Some generalizations of ascent and descent for linear operators

The purpose of this paper is to present in linear spaces some results for new notions called A-left (resp., A-right) ascent and A-left (resp., A-right) descent of linear operators (where A is a given operator) which generalize two important notions in operator theory: ascent and descent. Moreover, if A is a positive operator, we obtain several properties of ascent and descent of an operator in semi-Hilbertian spaces. Some basic properties and many results related to the ascent and descent for a linear operator on a linear space Kaashoek (Math Ann 172:105–115, 1967), Taylor (Math Ann 163:18–49, 1966) are extended to these notions. Some stability results under perturbations by compact operators and operators having some finite rank power are also given for these notions.


Introduction and terminologies
The notions ascent and descent of a linear operator were introduced by Riesz [12] in connection with his investigation of compact linear operators. Heuser [7] considered these notions for a linear operator T in a linear space X under the condition that T is defined everywhere. The last condition was lifted by Taylor [18], whose treatment was completed by Kaashoek [8]. Kaashoek also provided a unified way of proving Taylor's results (see also [11,19]). Especially in recent years, various aspects of the problem related to generalization of these notions have appeared in the literature. Let us mention, for example, [14] for linear relations. In this paper, closely related to this problem, some new notions are considered and many properties are investigated. These new notions will be called A-left (resp., A-right) ascent and A-left (resp., A-right) descent, where A is a given linear operator. Many results from [8,9,18] are extended to these notions.
The class of operators with finite ascent and descent has been studied in a large number of papers. For example, the class of partial isometries on a complex Hilbert space with finite ascent and descent recently studied in [1,13]. The classes of partial isometries, quasi-isometries, contractions, and m-isometries in semi-Hilbertian spaces have been the object of some intensive study, especially by Arias et al. [2,3], Suciu [16,17], Sid Ahmed et al. [15] but also by Fongi et al. [5]. In this paper, if A is a positive operator, we obtain several properties of ascent and descent of an operator in semi-Hilbertian spaces.
In this section, we set up notation and terminology and we introduce the notions of A-left (resp., A-right) ascent and A-left (resp., A-right) descent.
Throughout this paper, X will be a linear space over the field of complex numbers. We will denote by L (X) the set of all linear operators on X. For an operator T ∈ L (X), write N(T ) for its kernel and R(T ) for its range. It is well known that N(T n ) ⊂ N(T n+1 ) and R(T n+1 ) ⊂ R(T n ), for all n ∈ N, where N = {0, 1, 2, · · · } denotes the set of all non-negative integers. The smallest non-negative integer for which there is equality is called the ascent of T and the descent of T, denoted by asc(T ) and des(T ), respectively. In case no such number exists the ascent or descent of T is said to be infinite. The nullity and the defect of a linear operator T are defined by α(T ) = dim N(T ), β(T ) = dim X/R(T ). The

We shall denote by α A (T ) = α(AT ), β A (T ) = β(AT ), α r A (T ) = α(T A) and β r A (T ) = β(T A) its A-left nullity, A-left defect, A-right nullity, and A-right defect of T, respectively.
Remark 1.1 Let H be a Hilbert space with inner product · , · . Given a positive bounded operator A on H, let · , · A be the semi-inner product defined by x , y A := Ax , y , ∀ x, y ∈ H.
By · A , we denote the semi-norm induced by · , · A , that is is called a semi-Hilbertian space (see [2,3]). We observe that if T is a bounded operator on H, and then where M is a linear subspace of X.
We present some useful properties concerning the A-left kernel, A-left range, A-right kernel, and A-right range of the sequence (T n ) n∈N ⊂ L (X).

Lemma 1.2 Let A, T ∈ L (X). Then
Proof The proof of this lemma is trivial.

