N such that m + n ≥ p(T). In particular Im(T n) + ker(T p(T)+ae(T)) is closed for every n ∈ N, so by [6, Lemma 2.9], Im(T n) + ker(T m) is closed, for all n ∈ N and m ≥ ae(T). (ii) From [6, Lemma 2.10], it follows that T n(0) and Im(T ae(T)+n) are closed, for all n ∈ N. Let S = T|Im(T p(T )) be the restriction of T to Im(T p(T)). Now, let n ∈ N, by Lemma 2.2, we have ãe(T) ≤ p(T), this implies that dim ker(T n) ∩ Im(T p(T)) = α(Sn) ≤ n α(S) = n α̃p(T)(T) < +∞ (see [9, Lemma 5.1]) and as Im(T p(T))+ker(T n) is closed, so by [7, Propositions 3.9, 3.10, 4.8, Lemma 4.2], ker(T n) is closed, for all n ∈ N. This completes the proof of the lemma. For T ∈ LR(H) and k ∈ N, T̃k denotes the following linear relation : T̃k : D(T)/ker(T k) ⊆ H/ker(T k) −→ H/ker(T k) x 7−→ Tx := {z : z ∈ Tx}. We will prove first that the linear relation T̃k is well-defined. To do this, let us choose x1, x2 ∈ D(T) such that x1 −x2 ∈ ker(T k). So, 0 ∈ T k(x1 −x2), and therefore there exists x ∈ T(x1 − x2) = Tx1 − Tx2, such that 0 ∈ T k−1(x). essential g-ascent and g-descent of a closed relation 135 From this, we get x ∈ ker(T k−1) and 0 ∈ Tx1 − Tx2. Let y ∈ T(x1), since T(x1) = y + T(0) then 0 ∈ y + T(0) − Tx2 = y − Tx2. Hence, T(x1) ⊆ T(x2) and by interchanging x1 and x2, we deduce that T(x1) = T(x2). Let M be a closed subspace of H, then H/M is a Hilbert space with the inner product < · , · >M : H/M × H/M −→ R (x , y) 7−→ < P(x) , P(y) >, where P is the orthogonal projection onto M⊥ and < · , · > is the standard inner product on H. Note, the Hilbert space topology of (H/M, < · , · >M) coincides with the quotient topology of H/M : ∥x∥ = √ < x, x >M = √ < P(x), P(x) > = dist(x, M), where dist(x, M) denotes, as usual, the distance of x to M. Now, let T ∈ CR(H) such that ãe(T) < +∞ and Im(T) + ker(T ãe(T)) is closed. From Lemma 3.2, it follows easily that H/ker(T n) is a Hilbert space, for all n ∈ N. In [2, Lemma 2.1], it was established that if T is a bounded operator and T admits a finite essential g-ascent such that Im(T) + ker(T ãe(T)) is closed, then T̃j is both regular and upper semi-Fredholm operator, for all j ≥ p(T), where T̃j is the operator induced by T on H/ker(T j). In [5, Lemma 3.4], this result was extended to the case of unbounded closed operators, and in the following lemma, we prove that this result remains valid even in the context of closed linear relations. Lemma 3.3. Let T ∈ CR(H) such that ãe(T) < +∞ and j ≥ p(T). If Im(T) + ker(T ãe(T)) is closed, then T̃ : D(T)/ker(T j) ⊆ H/ker(T j) −→ H/ker(T j) x 7−→ Tx := {y : y ∈ Tx} is both regular and upper semi-Fredholm relation. Proof. First, recall that from Lemma 3.2, we have ker(T n) is closed for all n ∈ N, and in particular, H/ker(T j) is a Hilbert space. Now, we will 136 z. garbouj, h. skhiri show that ker(T̃) = ker(T j+1)/ker(T j). To do this let x ∈ ker(T j+1), so y ∈ Tx and 0 ∈ T jy for some y ∈ H. Since 0H/ker(T j) = y ∈ Tx, it fol- lows that ker(T j+1)/ker(T j) ⊆ ker(T̃). In order to prove the converse in- clusion, assume that x ∈ ker(T̃). From 0 ∈ Tx, we deduce that T(x) ∩ ker(T j) ̸= ∅, which shows that x ∈ T −1(ker(T j)) = ker(T j+1). Consequently ker(T̃) ⊆ ker(T j+1)/ker(T j). Moreover, it is clear that Im(T̃) = [Im(T) + ker(T j)]/ker(T j), so from Lemma 3.2, we get Im(T̃) is closed. Let π be the natural quotient map with domain H×H and null space ker(T j)×ker(T j), then G(T̃) = π(G(T)). From [7, Corollary 4.9], we know that there exist a Hilbert space Z and a bounded operator θ from Z to H × H such that Im(θ) = G(T). This implies that πθ(Z) = G(T̃). Therefore T̃ is a range space relation. On the other hand, we have α(T̃) = αj(T) < +∞ and Im(T̃) is closed. So from [7, Lemma 4.6], T̃ is closed, and hence T̃ is upper semi-Fredholm. Furthermore, by [7, Lemma 2.5], it follows that ker(T j+1) ⊆ Im(T n)+ker(T j), for all n ∈ N. Consequently, ker(T̃) ⊆ Im(T̃ n), for all n ∈ N, and the proof is therefore complete. Now, we are ready to state our main result of this section, which is an extension of [2, Theorem 2.3] and [5, Theorem 3.8] to closed multivalued linear operators. Theorem 3.4. Let T ∈ CR(H) such that ãe(T) < +∞ and Im(T) + ker(T ãe(T)) is closed. Then there is ε > 0 such that for every 0 < |λ| < ε, we have : (i) Tλ is both regular and upper semi-Fredholm, (ii) αp(T)(T) ≤ α(Tλ) ≤ (p(T) + 1) α̃p(T)(T), (iii) β(Tλ) = dim H/[Im(T) + ker(T p(T))]. Proof. Let k = p(T) and let T̃k be the relation induced by T on H/ker(T k). First, from Lemma 3.3, T̃k is both regular and upper semi-Fredholm relation. Using [1, Theorems 23, 25], we deduce that there exists ε > 0, such that λI−T̃k is both regular and upper semi-Fredholm relation for every 0 < |λ| < ε. Furthermore, by [4, Corollary V.15.7] and [1, Theorem 27], we have α(T̃k) = α(λI − T̃k), β(T̃k) = β(λI − T̃k), ∀ λ ∈ C, 0 < |λ| < ε. (1) We prove first that ker(λI − T̃k) = [ker(λI − T) + ker(T k)]/ker(T k), (2) essential g-ascent and g-descent of a closed relation 137 Im[(λI − T̃k)n] = Im[(λI − T)n]/ker(T k), ∀ n ∈ N. (3) If x ∈ ker(λI − T̃k), then (λI − T)(0) = (λI − T)(x), and so T(0) + ker(T k) = (λI−T)x+ker(T k). It is clear that x ∈ D(T i), for all i ∈ N and T(0)+T −k(0) = (λI −T)x+T −k(0). From [8, Corollary 2.1] and [4, Corollary I.2.10], we obtain T k+1(0) + T k(0) = (λI − T)T kx + T k(0). Since T k(0) ⊆ T k+1(0) = (λI − T)T k(0) (see [8, Theorem 3.6 ]), it follows that (λI −T)T kx = (λI −T)T k(0), which implies that x ∈ ker[(λI − T)T k] and x ∈ ker[(λI − T)T k]/ker(T k). Consequently ker(λI − T̃k) ⊆ ker[(λI − T)T k]/ker(T k). To prove the converse inclusion, let x ∈ ker[(λI − T)T k]/ker(T k), so 0 ∈ T k(λI − T)x. This implies that there exists z ∈ H such that z ∈ (λI − T)x and 0 ∈ T kz. Hence 0 = z ∈ (λI − T)x, and x ∈ ker(λI − T̃k). Now from [8, Theorem 3.4], we obtain ker(λI − T̃k) = [ker(λI − T) + ker(T k)]/ker(T k). Since ker(T k) ⊆ Im[(λI − T)n], it follows that Im[(λI − T̃k)n] = [Im[(λI − T)n] + ker(T k)]/ker(T k) = Im[(λI − T)n]/ker(T k). (i) Let S = T|Im(T k) be the restriction of T to Im(T k). By (1) and (2), we obtain dim ker(λI − T)/ker(λI − T) ∩ ker(T k) = dim[ker(λI − T) + ker(T k)]/ker(T k) =α(T̃k) < +∞ (4) and dim ker(λI − T) ∩ ker(T k) ≤ dim Im(T k) ∩ ker(T k) = α(Sk) ≤ kα(S) = k α̃k(T) < +∞ (5) (see [9, Lemma 5.1]). In particular, this proves that α(λI − T) < +∞. From (3), we infer that Im(λI − T) is closed. Finally, since λI − T̃k is regular, by (2) and (3), we deduce that λI − T is regular. (ii) From (4) and (5), we get αk(T) = α(T̃k) ≤ α(λI − T) = αk(T) + dim ker(λI − T) ∩ ker(T k) ≤ α̃k(T) + k α̃k(T) ≤ (k + 1) α̃k(T). Assertion (iii) follows from (1) and (3), which completes the proof. 138 z. garbouj, h. skhiri The following theorem is a simple consequence of Theorem 3.4. Theorem 3.5. Let T ∈ CR(H) such that ã(T) < +∞ and Im(T) + ker(T ã(T)) is closed. Then there is ε > 0 such that for every 0 < |λ| < ε, we have (i) Tλ is injective with closed range, (ii) β(Tλ) = dim H/[Im(T) + ker(T p(T))]. Corollary 3.6. Let T ∈ CR(H), then σ g asc(T) and σ e, g asc(T) are closed. For T ∈ CR(H), we consider the set : E(T) = {λ ∈ σ(T) : λ an isolated point, ã(Tλ) < +∞, d̃(Tλ) = m < +∞ and Im[(Tλ)m] is closed}. The following lemma is the key to prove Theorem 3.8. Lemma 3.7. Let T ∈ CR(H), then E(T) = {λ ∈ σ(T) : ã(Tλ) < +∞ and d̃(Tλ) < +∞}. Proof. Assume that T has finite g-ascent and g-descent. First, by Lemma 2.2 and Lemma 2.4, we note that ã(T) = d̃(T) = p(T). Let m = d̃(T), S = T|Im(T m) be the restriction of T to Im(T m) and T̃m be the relation induced by T on H/ker(T m), then by [9, Lemma 5.1], dim Im(T m) ∩ ker(T k) = α(Sk) ≤ k α(S) = kα̃m(T) = 0 and dim H/[Im(T k) + ker(T m)] = β(T̃m k ) ≤ k β(T̃m) = k β̃m(T) = 0, for all k ∈ N. Therefore Im(T m) ∩ ker(T k) = {0} and Im(T k) + ker(T m) = H, ∀ k ≥ 0. (1) Using now the equality (1) for k = m and [7, Propositions 3.10 and 4.8, Lemma 4.2], we get Im(T m) is closed. From (1) and Theorem 3.5, we deduce that there exists ε > 0 such that λI − T is bijective, for every 0 < |λ| < ε. Consequently if 0 ∈ σ(T), then 0 is an isolated point of σ(T). This completes the proof of Lemma 3.7. essential g-ascent and g-descent of a closed relation 139 The following theorem is an extension of [2, Theorem 2.7] and [5, Theorem 3.10], to case of closed linear relations. Theorem 3.8. Let T ∈ CR(H), then ϱe, gasc(T) ∩ ∂σ(T) = ϱ g asc(T) ∩ ∂σ(T) = ϱeasc(T) ∩ ∂σ(T) = ϱasc(T) ∩ ∂σ(T) = E(T). Proof. By arguing as in the proof of [5, Theorem 3.10], with Theorem 3.4 and Lemma 3.7, we get the result. As an immediate consequence of Theorem 3.8, we have the following result. Corollary 3.9. Let T ∈ CR(H) such that ϱ(T) ̸= ∅. Then the following assertions are equivalent : (i) σasc(T) = ∅; (ii) σeasc(T) = ∅; (iii) ∂σ(T) ⊆ ϱasc(T); (iv) ∂σ(T) ⊆ ϱeasc(T); (v) σ(T) = E(T). Remark 3.10. We note that Corollary 3.9 in general is not true if ϱ(T) = ∅. Indeed, let M ̸= {0} be a closed subspace of H and let T : H −→ H be the linear relation defined by Tx = M for all x ∈ H. It is clear that T ∈ CR(H). For each λ ∈ C and for each n ∈ N\{0}, we have (Tλ) nx = λnx + M, ∀ x ∈ H. Since ker[(Tλ) n] = { H if λ = 0 M if λ ̸= 0 ̸= {0} and Im[(Tλ)n] = { H if λ ̸= 0 M if λ = 0, it follows that a(Tλ) = 1 and Im(Tλ) + ker(Tλ) = H, for all λ ∈ C. This implies that ϱ(T) = ∅ and ϱasc(T) = C. Hence, ϱeasc(T) = C and T satisfies the conditions (i)-(iv) of Corollary 3.9. Since E(T) consists of isolated points of σ(T), we deduce that E(T) = ∅, and thus T does not satisfy the condition (v). 140 z. garbouj, h. skhiri 4. Spectral mapping theorems of g-ascent and essential g-ascent spectrums We start this section with the following proposition. Proposition 4.1. Let T ∈ CR(H) be everywhere defined and suppose that T k(0) is closed for all k ∈ N. If n = ae(T) (resp. a(T), ãe(T), ã(T)) is finite, then Im(T) + ker(T n) is closed if and only if Im(T n+1) is closed. Proof. First, from [7, Lemma 4.6, Corollary 4.9], we get T k ∈ CR(H), for all k ∈ N. Put n = ae(T) (resp. a(T), ãe(T), ã(T)). If Im(T) + ker(T n) is closed, by [6, Lemma 2.9] and Lemma 3.2, we get Im(T n+1)+ker(T n) is closed and by [9, Lemma 3.2], we have dim[Im(T n+1) ∩ ker(T n)]/[T n+1(0) ∩ ker(T n)] = dim ker(T 2n+1)/ker(T n+1) = n∑ i=1 αn+i(T) ≤ n αn+1(T) < +∞. Then from [7, Propositions 3.9, 3.10, 4.8, Lemma 4.2], it follows that Im(T n+1) is closed. Now, suppose that Im(T n+1) is closed, from [3, Lemma 2.4], we obtain Im(T) + ker(T n) = T −n[Im(T n+1)] is closed, which completes the proof. Corollary 4.2. Let T ∈ CR(H) be everywhere defined and suppose that T n(0) is closed for all n ∈ N. Then (i) ϱ g asc(T) = {λ ∈ C : ã(Tλ) < +∞ and Im[(Tλ)ã(Tλ)+1] is closed}, (ii) ϱ e, g asc(T) = {λ ∈ C : ãe(Tλ) < +∞ and Im[(Tλ)ãe(Tλ)+1] is closed}, (iii) ϱasc(T) = {λ ∈ C : a(Tλ) < +∞ and Im[(Tλ)a(Tλ)+1] is closed}, (iv) ϱeasc(T) = {λ ∈ C : ae(Tλ) < +∞ and Im[(Tλ)ae(Tλ)+1] is closed}. Let T ∈ LR(H), for every non-constant polynomial P = n∏ i=1 (λi−X)αi, with coefficients in C, we can associate the linear relation P(T) ∈ LR(H) defined by : P(T) := n∏ i=1 (λiI − T)αi. essential g-ascent and g-descent of a closed relation 141 Remark 4.3. Let T ∈ CR(H) be everywhere defined such that ϱ(T) ̸= ∅. If P is a non-constant complex polynomial, from Corollary 4.2 and [3, Theorem 4.7], it follows that σasc(P(T)) = P(σasc(T)) and σ e asc(P(T)) = P(σ e asc(T)). For T ∈ LR(H), we remark that d(T −1) = inf{n ∈ N : D(T n) = D(T n+1)}, where as usual the infimum over the empty set is taken to be +∞. Hence, if d(T −1) < +∞ then D(T d(T −1)) = D(T d(T −1)+n) ⊆ D(T n), ∀ n ∈ N. Example 4.4. Let H be a separable Hilbert space and let K ∈ LR(H). Consider the linear relation T : ⊕∞ i=0 H −→ ⊕∞ i=0 H defined by T(h0 ⊕ h1 ⊕ h2 ⊕ . . .) = K(h1) ⊕ h2 ⊕ h3 ⊕ . . . . Clearly, Im(T 2) = Im(T) and D(T k) = H ⊕ i=k⊕ i=1 D(K) ⊕ ∞⊕ i=k+1 H, ∀k ≥ 1. Hence, if D(K) H then d(T −1) = +∞, and, if D(K) = H then d(T −1) = 0. Let S = T −1, so if Im(K) H then d(S−1) = d(T) = 1, and, d(S−1) = 0 when Im(K) = H. Let us assume that T is a range space relation (see, Definition 3.1) such that q = d(T −1). It is clear that if P = (λ1−X)α1(λ2−X)α2 · · · (λm−X)αm is a complex polynomial then P(T) is a range space relation (see, [7, Propositions 4.7, 4.8]) and j = d(P(T)−1) ≤ q according to [8, Theorem 3.2]. Furthermore, if P is a non-constant polynomial then D([P(T)]j) = D(T q). Let us assume that P is a non-constant complex polynomial and T(0) ⊆ D(T q), then for all n ≥ q and m ∈ N, we have Im ( [P(T)]n ) ⊆ D(T m). Indeed, we prove by induction that T n(0) ⊆ D(T q). The cases n = 0, 1 are obvious. Suppose that T n(0) ⊆ D(T q), then T n+1(0) = T ( T n(0) ) ⊆ T ( D(T q+1) ) = TT −1(D(T q)) = D(T q) ∩ Im(T) + T(0) ⊆ D(T q). This implies that T n(0) ⊆ D(T q) for all n ∈ N, and consequently T n(0) ⊆ D(T m) for all n, m ∈ N. Finally, by [10, Lemma 4.1], we get Im ( [P(T)]n ) ⊆ D(T q) ⊆ D(T m), for every n ≥ q and m ∈ N. 142 z. garbouj, h. skhiri In the following, we define Υ(H) = { T ∈ LR(H) : T is a range space relation, q = d(T −1) < +∞, T(0) ⊆ D(T q), T n(0), D(T n+2) and Im(Tλ) + D(T q) are closed, ∀ λ ∈ C, ∀ n ∈ N } . Clearly, Υ(H) ̸= ∅, because T ∈ Υ(H), when T is a closed linear relation everywhere defined such that T n(0) is closed for all n ∈ N. For family of vectors (xi)i∈I in H, we denote by Vect(xi, i ∈ I), the vector subspace generated by (xi)i∈I. Example 4.5. (i) Let H be a separable Hilbert space and let K ∈ CR(H) such that D(K) H is closed. Let H = ⊕3 i=0 H and consider the linear relation T : H −→ H defined by T(h0 ⊕ h1 ⊕ h2 ⊕ h3) = K(h1) ⊕ h2 ⊕ h3 ⊕ h3. Clearly, T n(0 ⊕ 0 ⊕ 0 ⊕ 0) = K(0) ⊕ 0 ⊕ 0 ⊕ 0 is closed for all n ≥ 1 and D(T k) =   H ⊕ D(K) ⊕ H ⊕ H if k = 1 H ⊕ D(K) ⊕ D(K) ⊕ H if k = 2 H ⊕ D(K) ⊕ D(K) ⊕ D(K) if k ≥ 3 is closed. Hence d(T −1) = 3 and T(0 ⊕ 0 ⊕ 0 ⊕ 0) ⊆ D(T 3). It is not difficult to see that Im(Tλ) + D(T 3) = { H ⊕ H ⊕ H ⊕ D(K) if λ = 1 H ⊕ H ⊕ H ⊕ H if λ ̸= 1 is closed. Since T ∈ CR(H), it follows that T ∈ Υ(H). Assume that K(0) H and let S = T −1. It is easy to see that T n(h0 ⊕ h1 ⊕ h2 ⊕ h3) = K(h3) ⊕ h3 ⊕ h3 ⊕ h3, ∀h0 ∈ H, ∀ h1, h2, h3 ∈ D(K), for all n ≥ 3. Therefore d(S−1) = d(T) ≤ 3. Let h ∈ H\K(0) and ξ = h ⊕ 0 ⊕ 0 ⊕ 0, then ξ ∈ S(0) = ker(T) = H ⊕ ker(K) ⊕ {0} ⊕ {0} and ξ ̸∈ D(S3) = Im(T 3). Consequently, S ̸∈ Υ(H). (ii) Let H be a separable Hilbert space and (en)n∈N be an orthonormal basis of H. Define the following operators T and L in H by D(T) = D(L) = Vect(en : n ≥ 2), T(en) = en+1 essential g-ascent and g-descent of a closed relation 143 and L(en) = en−1, for all n ≥ 2. It is clear that D(T k) = D(T) and D(Lk) = Vect(en : n ≥ 1 + k), for all k ≥ 1 and hence d(T −1) = 1 and d(L−1) = +∞ (L ̸∈ Υ(H)). Since T ∈ CR(H), Im(Tλ) ⊆ D(T) for all λ ∈ C and D(T) is closed, then T ∈ Υ(H). (iii) Let T be defined as in (ii) and k ≥ 2. Now, we define the following relation S := Im(T k) + T (i.e., S(x) = {y + z : y ∈ Im(T k), z ∈ T(x)}, for all x ∈ D(T)). Since S(0) = Im(T k) is closed (because β(T) = 3), T is a closed operator and D(S) = D(T) is closed, then ∥QSS(x)∥ = ∥Tx∥ ≤ ∥Tx∥ ≤ ∥T∥ ∥x∥, for all x ∈ D(S). This proves that QSS is closed and by [4, Proposition II.5.3], we get that S ∈ CR(H). It is clear that Sj = { Im(T k) + T j if j < k Im(T k) if j ≥ k. For all j ≥ 1, we have D(Sj) = D(T) and Sj(0) = Im(T k) are closed in H, and from this we get that d(S−1) = 1 and S(0) ⊆ D(S). Moreover, for all λ ∈ C, we see that Im(Sλ) + D(S) = Im(Tλ) + Im(T k) + D(T) = D(T) is closed. Now, we can conclude that S ∈ Υ(H). (iv) Let T be defined as in Example 2.1, where M is a closed subspace of H and M H. It is easy to see that G(T) = M × M and T n(0) = D(T n) = Im(Tλ) = M, ∀ n ≥ 1, λ ∈ C. From this it follows that T ∈ Υ(H). (v) Let L be defined as in Example 2.1. It is not difficult to see that L is a closed relation, d(L−1) = 1, D(Ln) = N is closed, D(Ln)+Im(Lλ) = H, ∀n ≥ 1, λ ∈ C. But L(0) = M ̸⊂ D(L), so that L ̸∈ Υ(H). 