Some results on Fredholm and semi-Fredholm perturbations

In this paper, we investigate some properties of semi-Fredholm operators on Banach spaces. These results are applied to the determination of the stability of various essential spectra of closed densely deﬁned linear operators. Also, we generalize some results in the literature and we extend and unify those obtained in Jeribi (J Math Anal Appl 271:343–358, 2002), Jeribi (J Math Anal Appl 275:222–237, 2002), Jeribi (Ser Math Inf 17:35–55, 2002), Jeribi (Arch Inequal Appl 2:123–140, 2004), Latrach and Dehici (J Math Anal Appl 259:277–301, 2001), Latrach and Paoli (J Aust Math Soc 77:73–89, 2004).

for example [11,29,33]). Irrespective of whether A is bounded or not on a Banach space X , there are several definitions of the essential spectrum, most are enlargement of the continuous spectrum. Define the sets  (.) and σ e2 (.) are the Gustafson and Weidman essential spectra [10]. σ e3 (.) is the Kato essential spectrum [22]. σ e4 (.) is the Wolf essential spectrum [10,30,33]. σ e5 (.) is the Schechter essential spectrum [10,30,31] and σ e6 (.) denotes the Browder essential spectrum [10,30]. Note that all these sets are closed and, in general, we have But if X is a Hilbert space and A is self-adjoint, then all these sets coincide.
One of the central questions in the study of the essential spectra of closed densely defined operator A on Banach space X consists of showing what are the conditions that we must impose on K ∈ C(X ) in order that σ ei (A + K ) = σ ei (A), i = 1, .., 6. In this direction some authors have been interested by this question concerning the stability of the Schechter essential spectrum and they have proved the following: If K is a strictly singular operator on L p -spaces p ≥ 1, then σ e5 (A + K ) = σ e5 (A) (see [26,Theorem 3.2]). If K is a weakly compact operator on Banach spaces which possess the Dunford-Pettis property (see Definition 2.2), then σ e5 (A + K ) = σ e5 (A) (see [23,Theorem 3.2]). If K ∈ L(X ) such that (λ − A) −1 K is a strictly singular (resp., weakly compact) operator on L p -spaces p > 1 (resp., on Banach spaces which possess the Dunford-Pettis property), then σ e5 (A + K ) = σ e5 (A) (see [14,15]). In [16] Jeribi extended this analysis to the case of general Banach spaces where a detailed treatment of the Schechter essential spectrum of a closed densely defined linear operator A subjected to additive perturbations K such that (λ − A) −1 K or K (λ− A) −1 belonging to arbitrary subsets of L(X ) contained in the ideal of Fredholm perturbations. In [17,18] Jeribi extended these results to unbounded perturbations. His approach consists principally in considering the class of A-closable operator K (not necessarily bounded) which contained in the set of A-resolvent Fredholm perturbations, and of proving that [19] this author has extended the analysis used above for the Wolf essential spectrum, the Schechter essential spectrum and the Browder essential spectrum. More precisely, let X be a Banach space and let A ∈ C(X ). Let I(X ) be an arbitrary two-sided ideal of L(X ) satisfying the condition F 0 (X ) ⊂ I(X ) ⊂ J (X ), where F 0 (X ) stands for the ideal of finite rank operators. Then σ ei (A) = σ ei (A + J ), i = 4, 5 for all J ∈ I(X ) and if Cσ e5 (A) the complement of σ e5 (A) is connected and neither ρ(A) nor ρ(A + J ) is empty, then σ e6 (A) = σ e6 (A + J ). The purpose of this work is to pursue the analysis started in [12,13,[16][17][18][19][20][24][25][26][27] and to extend it to general Banach spaces.
We organize the paper in the following way : the next section is devoted to some Fredholm perturbation results and stability of the essential spectra of closed densely defined linear operators on a Banach space.

Perturbation results
In the beginning of this section we introduce some definitions.

Definition 2.1 Let X and Y be two Banach spaces and let
. These classes of operators were introduced and investigated in [6]. In particular, it is shown that F b (X, Y ) is a closed subset of L(X, Y ) and F b (X ) is a closed two-sided ideal of L(X ). In general we have In this paper, we prove that the two sets F b (X, Y ) and F(X, Y ) are equal. In fact, the inclusion The other inclusion is based on some results of Fredholm theory which are found in [31].
It must be noted that the problem of the equalities between the sets F b Definition 2.2 Let X be a Banach space. X is said to have the Dunford-Pettis property (for short property DP) if for each Banach space Y every weakly compact operator T : X −→ Y takes weakly compact sets in X into norm compact sets of Y .
The Dunford-Pettis property as defined above was explicitly defined by Grothendieck [9] who undertook an extensive study of this and related properties. It is well known that any L 1 space has the property DP [3]. Also, if Ø is a compact Hausdorff space C(Ø) has the property DP [9]. For further examples, we refer to [1] or [4, pp. 494, 479, 508 and 511]. Note that the property DP is not conserved under conjugation. However, if X is a Banach space whose dual has the property DP then X has the property DP (see, e.g., [9]). For more information, we refer to the paper by Diestel [2] which contains a survey and exposition of the Dunford-Pettis property and related topics.