Lemma 1.3 Let A ∈ L (X) and T ∈ L R(A) (X).
(1) If N r A (T k ) = N r A (T k+1 ) for some k ∈ N, then N r A (T n ) = N r A (T k ), for all non-negative integers n ≥ k. (2) If R r A (T k ) = R r A (T k+1 ) for some k ∈ N, then R r A (T n ) = R r A (T k ), for all non-negative integers n ≥ k.
Proof (1) Assume that N r A (T n+1 ) = N r A (T n ). It will be shown that N r A (T n+2 ) = N r A (T n+1 ), and then, the statement will follow by induction. The inclusion N r A (T n+1 ) ⊂ N r A (T n+2 ) is clear, so that only the converse inclusion remains to be proved. Let x ∈ N r A (T n+2 ) and let y = T Ax ∈ R(A), so that y = Ax 0 for some x 0 ∈ X and T n+1 Ax 0 = T n+2 Ax = 0. Thus, x 0 ∈ N r A (T n+1 ) = N r A (T n ). Therefore, T n+1 Ax = T n y = T n Ax 0 = 0, so that x ∈ N r A (A n+1 ), which implies (1). (2) Assume that R r A (T n ) = R r A (T n+1 ). It suffices to show that R r A (T n+1 ) = R r A (T n+2 ). Clearly, (1.2) shows that R r A (T n+2 ) ⊂ R r A (T n+1 ). To see the converse inclusion, let y ∈ R r A (T n+1 ), and then, y = T n+1 Ax, for some x ∈ X. Since T n Ax ∈ R r A (T n ) = R r A (T n+1 ), it follows that there exists z ∈ X, such that T n Ax = T n+1 Az, and hence, y = T (T n Ax) = T n+2 Az ∈ R r A (T n+2 ). This leads to R r A (T n+1 ) ⊂ R r A (T n+2 ) and the proof is complete.
, and then, the statement will follow by induction.
it follows that T n x = T n+1 z + w, for some z ∈ X and w ∈ N(A). Therefore . This completes the proof.
For an operator A ∈ L (X), the statements in Lemmas 1.2 and 1.3 lead to the introduction of the A-right ascent and the A-right descent of T ∈ L R(A) (X) by )}, respectively, whenever these minima exist. If no such numbers exist, the A-right ascent and A-right descent of T are defined to be ∞. Likewise, the statements in Lemmas 1.2 and 1.4 lead to the introduction of the A-left ascent and the A-left descent of T ∈ L N(A) (X) by respectively, where the infimum over the empty set is taken to be infinite. The case when A = I, represents the ascent and descent of T , that is . Remark 1.5 Suppose that X = H is a Hilbert space, T is a bounded operator on H, and A is a positive operator on H.
(1) Recall that T is an A-isometry (see [2,Definition 3.1]) if T x A = x A , for all x ∈ H. We say that T is an isometry if T is an I -isometry. Obviously, if T is an isometry, then T is injective, i.e., asc(T ) = 0. From [2, Proposition 3.2], we know also that if T is an A-isometry, then T (N(A)) ⊂ N(A) and asc A (T ) = 0.
(2) T is said to be an A-quasi-isometry if T * AT = T * 2 AT 2 (see [16]). If the relation is verified with A = I, T is called a quasi-isometry. Suppose that T is an A-quasi-isometry. Since , it follows by Remark 1.1 that asc A (T ) ≤ 1.