144 z. garbouj, h. skhiri Let us show that if T ∈ Υ(H) and P = (λ1−X)α1(λ2−X)α2 · · · (λm−X)αm is a non-constant complex polynomial, then A = P(T) ∈ Υ(H). Put q = d(T −1) and define the following relation T : D(T) ⊆ H/D(T q) −→ H/D(T q) x 7−→ Tx. Note that the linear relation T is well-defined. Indeed, since T(0) ⊆ D(T q), T(D(T q)) = T(D(T q+1)) = TT −1(D(T q)) = D(T q) ∩ Im(T) + T(0) ⊆ D(T q), which implies that if x1, x2 ∈ D(T) such that x1 = x2, then Tx1 = Tx2. We remark also that T(0) = T(0) = 0 and thus T is an unbounded operator. Now, let λ ∈ C and x ∈ ker(λI − T), then Tλx = Tλ(0) = 0, which implies that Tλx ⊆ D(T q) and x ∈ D(T q+1) = D(T q). Consequently, x = 0, and thus ker(λI − T) = {0}. On the other hand, it is clear that Im(λI − T) = ( Im(Tλ) + D(T q) ) /D(T q) is closed. As in the proof of Lemma 3.3, we obtain λI − T is a range space operator. Now, applying [7, Lemma 4.6], we get λI − T is a closed operator, and thus λI − T ∈ Φ+(H/D(T q)). Recall that, if S, L ∈ CR(H) are two opera- tors such that L ∈ Φ+(H) and Im(S) is closed, then LS ∈ CR(H) and Im(LS) is closed. Since, for every i, j ∈ {1, 2, · · · , m}, λiI − T ∈ Φ+(H/D(T q)) and Im(λjI − T) is closed, we deduce that (λiI − T)(λjI − T) ∈ CR(H/D(T q)) and Im[(λiI − T)(λjI − T)] is closed. Consequently, from ker ( (λiI − T)(λjI − T) ) = {0}, it follows that (λiI − T)(λjI − T) ∈ Φ+(H/D(T q)). Therefore Im(P(T)) = ( Im(P(T)) + D(T q) ) /D(T q) is closed, and finally we obtain Im(A) + D(Ad(A −1)) = Im ( P(T) ) + D(T q) is closed. Now, let λ ∈ C and put Q = λ −P = a m∏ i=1 (µi −X)βi. Arguing in the same way as previous, we can conclude that Im(Aλ) + D(Ad(A −1)) is closed. Let k = m∑ i=0 αi and n ∈ N, by [8, Theorems 3.2, 3.6], we get An(0) = T k n(0) and D(An+2) = D(T (n+2)k) are closed, and A(0) ⊆ D(T q) = D(Ad(A −1)). This proves that, A ∈ Υ(H). The aim of this section is to establish the spectral mapping theorem of g- ascent and essential g-ascent spectrums of a closed linear relation T ∈ Υ(H). First, we have the following remark. essential g-ascent and g-descent of a closed relation 145 Remark 4.6. (i) If T ∈ Υ(H), then P(T) ∈ CR(H), for any complex polynomial P of degree n ≥ min{d(T −1), 2}. Indeed, we have D[P(T)] = D(T n) and P(T)(0) = T n(0) are closed, and P(T) is a range space relation. So, by [7, Lemma 4.6], it follows that P(T) ∈ CR(H). (ii) Let T ∈ Υ(H) such that D(T) is closed or T be a closed relation. By (i) and [7, Lemma 4.6], we deduce that P(T) ∈ CR(H), for any complex polynomial P. The next lemma is used to prove Lemma 4.8. Lemma 4.7. Let T ∈ LR(H), P = m∑ i=0 aiX i = α m∏ i=1 (λi − X) and Q = n∑ i=0 biX i = β n∏ i=1 (λi − X) be non-constants complex polynomials. Then (i) P(T) = m∑ i=0 aiT i, (ii) P(T) + Q(T) = (P + Q)(T) + T s − T s, where s = max{n, m}. Proof. It is easy to see that if ξ ∈ C and i, j ∈ N such that j ≤ i, then ξT j(x) + T i(0) = T j(ξx) + T i(0), ∀ x ∈ D(T j). (1) It follows from this that for all µ ∈ C\{0}, ξT j(x) + µT i(x) = T j(ξx) + µT i(x), ∀ x ∈ D(T i). (2) (i) We will prove that P(T) = m∑ i=0 aiT i and am = (−1)mα. By [4, Proposition I.4.2], we know that if R, S, L ∈ LR(H) such that D[L(R+S)] = D(LR+LS), then L(R + S) = LR + LS. So by [8, Theorem 3.2] and (2), we get (λ1I − T)(λ2I − T)x = (λ1I − T)(λ2x) − (λ1I − T)Tx = λ1λ2x − T(λ2x) − T(λ1x) + T 2(x) = λ1λ2x − T [(λ2 + λ1)x] + T 2(x) = λ1λ2x − (λ2 + λ1)T(x) + T 2(x), ∀ x ∈ D(T 2). Suppose that α(λ1I − T)(λ2I − T) · · · (λm−1I − T) = m−2∑ i=0 αiT i + α (−1)m−1T m−1 146 z. garbouj, h. skhiri and let us show that α(λ1I − T)(λ2I − T) · · · (λmI − T) = m−1∑ i=0 γiT i + α (−1)mT m. By [8, Corollary 2.1] and (2), we obtain P(T)x = α(λ1I − T)(λ2I − T) · · · (λmI − T)x = (λmI − T)[α(λ1I − T)(λ2I − T) · · · (λm−1I − T)]x = (λmI − T) ( m−2∑ i=0 αiT i + α (−1)m−1T m−1 ) x = (λmI − T) ( m−2∑ i=0 T i(αix) + α (−1)m−1T m−1x ) = m−2∑ i=0 T i(λmI − T)(αix) + α (−1)m−1T m−1(λmI − T)x = m−2∑ i=0 T i(λmαix) − m−2∑ i=0 T i+1(αix) + α (−1)m−1T m−1(λmx) + α (−1)mT mx = m−2∑ i=0 T i(λmαix) − m−1∑ i=1 T i(αi−1x) + α (−1)m−1T m−1(λmx) + α (−1)mT mx So, P(T)x = λmα0x + m−2∑ i=1 T i[(λmαi − αi−1)x] + T m−1[(α (−1)m−1λm − αm−2)x] + α (−1)mT mx = λmα0︸ ︷︷ ︸ γ0 x + m−2∑ i=1 (λmαi − αi−1)︸ ︷︷ ︸ γi T i(x) + (α (−1)m−1λm − αm−2)︸ ︷︷ ︸ γm−1 T m−1(x) + α (−1)mT mx = m−1∑ i=0 γiT ix + α (−1)mT mx, ∀ x ∈ D(T m). essential g-ascent and g-descent of a closed relation 147 This shows that P = m−1∑ i=0 γiX i + α (−1)mXm and hence ai = γi, for all i = 1, · · · , m − 1 and am = α (−1)m. (ii) Assume that m ≤ n, from (i) and (1), we see that [P(T) + Q(T)](x) = [P(T) + Q(T)](x) + [P(T) + Q(T)](0) = m∑ i=0 aiT ix + n∑ i=0 biT ix + T n(0) = m∑ i=0 (aiT ix + biT ix) + n∑ i=m+1 biT ix + T n(0) = m∑ i=0 [T i(aix) + T i(bix)] + n∑ i=m+1 biT i(x) + T n(0) = m∑ i=0 T i[(ai + bi)x] + n∑ i=m+1 biT i(x) + T n(0) = m∑ i=0 (ai + bi)T i(x) + n∑ i=m+1 biT i(x) + T n(0) = n∑ i=0 ωiT i(x) + T n(0), ∀ x ∈ D(T n), Since P + Q = n∑ i=0 ωiX i, then P(T) + Q(T) = (P + Q)(T) + T n − T n. This completes the proof. The following result is an improvement of [5, Lemma 4.4] to closed linear relations. Lemma 4.8. Let T ∈ LR(H) and let P and Q be two relatively prime complex polynomials. If A = P(T) and B = Q(T), then (i) Im(AnBn) = Im(An) ∩ Im(Bn), for all n ∈ N, (ii) ker(AnBn) = ker(An) + ker(Bn), for all n ∈ N, (iii) ker(An) ⊆ Im(Bm) and ker(Bn) ⊆ Im(Am), for all n, m ∈ N, (iv) ãe(AB) = max{ãe(A), ãe(B)} and ã(AB) = max{ã(A), ã(B)}. In addition, assume that T ∈ Υ(H), 148 z. garbouj, h. skhiri (v) if max{ãe(A), ãe(B)} < +∞, then Im(A) + ker(Aãe(A)) and Im(B) + ker(Bãe(B)) are both closed if and only if Im(AB) + ker[(AB)ãe(AB)] is closed, (vi) if max{ã(A), ã(B)} < +∞, then Im(A) + ker(Aã(A)) and Im(B) + ker(Bã(B)) are both closed if and only if Im(AB) + ker[(AB)ã(AB)] is closed. Proof. The proof is trivial when P or Q is a constant polynomial. Assume that P and Q are non-constants polynomials. So, P = (λ1 − X)α1(λ2 − X)α2 · · · (λm − X)αm and Q = (µ1 − X)β1(µ2 − X)β2 · · · (µs − X)βs. First, it is clear that the assertions (i) and (ii) follow immediately from [8, Theorems 3.3, 3.4]. (iii) By [10, Lemma 7.2], we know that ker[(λiI − T)αi] ⊆ Im[(µjI − T)βj ], so from [8, Theorems 3.3, 3.4], ker(An) ⊆ Im(Bm), for all n, m ∈ N. (iv) Let n ∈ N, we have Im(AnBn) ∩ ker(AB) = Im(An) ∩ Im(Bn) ∩ [ker(A) + ker(B)] = Im(An) ∩ [ker(A) + Im(Bn) ∩ ker(B)] = [Im(An) ∩ ker(A)] + [Im(Bn) ∩ ker(B)]. Therefore α̃n(AB) = 0 ⇐⇒ max{α̃n(B), α̃n(A)} = 0, α̃n(AB) < +∞ ⇐⇒ max{α̃n(B), α̃n(A)} < +∞. (v) Let n ∈ N\{0}. Since P n and Qn are relatively prime, we know that there exist two polynomials Pn = a p∏ i=1 (νi − X)ri and Qn = b r∏ i=1 (ωi − X)ji such that P nPn + Q nQn = 1. Let pn (resp. k) be the degree of Pn (resp. P) and α(n) = nk + pn. Then, the degree of P nPn (resp. Q nQn) is α(n) and by [8, Theorem 3.2], we get D[AnPn(T) + BnQn(T)] = D[AnPn(T)] ∩ D[BnQn(T)] = D(T α(n)). Now, by Lemma 4.7, we obtain AnPn(T)x + B nQn(T)x = x + T α(n)(0), ∀ x ∈ D(T α(n)). (1) If n ≥ q = d(T −1), then α(n) ≥ q, and by (1), it is clear that D(T q) = D(T α(n)) ⊆ Im(An) + Im(Bn). essential g-ascent and g-descent of a closed relation 149 Since Im(An) ⊆ D(T q) and Im(Bn) ⊆ D(T q), for every j ∈ N, it follows that D(T q) = Im(An) + Im(Bn) = [ker(Aj) + Im(An)] + [Im(Bn) + ker(Bj)] (2) and ker(AjBj) + Im(AnBn) = ker(Aj) + ker(Bj) + Im(An) ∩ Im(Bn) = [ker(Aj) + Im(An)] ∩ Im(Bn) + ker(Bj) = [ker(Aj) + Im(An)] ∩ [Im(Bn) + ker(Bj)]. (3) From [7, Propositions 3.9, 3.10, 4.8, Lemma 4.2] and by using (2) and (3), we get ker(AjBj) + Im(AnBn) is closed ⇔ { ker(Aj) + Im(An), ker(Bj) + Im(Bn) are both closed.(4) Let n > q = d(T −1) and j = ãe(A) + ãe(B) + ãe(AB) + p(A) + p(B) + p(AB). Let us assume that Im(AB) + ker(AjBj) is closed. Since the degree of PQ is greater than or equal to two, then from Remark 4.6, AB ∈ CR(H). It follows from [6, Lemma 2.9] and Lemma 3.2 that Im(AnBn) + ker(AjBj) is closed. Let us assume that N = Im(AnBn)+ker(AjBj) is closed. As D(Bn−1An−1) = D(T q) is closed and Im(Bn−1An−1) ⊆ D(T q), S : D(T q) −→ D(T q) x 7−→ Bn−1An−1x is well-defined. From [7, Lemmas 4.6, 4.10, Proposition 4.8], it follows that S ∈ CR(D(T q)). Because D(S) = D(T q) is closed and S(0) ⊆ N ⊆ D(T q), by [3, Lemma 2.4],( Im(AB) + ker[(AB)j+n−1] ) ∩ D(T q) = S−1(N) is closed. Since (Im(AB) + ker[(AB)j+n−1]) + D(T q) = Im(AB) + D[(AB)d(A −1B−1)] is closed, from [7, Propositions 3.9, 3.10, 4.8, Lemma 4.2], it follows that Im(AB) + ker[(AB)j+n−1] is closed. Now, by [6, Lemma 2.9], Im(AB) + ker[(AB)j] is closed, and thus Im(AB) + ker(AjBj) is closed ⇐⇒ Im(AnBn) + ker(AjBj) is closed. (5) 150 z. garbouj, h. skhiri Arguing in the same way as previous, we can conclude that Im(A) + ker(Aj) is closed ⇐⇒ Im(An) + ker(Aj) is closed, Im(B) + ker(Bj) is closed ⇐⇒ Im(Bn) + ker(Bj) is closed. (6) Finally, it follows from (4), (5), (6) and [6, Lemma 2.9] that Im(A)+ker(Aãe(A)) and Im(B)+ker(Bãe(B)) are both closed if and only if Im(AB)+ker[(AB)ãe(AB)] is closed. (vi) Finally, by [6, Lemma 2.9] and the assertion (v), we see that the following assertions are equivalent : a) Im(A) + ker(Aã(A)) is closed and Im(B) + ker(Bã(B)) is closed, b) Im(A) + ker(Aãe(A)) is closed and Im(B) + ker(Bãe(B)) is closed, c) Im(AB) + ker[(AB)ãe(AB)] is closed, d) Im(AB) + ker[(AB)ã(AB)] is closed. The proof is complete. Theorem 4.9. Let T ∈ Υ(H) be a closed linear relation. If A and B are defined as in Lemma 4.8, then 0 ∈ ϱe, gasc(AB) ⇐⇒ 0 ∈ ϱ e, g asc(A) ∩ ϱ e, g asc(B) and 0 ∈ ϱgasc(AB) ⇐⇒ 0 ∈ ϱ g asc(A) ∩ ϱ g asc(B). Proof. This is an obvious consequence of Remark 4.6 and Lemma 4.8. Theorem 4.10. Let T ∈ Υ(H) be a closed linear relation and m ∈ N\{0}. Then 0 ∈ ϱe, gasc(T) ⇐⇒ 0 ∈ ϱ e, g asc(T m) and 0 ∈ ϱgasc(T) ⇐⇒ 0 ∈ ϱ g asc(T m). Proof. First, since T ∈ Υ(H) is closed, we obtain T m is closed (see Remark 4.6). Let n, m ∈ N\{0} and S = T|Im(T mn) be the restriction of T to Im(T mn). From [9, Lemma 5.1], it follows that, α̃nm(T) ≤ α̃n(T m) = α(Sm) ≤ mα(S) = mα̃nm(T), essential g-ascent and g-descent of a closed relation 151 and this proves that ãe(T) < +∞ if and only if ãe(T m) < +∞. Now, put k = max{p(T m), p(T)} < +∞, then by Lemma 2.2, k ≥ max{ãe(T), ãe(T m)}. Let n > d(T −1), as in the proof of Lemma 4.8 and according to [6, Lemma 2.9] and Lemma 3.2, we deduce that Im(T m) + ker(T m k) is closed =⇒ Im(T m n) + ker(T m k) is closed, =⇒ Im(T) + ker(T m k) is closed. Hence, from [6, Lemma 2.9] and Lemma 3.2, Im(T) + ker(T ãe(T)) is closed ⇐⇒ Im(T) + ker(T m k) is closed, ⇐⇒ Im(T m) + ker(T m k) is closed, ⇐⇒ Im(T m) + ker[(T m)ãe(T m)] is closed. This gives 0 ∈ ϱe, gasc(T) ⇐⇒ 0 ∈ ϱ e, g asc(T m). Furthermore, for every m ∈ N\{0}, ã(T) < +∞ if and only if ã(T m) < +∞. So that Im(T) + ker(T ã(T)) is closed ⇐⇒ Im(T) + ker(T ãe(T)) is closed, ⇐⇒ Im(T m) + ker(T m ãe(T m)) is closed, ⇐⇒ Im(T m) + ker(T m ã(T m)) is closed. Consequently, 0 ∈ ϱgasc(T) ⇐⇒ 0 ∈ ϱ g asc(T m), which completes the proof. Corollary 4.11. Let T ∈ Υ(H) be a closed linear relation and let P = (λ1 − X)m1(λ2 − X)m2 · · · (λn − X)mn be a complex polynomial such that mi ̸= 0 for every i = 1, 2, · · · , n. Then 0 ∈ ϱe, gasc(P(T)) ⇐⇒ λi ∈ ϱ e, g asc(T), ∀ 1 ≤ i ≤ n and 0 ∈ ϱgasc(P(T)) ⇐⇒ λi ∈ ϱ g asc(T), ∀ 1 ≤ i ≤ n. Proof. First, since T ∈ Υ(H) is closed, P(T) is closed (see Remark 4.6). Now, from Theorem 4.9 and Theorem 4.10, we deduce 0 ∈ ϱe, gasc(P(T)) ⇐⇒ 0 ∈ ∩ 1≤i≤n ϱ e, g asc[(λiI − T)mi], ⇐⇒ 0 ∈ ∩ 1≤i≤n ϱ e, g asc(λiI − T), ⇐⇒ λi ∈ ϱ e, g asc(T), ∀ 1 ≤ i ≤ n. 152 z. garbouj, h. skhiri In the same way, we prove that 0 ∈ ϱgasc(P(T)) ⇐⇒ λi ∈ ϱ g asc(T), ∀ 1 ≤ i ≤ n, and the proof is therefore complete. The following extends [3, Theorem 4.7], we do not require that the relation T be everywhere defined and ϱ(T) ̸= ∅. Theorem 4.12. Let T ∈ Υ(H) be a closed linear relation and P = (λ1 − X)m1(λ2 − X)m2 · · · (λn − X)mn be a complex polynomial such that mi ̸= 0 for all i = 1, 2, · · · , n. Then P(σe, gasc(T)) = σ e, g asc(P(T)) and P(σ g asc(T)) = σ g asc(P(T)). Proof. First note, by Remark 4.6, P(T) is a closed linear relation. Now, from Corollary 4.11, it follows that λ ∈ P(σe, gasc(T)) ⇐⇒ λ = P(µ), where µ ∈ σ e, g asc(T), ⇐⇒ λ − P(Z) = (µ − Z)kQ(Z), where Q(µ) ̸= 0, ⇐⇒ λ ∈ σe, gasc(P(T)). The second equality, can be proved in the same way as the first one. This finishes the proof of the theorem. Example 4.13. (i) Let T as in (i) of Example 4.5, we have T ∈ Υ(H) is a closed linear relation and D(T) H. Hence, if P is a non-constant complex polyno- mial, by [3, Theorem 4.7], it is not possible to deduce that P(σ g asc(T)) = σ g asc(P(T)) or P(σ e, g asc(T)) = σ e, g asc(P(T)). However, from Theorem 4.12, we can do this. (ii) Let B = (e1, e2, e3) be a basis of C3. Consider the linear relation : T ( 3∑ i=1 αi ei ) = Vect(e1, e2) + α3 e3, ∀ α1, α2, α3 ∈ C. Clearly, T ∈ Υ(H), because T is a closed linear relation everywhere defined and T n(0) is closed for all n ∈ N. For all λ ∈ C, we note that 0 ∈ (λI − T)(e1) = Vect(e1, e2), this implies that ϱ(T) = ∅. Hence, if P is a non-constant complex polynomial, by [3, Theorem 4.7], it is not possible to deduce that P(σ g asc(T)) = σ g asc(P(T)) or P(σ e, g asc(T)) = σ e, g asc(P(T)). However, from Theorem 4.12, we can do this. essential g-ascent and g-descent of a closed relation 153 Remark 4.14. We note that Theorem 4.12 is false in general without the assumption that P is a non-constant polynomial, even if T is a bounded linear operator. For example, if H is a separable Hilbert space and (en)n∈N is an orthonormal basis of H, we define the following bounded operator on H, T ( +∞∑ n=0 xnen ) = +∞∑ n=0 xn n + 1 en. Since ker(T) = {0} (which gives ae(T) = 0) and Im(T) is not closed in H, it follows that 0 ∈ σeasc(T) ⊆ σasc(T). Put P = c ∈ C, since σeasc(T) and σasc(T) are non-empty sets, it follows that P(σeasc(T)) = P(σasc(T)) = {c}. However, ϱasc(P(T)) = ϱasc(cI) = C. Indeed, C\{c} = ϱ(cI) ⊆ ϱasc(cI), and cI − cI is the zero operator on H with ascent is equal to 1 and kernel is equal to H. Hence σeasc(P(T)) = σasc(P(T)) = ∅. 5. A spectral mapping theorems for essential g-descent and g-descent spectrums We start this section with the following definitions. The descent and the essential descent resolvent sets of T ∈ LR(H) are respectively defined by ϱdes(T) = {λ ∈ C : d(Tλ) < +∞} and ϱedes(T) = {λ ∈ C : de(Tλ) < +∞}. The descent and the essential descent spectrums of T are respectively σdes(T) := C\ϱdes(T) and σedes(T) := C\ϱ e des(T). The g-descent resolvent set and the essential g-descent resolvent set of T ∈ LR(H) are respectively defined by ϱ g des(T) = {λ ∈ C : d̃(Tλ) < +∞} and ϱ e, g des(T) = {λ ∈ C : d̃e(Tλ) < +∞}. The g-descent and the essential g-descent spectrums of T are respectively σ g des(T) := C\ϱ g des(T) and σ e, g des(T) := C\ϱ e, g des(T). Evidently ϱ(T) ⊆ ϱgdes(T) ⊆ ϱdes(T) ⊆ ϱ e des(T), ϱ(T) ⊆ ϱ g des(T) ⊆ ϱ e, g des(T) ⊆ ϱ e des(T). This section will be devoted to study the spectral mapping theorems of the g-descent and the essential g-descent spectrums of linear relations. The following lemmas will be used to prove the main result of this section. 154 z. garbouj, h. skhiri Lemma 5.1. Let T ∈ LR(H), P and Q are relatively prime complex poly- nomials. Let n ∈ N\{0}, A = P(T) and B = Q(T). Then (i) d̃e(AB) = max{d̃e(A), d̃e(B)} and d̃(AB) = max{d̃(A), d̃(B)}, (ii) T possess a finite g-descent (resp. essential g-descent) if and only if the same holds for T n. Proof. (i) Let n ∈ N, from Lemma 4.8, we have ker(AnBn) + Im(AB) = ker(An) + ker(Bn) + Im(A) ∩ Im(B), = ( ker(An) + Im(A) ) ∩ Im(B) + ker(Bn), = ( ker(An) + Im(A) ) ∩ ( Im(B) + ker(Bn) ) . Therefore ker(AnBn) + Im(AB) ⊆ ker(An) + Im(A) ⊆ H, and consequently, β̃n(AB) = β̃n(A) + dim ( ker(An) + Im(A) ) / ( [ker(An) + Im(A)] ∩ [Im(B) + ker(Bn)] ) , max{β̃n(A), β̃n(B)} ≤ β̃n(AB) ≤ β̃n(A) + β̃n(B). (ii) Let m ∈ N and S be the relation induced by T on H/ker(T nm). Thus from [9, Lemma 5.1], we obtain β̃nm(T) = β(S) ≤ β(Sn) = β̃m(T n) ≤ n β(S) = n β̃nm(T). This completes the proof. Lemma 5.2. Let T ∈ LR(H) and m ∈ N\{0}. Assume that A and B are defined as in Lemma 5.1. Then (i) 0 ∈ ϱe, gdes(T) if and only if 0 ∈ ϱ e, g des(T m), (ii) 0 ∈ ϱgdes(T) if and only if 0 ∈ ϱ g des(T m), (iii) 0 ∈ ϱe, gdes(AB) if and only if 0 ∈ ϱ e, g des(A) ∩ ϱ e, g des(B), (iv) 0 ∈ ϱgdes(AB) if and only if 0 ∈ ϱ g des(A) ∩ ϱ g des(B). Proof. It is an obvious consequence of Lemma 5.1. essential g-ascent and g-descent of a closed relation 155 Corollary 5.3. Let T ∈ LR(H) and let P = (λ1 − X)m1(λ2 − X)m2 · · · (λn − X)mn be a complex polynomial such that mi ̸= 0 for every i = 1, 2, · · · , n. Then 0 ∈ ϱe, gdes(P(T)) ⇐⇒ λi ∈ ϱ e, g des(T), ∀ 1 ≤ i ≤ n and 0 ∈ ϱgdes(P(T)) ⇐⇒ λi ∈ ϱ g des(T), ∀ 1 ≤ i ≤ n. Proof. From Lemma 5.2, it follows that 0 ∈ ϱe, gdes(P(T)) ⇐⇒ 0 ∈ ∩ 1≤i≤n ϱ e, g des[(λiI − T) mi], ⇐⇒ 0 ∈ ∩ 1≤i≤n ϱ e, g des(λiI − T), ⇐⇒ λi ∈ ϱ e, g des(T), ∀ 1 ≤ i ≤ n. In the same way, we obtain 0 ∈ ϱgdes(P(T)) ⇐⇒ λi ∈ ϱ g des(T), ∀ 1 ≤ i ≤ n. The proof is complete. First note, that the results of this section are true for Banach spaces. Hence, the following extends [3, Theorem 3.4], we do not require that the relation T be everywhere defined and dim T(0) < +∞. Theorem 5.4. Let T ∈ LR(H) and P = (λ1 − X)α1 · · · (λm − X)αm be a complex polynomial such that αi ̸= 0 for all i = 1, 2, · · · , m. Then P(σ e, g des(T)) = σ e, g des(P(T)) and P(σ g des(T)) = σ g des(P(T)). Proof. From Corollary 5.3, it follows that λ ∈ P(σe, gdes(T)) ⇐⇒ λ = P(µ), where µ ∈ σ e, g des(T), ⇐⇒ λ − P(Z) = (µ − Z)kQ(Z), where Q(µ) ̸= 0, ⇐⇒ λ ∈ σe, gdes(P(T)). The second equality, can be proved in the same way as the first one. This finishes the proof of the theorem. 