Remark 2.3 It is proved in [23, Proposition 3.1] that if X is a Banach space with the property DP, then
Let A ∈ C(X, Y ), the graph norm of A is defined by where D( A) denotes the domain of A. It follows from the closedness of A that D( A) endowed with the norm . A is a Banach space. In this new space, denoted by X A the operator A satisfies Ax ≤ x A and consequently,

Definition 2.4 We say that an operator J is
Let J be an arbitrary A-bounded operator. Hence, we can regard A and J as operators from X A into Y . They will denoted by A and J , respectively. These belong to L(X A , Y ). Furthermore, we have obvious relations

Definition 2.6
Let X be a Banach space, A ∈ C(X ) and F be an arbitrary A-defined linear operator on X . We Let AF(X ), AF + (X ) and AF − (X ) designate the sets of A-Fredholm, upper A-Fredholm and lower A-Fredholm perturbations, respectively.
Proposition 2.10 Let X , Y and Z be three Banach spaces. Z )). Next, using the relation together with the compactness of the operator K J we get ( Z )). (ii) The proof of (ii) is obtained as like as the proof of (i).

Theorem 2.11 Let X , Y and Z be three Banach spaces.
(i) If the set b (Y, Z ) is not empty, then Y ) is not empty, then The proof may be sketched in a similar way to (i), it suffices to replace Proposition 2.10 (i) by Proposition 2.10 (ii).

Proposition 2.13 Let X , Y and Z be three Banach spaces. If b (Y, Z ) is not empty, then
Hence, using the last inequality we have So, K J is (AE 1 + J )-compact, which proves the claim. Next, using the relation  Y ) and S ∈ C(Z , X ). If T ∈ b (X, Y ) and S ∈ − (Z , X ), then T S ∈ − (Z , Y ).

Proof By Proposition 2.14 the operator T S is closed. Put N 1 = R(S)∩ N (T ). Since N (T ) is finite dimensional,
for some finite dimensional subspace N 2 . Obviously, R(S) ∩ N 2 = {0}. Furthermore R(S) ⊕ N 2 is closed, because R(S) is closed and dim N 2 < ∞ (see [31, Lemma 2.1, p. 107]). Next, we prove that there exists a finite dimensional subspace N 3 such that Then X 1 is closed and dim X/ X 1 = k − 1. Thus we can repeat the above reasoning for X 1 in place of X 0 . Proceeding in this way, we find k steps vectors and hence u = 0. So u −→ [T u] has the desired properties, and thus  Z ). Now arguing as in the proof of Proposition 2.13, we prove that K J is (AE 1 + J )-compact. Next, using the relation

Theorem 2.19 Let X , Y and Z be three Banach spaces. If b (Y, Z ) is not empty, then
Theorem 2.20 Let X and Y be two Banach spaces. Then

Remark 2.21
Note that for X = Y , Theorem 2.20 is nothing but Lemma 2.3 (ii) in [24].

Proof of Theorem 2.4 Clearly
, then by [31, Theorem 1.1, p. 162] there exist A 0 ∈ L(Y, X ) and K ∈ L(Y ), of finite rank such that  Y ). Now by [31, Lemma 1.7, p. 166] we see that A + F ∈ (X, Y ). This shows that F ∈ F(X, Y ) which ends the proof.

Corollary 2.22 Let X be a Banach space and A ∈ C(X ), then U AF(X ) = AF(X ). Then
Proposition 2.23 Let X and Y be two Banach spaces, A ∈ C(X, Y ). The following statements are satisfied.
Since, X A is continuously embedded in X and is dense in X , we have by [ The proof of (ii) and (iii) may be sketched in a similar way to (i).

Lemma 2.24
Let A ∈ C(X, Y ) and let J : X −→ Y be a linear operator. Assume that J ∈ U AF(X, Y ). Then Proof Since A ∈ C(X, Y ) and J ∈ U AF(X, Y ), hence as mentioned above we can regard A and J as operators from X A into Y . They will be denoted by A and J , respectively. These belong to L(X A , Y ) and we have (2.5) Observe that the assertion (ii), (iii) and (iv) are immediate. (i) Assume that A ∈ (X, Y ). Then using (2.5) we infer that A ∈ b (X A , Y ). Hence it follows from [31,Theorem 1.4 p. 108] that there A 0 ∈ L(Y, X A ) and K ∈ K(X A ) such that: This leads to Next, it follows from (2.6) that On the other hand, since J ∈ U AF(X, Y ) and A 0 ∈ L(Y, X A ), applying Theorem 2.19 we get A 0 J ∈ F(X A ).
Using the fact that K(X A ) ⊂ F(X A ) we infer that Q ∈ F(X A ). Therefore applying Proposition 3.1 (i) in [24] to (2.7), we get  Further, Proof The proofs of the items (ii), (iii), (iv) and the first part of (i) for i = 4 use Lemma 2.24 and are immediate. So, they are omitted. The proof of (i) for i = 5 is similar to (i) for i = 5 of Theorem 2.11 in [27].
Let X be a Banach space and A ∈ C(X ). The point (resp., residual, continuous) spectrum of A will be denoted by σ p (A) (resp., σ r (A), σ c (A)).

Corollary 2.27
Let X be a Banach space and A ∈ C(X ). The following statements are satisfied.
This ends the proof.