. This gives that asc r Indeed, assume that asc(T ) = n < +∞ and x ∈ N r A (T n+1 ), then Ax ∈ N(T n+1 ) = N(T n ). Thus, x ∈ N r A (T n ) and by (1.1), we deduce that N r A (T n ) = N r A (T n+1 ). Hence, asc r A (T ) ≤ n. Next, we give examples of quantities introduced above. Example 1.7 Let L , S ∈ L (X) and let T ∈ L (X × X) be the operator defined by The ascent and descent of T are given by asc(T ) = max{asc(S), asc(L)}, des(T ) = max{des(S), des(L)}. (1.4) Let P ∈ L (X) be a projection onto N(L) and Q ∈ L (X) be a projection with kernel R(L). Consider the operators A and B on X × X defined by Clearly, R(A) and N(B) are invariant subspaces of T and for all n ∈ N\{0}. This shows that (1.6) We will distinguish four cases : • Case 1 : If asc(L) = 0 (i.e., N(L) = {0} and P = 0), then in (1.5), the equality T n A(x, y) = (S n (x), 0) remains valid for n = 0, and hence asc r A (T ) = asc(S) = asc(T ), des r A (T ) = des(S) ≤ des(T ).
For the relationships between the descent and the B-left descent of T, we have des B (T ) = des(T ) if des(L) = 1 or des(L) ≤ des(S) and des B (T ) < des(T ) when 1 ≤ des(S) < des(L). We also have des B (T ) < +∞, but des(T ) = +∞ when des(S) < des(L) = +∞.  (3) Let T be a n-left generalized partial isometry in H (see [6]), i.e., T n T * T = T n for some n ∈ N, where T * is the adjoint of T. Let A = T * T, we suppose that T (R(A)) ⊂ R(A). First of all, we see that T k T * T = T k , for all k ≥ n, which shows that asc r A (T ) < +∞ (resp., des r A (T ) < +∞) if and only if asc(T ) < +∞ (resp., des(T ) < +∞). We also see that des r A (T ) ≤ max{n, des(T )}, and by Remark 1.6, we have asc r A (T ) ≤ asc(T ).
(4) Let T be a n-right generalized partial isometry in H (see [6]), i.e., T T * T n = T n for some n ∈ N. Let A = T T * , and assume that T (N(A)) ⊂ N(A). First, from Remark 1.6, we have des A (T ) ≤ des(T ).
Since AT i = T i , for all i ≥ n, it follows that asc A (T ) = +∞ (resp., des A (T ) = +∞) if and only if asc(T ) = +∞ (resp., des(T ) = +∞). We also deduce that The paper is organized as follows. In the next section, we first established some algebraic lemmas that will be used throughout this work. More precisely, the section presents isomorphism type results for the A-left (resp., A-right) kernel and the A-left (resp., A-right) range of a non-negative power of a linear operator (see [8,18] for the case A is injective or surjective). In Remark 2.4, we show that if A an operator injective (resp., surjective), then asc A (T ) = asc(T ) and des A (T ) = des(T ) (resp., asc r A (T ) = asc(T ) and des r A (T ) = des(T )), for all linear operator T. In the third section, several results related to the ascent and descent for a linear operator on a linear space [8,18] are extended to A-left (resp., A-right) ascent and descent. In Sect. 4, we are concerned with the stability of the A-left (resp., A-right) ascent and descent of an operator T, under perturbations by compact operators and operators having some finite rank power commuting with T. Finally, Sect. 5 is devoted to the study of relationship between the ascent and descent of an A-adjoint of an operator T and A-left (resp., A-right) ascent and descent of T.