156 z. garbouj, h. skhiri Example 5.5. Let T and K as in (i) of Example 4.5 (resp. suppose that dim K(0) = +∞ and we replace the condition D(K) H by D(K) ⊆ H), we have T ∈ Υ(H) is a closed linear relation and D(T) H (resp. dim T(0) = +∞). Hence, if P is a non-constant complex polynomial, by [3, Theorem 3.4], it is not possible to deduce that P(σ g des(T)) = σ g des(P(T)) (resp. P(σ e, g des(T)) = σ e, g des(P(T))). However, from Theorem 5.4, we can do this. Remark 5.6. We note that Theorem 5.4 is false in general without the assumption that P is a non-constant polynomial, even if T is a bounded linear operator. For example, let T be defined as in Remark 4.14. Since ker(T) = {0} and Im(T) is not closed, then β̃n(T) = β(T) = +∞, for all n ∈ N, and thus 0 ∈ σe, gdes(T) ⊆ σ g des(T). Put P = c ∈ C. Since σ e, g des(T) and σ g des(T) are non-empty sets, it follows that P(σ e, g des(T)) = P(σ g des(T)) = {c}. However, ϱ g des(P(T)) = ϱ g des(cI) = C. Indeed, C\{c} = ϱ(cI) ⊆ ϱ g des(cI), and cI − cI is the zero operator with g-descent is equal to 1. Therefore σ g des(P(T)) = σ e, g des(P(T)) = ∅. 6. Decomposition theorems First observe that if T ∈ LR(H) is a range space relation such that ãe(T) < +∞ and Im(T)+ker(T ãe(T)) is closed in H, then T is a quasi-Fredholm relation (see, [7, Definition 5.1]). In the following, we prove a decomposition theorem of linear relations with finite essential g-ascent such that Im(T) + ker(T ãe(T)) is closed in H. Theorem 6.1. Let T ∈ CR(H). Then there exists n ∈ N such that ãe(T) ≤ n and Im(T) + ker(T n) is closed in H if and only if there exist d ∈ N and two closed subspaces M and N such that : (i) H = M u N; (ii) Im(T d) ⊆ M, T(M) ⊆ M, N ⊆ ker(T d) and, if d > 0, N ̸⊂ ker(T d−1); (iii) G(T) = [G(T) ∩ (M × M)] u [G(T) ∩ (N × N)]; (iv) the restriction of T to M is both upper semi-Fredholm and regular rela- tion; essential g-ascent and g-descent of a closed relation 157 (v) A ∈ LR(N) such that its graph is the subspace G(T) ∩ (N × N), then A is a bounded operator everywhere defined and G(Ad) = N × {0}. Proof. ” =⇒ ” First, the assertions (i)-(iii) and (v) follow from [7, Theo- rem 5.2], and by the same theorem we know that S = T|M the restriction of T to M is regular. Let m = max{d, n}, since S is regular and Im(T m) ⊆ M, it follows that ker(T) ∩ Im(T m) = ker(S) ∩ Im(Sm) = ker(S), and hence α(S) = α̃m(T) < +∞. Therefore S ∈ Φ+(M). ” ⇐= ” Let S = T|M be the restriction of T to M and A ∈ LR(N) such that G(A) = G(T) ∩ (N × N), so ker(T) ∩ Im(T d) = ker(S) ∩ Im(Sd) = ker(S) and α̃d(T) < +∞. This implies that ãe(T) ≤ d. By [7, Theorem 6.4], Im(T) + ker(T d) is closed and from [6, Lemma 2.9], it follows that Im(T) + ker(T n) is closed for all n ≥ ãe(T), which completes the proof. Now from [6, Lemma 2.9], we know that if ã(T) = n < +∞ and Im(T) + ker(T n) is closed, then ãe(T) ≤ n and Im(T) + ker(T n) is closed. So, as a consequence of Theorem 6.1, we obtain the following theorem. Theorem 6.2. Let T ∈ CR(H). Then there exists n ∈ N such that ã(T) = n < +∞ and Im(T) + ker(T n) is closed if and only if there exist d ∈ N and two closed subspaces M and N such that : (i) H = M u N, (ii) T(M ∩ D(T)) ⊆ M, Im(T d) ⊆ M, N ⊆ ker(T d) and, if d > 0, N ̸⊂ ker(T d−1), (iii) G(T) = [G(T) ∩ (M × M)] u [G(T) ∩ (N × N)], (iv) the restriction of T to M is injective with closed range, (v) if A ∈ LR(N) such that its graph is the subspace G(T) ∩ (N × N), then A is a bounded operator everywhere defined and G(Ad) = N × {0}. The following lemma will be needed in the proof of Theorem 6.4. 158 z. garbouj, h. skhiri Lemma 6.3. Let T ∈ CR(H) such that de(T) < +∞. The following state- ments are equivalent : (i) Im(T n) ∩ ker(T) is closed for some n ≥ de(T), (ii) Im(T n) ∩ ker(T) is closed for all n ≥ de(T). Proof. It is clear that only the implication ”(i) =⇒ (ii)” needs to prove. Let n0 ≥ de(T) such that Im(T n0) ∩ ker(T) is closed. First, we prove that Im(T n0+1) ∩ ker(T) is closed. By the equality (1) in the proof of Lemma 2.4, we get βn0(T) ≥ Sn0(T) = dim ( Im(T n0) ∩ ker(T) ) / ( Im(T n0+1) ∩ ker(T) ) , and hence from [7, Propositions 3.9, 3.10 and 4.8, Lemma 4.2], Im(T n0+1) ∩ ker(T) is closed. Now if n0 > de(T), then n0 − 1 ≥ de(T) and so dim ( Im(T n0−1) ∩ ker(T) ) / ( Im(T n0) ∩ ker(T) ) < +∞. Therefore Im(T n0−1) ∩ ker(T) is also closed. This completes the proof. In the following result, we prove a decomposition theorem for T ∈ CR(H), with n = d̃e(T) < +∞ and Im(T n) ∩ ker(T) is closed in H. Theorem 6.4. Let T ∈ CR(H). Then d̃e(T) ≤ n and Im(T n) ∩ ker(T) is closed for some n ∈ N if and only if there exist d ∈ N and two closed subspaces M and N such that : (i) H = M u N, (ii) T(M ∩ D(T)) ⊆ M, Im(T d) ⊆ M, N ⊆ ker(T d) and, if d > 0, N ̸⊂ ker(T d−1), (iii) G(T) = [G(T) ∩ (M × M)] u [G(T) ∩ (N × N)], (iv) the restriction of T to M is both lower semi-Fredholm and regular rela- tion, (v) if A ∈ LR(N) such that its graph is the subspace G(T) ∩ (N × N), then A is a bounded operator everywhere defined and G(Ad) = N × {0}. Proof. ” =⇒ ” First, from Lemma 2.4, we have d̃e(T) ≤ p(T) < +∞ and β̃p(T)(T) is finite, which implies that Im(T) + ker(T p(T)) is closed in H. Moreover, by Lemma 6.3, T is a quasi-Fredholm relation and so from [7, essential g-ascent and g-descent of a closed relation 159 Theorem 5.2], it follows that there exist d ∈ N and two closed subspaces M and N such that H = MuN, T(M∩D(T)) ⊆ M, N ⊆ ker(T d) ⊆ D(T), if d > 0, N ̸⊂ ker(T d−1) and the restriction of T to M, S = T|M, is regular. Now, let m = max{d, n}, then Im(T) + ker(T m) = Im(S) u N (see the equality (6.7) in the proof of [7, Theorem 6.4]). This implies that dim M/Im(S) = dim[M u N]/[Im(S) u N] = β̃m(T) < +∞ and consequently S ∈ Φ−(M). ” ⇐= ” Let S = T|M be the restriction of T to M and A ∈ LR(N) such that G(A) = G(T) ∩ (N × N), so that Im(T) + ker(T d) = Im(S) u N, ker(T) ∩ Im(T d) = ker(S) ∩ Im(Sd) = ker(S). This implies that β̃d(T) ≤ β(S) < +∞ and ker(T) ∩ Im(T d) is closed. This completes the proof of the theorem. Now from Lemma 6.3, we know that if d̃(T) = n < +∞ and Im(T n) ∩ ker(T) is closed, then d̃e(T) ≤ n and Im(T n) ∩ ker(T) is closed. Therefore, we can prove the following theorem similarly as Theorem 6.4. Theorem 6.5. Let T ∈ CR(H). Then d̃(T) ≤ n and Im(T n) ∩ ker(T) is closed for some n ∈ N if and only if there exist d ∈ N and two closed subspaces M and N such that : (i) H = M u N, (ii) T(M ∩ D(T)) ⊆ M, Im(T d) ⊆ M, N ⊆ ker(T d) and, if d > 0, N ̸⊂ ker(T d−1), (iii) G(T) = [G(T) ∩ (M × M)] u [G(T) ∩ (N × N)], (iv) the restriction of T to M is surjective, (v) if A ∈ LR(N) such that its graph is the subspace G(T) ∩ (N × N), then A is a bounded operator everywhere defined and G(Ad) = N × {0}. Remark 6.6. Let T ∈ CR(H) such that max{ã(T), d̃(T)} ≤ m, for some m ∈ N. It is clear that S = T|Im(T m), the restriction of T to Im(T m) is bijective, H = Im(T m) + ker(T m) and Im(T m) ∩ ker(T m) = {0} (see the equality (1) in the proof of Lemma 3.7), so from [7, Propositions 3.10, 4.8, Lemma 4.2], it 160 z. garbouj, h. skhiri follows that Im(T m) and ker(T m) are both closed. Let A ∈ LR(ker(T m)) such that G(A) = G(T) ∩ (ker(T m) × ker(T m)). First note that A(0) ⊆ ker(T m) ∩ Im(T m) = {0} and G(A) is closed, which implies that A is a closed operator. Since D(A) = {x ∈ ker(T m) : Ax ̸= ∅} = {x ∈ ker(T m) : Tx ̸= ∅ in ker(T m)} = {x ∈ ker(T m) : ∃ y ∈ Tx and y ∈ ker(T m)} = {x ∈ ker(T m) : ∃ y ∈ Tx and 0 ∈ T m(y)} = {x ∈ ker(T m) : 0 ∈ T m+1(x)} = {x ∈ ker(T m) : x ∈ ker(T m+1)} = ker(T m), then A is a bounded operator everywhere defined. But Am(ker(T m)) ⊆ ker(T m) ∩ Im(T m) = {0}, so G(Am) = G(T m) ∩ (ker(T m) × ker(T m)) = ker(T m) × {0}. Now, we will show that G(T) = [G(T) ∩ (Im(T m) × Im(T m))] u [G(T) ∩ (ker(T m) × ker(T m))]. Let (x, y) ∈ G(T), then x = x1 + x2 with x1 ∈ Im(T m) and x2 ∈ ker(T m). Therefore, there exist y1 ∈ T(x1) ⊆ Im(T m) and y2 ∈ T(x2) such that y = y1 + y2. Clearly, y2 ∈ T(x2) ⊆ T ( ker(T m) ) = T ( ker(T m+1) ) = TT −1 ( ker(T m) ) = ker(T m) ∩ Im(T) + T(0) ⊆ ker(T m) + T(0) and hence y2 = y ′ 2 + y ′′ 2, for some y ′ 2 ∈ ker(T m) and y′′2 ∈ T(0). We have, y1 + y ′′ 2 ∈ T(x1) + T(0) = T(x1) ⊆ Im(T m) and y′2 = y2 − y ′′ 2 ∈ T(x2) + T(0) = T(x2), so (x, y) = (x1, y1 + y ′′ 2) + (x2, y ′ 2) ∈ [G(T) ∩ (Im(T m) × Im(T m))] u [G(T) ∩ (ker(T m) × ker(T m))]. This implies that G(T) = [G(T) ∩ (Im(T m) × Im(T m))] u [G(T) ∩ (ker(T m) × ker(T m))]. Finally, if we put M = Im(T m) and N = ker(T m), then M and N satisfy the conditions (i)-(v) of Theorems 6.2 and 6.5. References [1] T. Álvarez, On regular linear relations, Acta Mathematica Sinica, English Series 28 (1) (2012), 183 – 194. essential g-ascent and g-descent of a closed relation 161 [2] O. Bel Hadj Fredj, M. Burgos, M. Oudghiri, Ascent spectrum and essential ascent spectrum, Studia Math. 187 (2008), 59 – 73. [3] E. Chafai, M. Mnif, Spectral mapping theorem for ascent, essential ascent, descent and essential descent spectrum of linear relations, Acta Mathematica Scientia, 34B (4) (2014), 1212 – 1224. [4] R. Cross, “Multivalued Linear Operators”, Marcel Dekker, New York, 1998. [5] Z. Garbouj, H. Skhiri, Essential ascent of closed operator and some de- composition theorems, Commun. Math. Anal. 16 (2) (2014), 19 – 47. [6] Z. Garbouj, H. Skhiri, Minimum modulus, perturbation for essential as- cent and descent of a closed linear relation in Hilbert spaces, Acta Mathe- matica Hungarica 151 (2) (2017), 328 – 360. [7] J.Ph. Labrousse, A. Sandovici, H.S.V. De Snoo, H. Winkler, The Kato decomposition of quasi-Fredholm relations, Operators and Matrices 4 (1) (2010), 1 – 51. [8] A. Sandovici, Some basic properties of polynomials in a linear relation in linear spaces, in “Oper. Theory Adv. Appl. 175”, Birkhäuser, Basel, (2007), 231 – 240. [9] A. Sandovici, H. De Snoo, An index formula for the product of linear relations, Linear Algebra Appl. 431 (11) (2009), 2160 – 2171. [10] A. Sandovici, H.S.V. De Snoo, H. Winkler, Ascent, descent, nullity, defect, and related notions for linear relations in linear spaces, Linear Algebra Appl. 423 (2-3) (2007), 456 – 497.