Some preliminary lemmas
The linear spaces Y and Z are said to be isomorphic whenever there exists a one-one linear mapping from Y onto Z. For abbreviation, we use the symbol Y ∼ = Z to denote that Y and Z are isomorphic. Let M and N be linear subspaces in the linear space X.
In this section, we prove some algebraic results needed in this paper. We start this section with the following remark.

Lemma 2.2 Let A, T ∈ L (X) and n, k ∈ N. Then
Proof Let n, k ∈ N, S = T n A and L = T k , by Remark 2.1, we get Suppose that T (N(A)) ⊂ N(A), and then, Denote by x the equivalence class of x relative to the quotient space Hence, θ(x) = T n x defines a linear operator from N A (T n+k ) to the quotient space Y. It is easy to see that θ is surjective and N(θ ) = N A (T n ). This proves that The proof is therefore complete.
Proof It follows from Remark 2.1 that: where x denotes the equivalence class in the quotient space R r The proof is complete.
according to Remark 2.1 and Lemma 2.3. This leads to Which shows that Proof Denote x the equivalence class of x relative to the quotient space Hence Next, it is shown that J is surjective and which implies that J induces an isomorphism between the spaces

This implies that
, it follows that: Thus, J is surjective, and the proof is complete.
Proof First of all, we observe that it follows from [8, Lemma 2.2] and Lemma 2.5 that: and so, once again by [8,Lemma 2.2], The proof is complete.
). To show that J is surjective and and the mapping J induces an isomorphism between the quotient spaces (N(T i+1 ) ∩ R(A))/N(J ) and R(J ). The proof is complete.
Finally, by applying [8, Lemma 2.2], we deduce that , and the proof is complete.

Lemma 2.9 Let A, T ∈ L (X).
Then Proof Let k be a non-negative integer. From Remark 2.1, it follows that: .
The proof is complete.
The following result relates the A-right (resp., A-left) nullity and A-right (resp., A-left) defect of a linear operator to that of its powers.
Suppose we have shown its validity for 1 ≤ k ≤ n. Then, we can complete the proof by showing To prove the second inequality in (1).
(2) First, we prove by induction that Then, we deduce from Lemma 2.2 that and apply Lemmas 2.3 and 2.9, and then Hence, the inequality β A (T k ) ≤ k β A (T ) is valid for k = n + 1, and our proof is complete.

A-left ascent, A-left descent, A-right ascent, and A-right descent
Part of the results proved in this section improve and generalize some results of A. E. Taylor and M. A.
We first present some remarks.
Indeed, assume that des r A (T ) ≥ k + 1, and then This leads to dim R(A) ≥ k + 1, contrary to assumption. Thus, des r A (T ) ≤ k. Now, assume that asc r A (T ) ≥ k + 1, and then, by Remark 2.1, we have It follows from this that dim R(A) ≥ k + 1, which contradicts the fact that dim R(A) = k. We must therefore conclude that asc r A (T ) ≤ k. Finally, as in ( * ), we can prove that des A (S) ≤ k.
Indeed, assume that asc A (T ) ≥ k + 1, and then, we obtain and hence, dim X/N(A) ≥ k + 1. This contradiction shows that asc A (T ) ≤ k. Now, assume that des A (T ) ≥ k + 1, and then, by Remark 2.1, we see that This gives that dim X/N(A) ≥ k + 1, contrary to assumption. Thus, des A (T ) ≤ k. Finally, as in ( * ), we can prove that asc r A (S) ≤ k. Arguing as in [18, Lemmas 3.1 and 3.2], with Lemma 1.2, we get the following remark.
Let p ∈ N and k ∈ N\{0}. Recall from [11, Theorems V.6.3 and V.6.4] that for a linear operator T ∈ L (X), one has asc(T ) ≤ p (resp., In the following, we will discuss these properties when T has finite A-left ascent or A-left descent or A-right ascent or A-right descent.
In the rest of this section, the interrelations between A-left (resp., A-right) nullity and A-left (resp., A-right) defect, and A-left (resp., A-right) ascent and A-left (resp., A-right) descent are studied.
, and hence, we have by However, this implies p = asc r A (T ) ≤ q, contradicting the assumption p > q. Hence, we must have p ≤ q. It remains to prove that the case in which p < q is impossible. From p < q, we infer that N r so that p ≤ des r A (T ) = q. This contradicts the fact that p < q. Hence, we must have asc r A (T ) = des r A (T ). (2) If p = asc A (T ) > des A (T ) = q, then by definition R A (T p ) = R A (T q ) and in particular which implies that p ≤ q. This contradiction shows that p ≤ q. It remains to prove that the case in which p < q is impossible. The inequality p < q implies that N A (T p ) = N A (T q ). Therefore, by Lemma 2.3, we obtain this gives that q = des A (T ) ≤ p, a contradiction. Hence, p = q, and this completes the proof.
A combination of Theorems 3.4, 3.5, and 3.6 leads to the following result.
(a) If there exists k ∈ N\{0}, such that (a) If there exists k ∈ N\{0}, such that If either the A-left (resp., A-right) ascent or the A-left (resp., A-right) descent is finite, it is possible to obtain inequalities involving the A-left (resp., A-right) nullity and the A-left (resp., A-right) defect.

Theorem 3.8 Let A ∈ L (X) and T ∈ L N(A) (X). Suppose that p = asc A (T ) < +∞, and then
.
(2) Let k be a non-negative integer. First of all, we observe that it follows from (1. According to Lemma 2.10, β A (T k ) < +∞, because β(A) ≤ β A (T ) < +∞. Now, by Lemma 2.3, we can see the following: On the other hand, since Then, from (3.10) and [8,Lemma 2.2], it follows that: Hence, by (3.9) and Lemma 2.2, we infer that Lemma 2.10), and so The proof is complete.
(2) Let k be a non-negative integer. .
Then, we deduce from Lemma 2.3 that . .

Since α r A (T ) = dim N r A (T )/N(A)+α(A) and N(T )∩R r A (T k ) ⊂ N(T )∩R(A), by Remark 2.1 and Lemma 2.2, we have
.
Combining these facts with formula (3.12) and using the hypothesis α r .

Furthermore, since R A (T ) ⊂ R(A) ⊂ X and N(A) ⊂ N A (T ), it follows that:
If, in addition, asc A (T ) < +∞, by Theorem 3.6, asc A (T ) = q and by Theorem 3.5, we deduce that and so, (3.14) gives and as a consequence of Lemma 2.2 and [8, Lemma 2.2], we have This implies that asc A (T ) < +∞ and the proof is complete.

Since R r A (T ) ⊂ R(A) ⊂ X and N(A) ⊂ N r A (T ), then β r A (T ) = dim R(A)/R r A (T ) + β(A) and α r A (T ) = dim N r A (T )/N(A) + α(A). Hence, by (3.17) and Remark 2.1, we infer that
and by (3.17), we get and Theorem 3.4 implies that asc r A (T ) < +∞. This completes the proof.

Compact perturbations
In this section, some known results related to the stability of the ascent and descent of an operator [9, Sections 2 and 3] are extended to A-left (resp., A-right) ascent and A-left (resp., A-right) descent.
We start our study with the following algebraic results for later use.

Lemma 4.1 Let A ∈ L (X) and C, T ∈ L N(A) (X). Suppose that C commutes with T. Then, for n, k
Proof Let n, k ∈ N\{0} and x ∈ N A (T n ), and then, there exist α 1 , · · · , α n+k−1 ∈ N, such that On the other hand, by (1.3), we have x ∈ N A (T n ) ⊂ N A (T i ), for all i ≥ n, this implies that T i x ∈ N(A), and as C(N(A)) ⊂ N(A), we infer that C n+k−1−i T i x ∈ N(A). Thus and hence It follows from this that Now, let us show the second inequality in Lemma 4.1. Let Since the a i are not all zero, , which completes the proof.

Lemma 4.2 Let
A ∈ L (X) and C, T ∈ L R(A) (X). Suppose that C commutes with T. Then, for n, k ∈ N\{0} Proof Let n, k ∈ N\{0} and x ∈ N r A (T n ). Then, there exist constants α 1 , · · · , α n+k−1 , such that However, by (1.1), we have N r A (T n ) ⊂ N r A (T i ), for all i ≥ n, and this implies that To prove the second inequality in Lemma 4.2, let x 1 , · · · , x m ∈ X, such that T n+k−1 Ax 1 , · · · , T n+k−1 Ax m are linearly independent in R r . Since −C commutes with T + C and C, T + C ∈ L R(A) (X), it follows that, for i = 1, · · · , m . Hence , then there exist constants a 1 , · · · , a m not all zero, such that Since the a i are not all zero, and the proof is complete.
The following two theorems generalize [9, Theorem 2.2]. Proof Suppose that asc A (T ) = p < +∞. For n ≥ p, let By Lemma 4.1, a n ≤ dim R A (C k ) < +∞. According to (1.3), the spaces N A [(T +C) i ] i∈N are an increasing nest of subspaces, it follows that there is an integer N ≥ p, such that a n = a N for n ≥ N . However, this implies On the other hand, we may interchange T by T + C and C by −C in Lemma 4.1 to conclude that For the converse implication, if asc A (T + C) < +∞, then by the direct implication, asc A (T ) = asc A (T + C − C) < +∞.
The proof for the case when T has finite, A-left descent is similar and will be omitted. This completes the proof.
Arguing as in the proof of [9,Theorem 2.2] or Theorem 4.3, with Lemma 4.2, we get the following theorem.

Theorem 4.4 Let A ∈ L (X) and C, T ∈ L R(A) (X). Suppose dim R r A (C k ) < +∞ for some integer k ≥ 1 and C commutes with T. Then, T has finite A-right ascent (resp., A-right descent) if and only if T + C has finite A-right ascent (resp., A-right descent).
For C, A ∈ L (X) and k ∈ N\{0}, we have dim R r A (C k ) < +∞ and dim R A (C k ) < +∞, when dim R(C k ) < +∞. Then, as a consequence of Theorems 4.3 and 4.4, we obtain the following corollary.

Corollary 4.5
Let A, C, T ∈ L (X), such that dim R(C k ) < +∞ for some integer k ≥ 1 and C commutes with T.
(1) If C, T ∈ L N(A) (X), then T has finite A-left ascent (resp., A-left descent) if and only if T + C has finite A-left ascent (resp., A-left descent).
, then T has finite A-right ascent (resp., A-right descent) if and only if T + C has finite A-right ascent (resp., A-right descent).
Now, we suppose that X is a Banach space, and let B(X) be the algebra of bounded linear operators on X. The next lemma is used to show Lemma 4.7.
is closed for all n ∈ N and λ ∈ C\{0}.
Proof We begin with n = 1. First, we see that R r A (λI − T ) = R(S λ ), where S λ : R(A) −→ X is the operator given by S λ (x) := (λI − T )x. We now argue as in the proof of [19,Theorem 5 where B is a certain bounded operator. Now, by [11,Theorem V.7.2], K := T B is a compact operator. The foregoing reasoning, applied to λ n − K , shows that R r A [(λI − T ) n ] is closed. This completes the proof. The next lemma is used to prove Theorems 4.8 and 4.9.  Proof (1) We can prove that A-left ascent of λI − T is finite similarly as in the proof of [11,Theorem V.7.9]. Indeed, suppose asc A (λI − T ) = +∞. Then, N A [(λI − T ) n−1 ] is a proper closed subset of N A [(λI − T ) n ] for n = 1, 2, · · · . By [11,Theorem II.3.5], there exists x n ∈ N A [(λI − T ) n ], such that x n = 1 and . Assume 1 ≤ m < n and let Then . Consequently x n − z ≥ 1 2 . However, we easily calculate that T x n − T x m = λ(x n − z), and so, T x n − T x m ≥ λ 2 > 0. This shows that {T x n } can have no convergent subsequence, in contradiction to the fact that T is compact. Thus, asc A (λI − T ) must be finite.
(2) The proof that des r A (λI − T ) is finite is similar. We first observe in Lemma 4.6 that R r A [(λI − T ) n ] is closed, for all n ∈ N. If des r A (λI − T ) = +∞, R r A [(λI − T ) n+1 ] would be a proper closed subset of R r A [(λI − T ) n ] for n = 0, 1, · · · . We choose y n ∈ R r A [(λI − T ) n ], so that y n = 1 and y n − y ≥ 1 We can write y k = (λI − T ) k Ax k for some x k ∈ X. Thus . Therefore, y m −w ≥ 1 2 . However, T y m − T y n = λ(y m −w), and so, T y m − T y n ≥ λ 2 , and we obtain a contradiction, just as in the earlier argument. This completes the proof.
Recall that if T and C are bounded operators, such that T is bijective and C is compact which commutes with T, then the ascent and descent of T + C are finite (see [9,Theorem 3.2]). We generalize this result to the A-left (resp., A-right) ascent and descent as in the following two theorems .

Theorem 4.8 Let A, C, T ∈ B(X), such that N(A) is an invariant subspace of T and C. Suppose that T is bijective and C is a compact operator which commutes with T.
(1) The A-left descent of T + C is finite.
(2) If asc A (T ) = 0, then the A-left ascent of T + C is finite.
(2) Let n be a non-negative integer. Since (T + C) n = T n (I + CT −1 ) n and asc A (T ) = 0, we get This shows that

Theorem 4.9 Let A, C, T ∈ B(X), such that R(A) is an invariant subspace of T and C. Suppose that T is bijective and C is a compact operator which commutes with T.
(1) The A-right ascent of T + C is finite.
(2) If R(A) is closed and des r A (T ) = 0, then the A-right descent of T + C is finite. Proof (1) Let us first observe from Remark 1.6 and [9, Theorem 3.2] that asc r A (T + C) ≤ asc(T + C) < +∞.
(2) Let n be a non-negative integer. Since (T + C) n = (I + CT −1 ) n T n and des r A (T ) = 0, we get

Application to an A-adjoint of an operator on a Hilbert space
This section is devoted to the study of the ascent and descent of an A-adjoint of an operator on a Hilbert space with an additional semi-inner product defined by a positive semidefinite operator A. We refer to [2,3] for more information about the A-adjoint of an operator T.
Throughout H denotes a Hilbert space with inner product · , · . For a given set M ⊆ H, the orthogonal subspace of M in H is denoted M ⊥ . For every T ∈ B(H), its adjoint is denoted by T * . Given A ∈ B(H), such that A * = A, then  if T = T. Let X be a linear space, for A, T ∈ L (X) and n ∈ N, we define the following quantities : . Lemma 1.2 shows that (A-α n (T )) n≥0 and (A-β r n (T )) n≥0 are decreasing sequences. By the same lemma, (A-α r n (T )) n≥0 (resp., (A-β n (T )) n≥0 ) is a decreasing sequence when T ∈ L R(A) (X) (resp., T ∈ L N(A) (X)). This leads to the introduction of the following new concepts : • The A-left g-ascent, g-asc A (T ), of T ∈ L (X) is defined by g-asc A (T ) = inf{n ∈ N : A-α n (T ) = 0}.
• The A-left g-descent, g-des A (T ), of T ∈ L N(A) (X) is defined by g-des A (T ) = inf{n ∈ N : A-β n (T ) = 0}.
• The A-right g-ascent, g-asc r A (T ), of T ∈ L R(A) (X) is defined by g-asc r A (T ) = inf{n ∈ N : A-α r n (T ) = 0}.
• The A-right g-descent, g-des r A (T ), of T ∈ L (X) is defined by g-des r A (T ) = inf{n ∈ N : A-β r n (T ) = 0}, where as usual the infimum over the empty set is taken to be +∞. It follows from [8, Lemmas 3.1 and 3.2] that the case when A = I represents the ascent and descent of T. Finally, we set some open problems for the generalization of ascent and descent of a linear operator in a linear space. Question 5.1 Can some results known for the ascent and descent of a linear operator be extended to the new concepts A-left (resp., A-right) g-ascent and g-descent?
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