Some Remarks on Perturbation Classes of Semi-Fredholm and Fredholm Operators

Volume 2007, Article ID 26254,10pages doi:10.1155/2007/26254

Research Article

Some Remarks on Perturbation Classes of Semi-Fredholm and Fredholm Operators

Abdelkader Dehici and Khaled Saoudi

Received 15 January 2007; Accepted 4 September 2007 Recommended by Jonathan M. Borwein

We show the existence of Banach spacesX,Y such that the set of strictly singular oper- ators᏿(X,Y) (resp., the set of strictly cosingular operatorsᏯ᏿(X,Y)) would be strictly included inF+(X,Y) (resp.,F−(X,Y)) for the nonempty class of closed densely defined upper semi-Fredholm operatorsΦ+(X,Y) (resp., for the nonempty class of closed densely defined lower semi-Fredholm operatorsΦ−(X,Y)).

Copyright © 2007 A. Dehici and K. Saoudi. This is an open access article distributed un- der the Creative Commons Attribution License, which permits unrestricted use, distri- bution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

It is well known that the work of T. Kato [1] on strictly singular operators has been the starting point of an interesting and complex domain in the operator theory, that is, Fred- holm and semi-Fredholm perturbations between two Banach spacesX andY denoted byᏲ(X,Y), (Ᏺ+(X,Y),Ᏺ−(X,Y)), it has been the object of many works studying and analysing these operators, especially the inclusion between all these classes and the sta- bility problem by passing to the dual (see [2–10]). The difficulty to study these ques- tions comes from the fact that their properties are related directly to the geometry of Banach spaces. The new thing in this paper consists in studying all these classes for closed densely defined perturbed semi-Fredholm and Fredholm operators which are not neces- sarily bounded. ForX=Y, Latrach and Dehici [11, Lemma 2.3] have shown thatᏲ(X)= Ᏺ^b(X), whereᏲ^b(X) refers to the class of Fredholm perturbations acting on Φ(X)∩ ᏸ(X), it forms the largest closed two-sided ideal in the set of Riesz operators᏾(X). The framework of the setsᏲ+(X),Ᏺ^b+(X),Ᏺ−(X), andᏲ^b₋(X) is quite different, we just notice the trivial inclusions (Ᏺ+(X)⊆Ᏺ^b+(X) andᏲ−(X)⊆Ᏺ^b₋(X)). These comments are also applicable ifX=Y. Here, by means of Heriditarily indecomposable Banach spaces of

Gowers-Maurey denoted byXGM, we will show that᏿(XGM×XGM,XGM)Ᏺ+(XGM× XGM,XGM) (resp.,Ꮿ᏿(X_GM^∗ ,X_GM^∗ ×X_GM^∗ )Ᏺ−(X_GM^∗ ,X_GM^∗ ×X_GM^∗ )), moreover, we will prove that the inclusions᏿(Z)Ᏺ^b+(Z) (resp.,Ꮿ᏿(Z^∗)Ᏺ^b₋(Z^∗)) are strict for an in- finity of Banach spacesZ.

2. Preliminaries and notations

First of all, let us start with recalling some definitions and results about Fredholm theory.

LetXandY be two Banach spaces, we denote byᏯ(X,Y) the space of closed densely defined operators fromX intoY, andᏸ(X,Y) denote the space of bounded linear op- erators fromXintoY. IfA∈Ꮿ(X,Y),N(A) (resp.,R(A)) denote the kernel (resp., the range) ofA. Setting

α(A) :=dimN(A), β(A) :=codimR(A). (2.1) The set of upper semi-Fredholm operators is defined by

Φ+(X,Y) :=

A∈Ꮿ(X,Y) :α(A)<∞

andR(A) is closed inY, (2.2) while the set of lower semi-Fredholm operators is given by

Φ−(X,Y) :=

A∈Ꮿ(X,Y) :β(A)<∞

thenR(A) is closed inY. (2.3) We denote byΦ(X,Y) the set Φ+(X,Y)Φ−(X,Y). IfA∈Φ(X,Y), the index ofA is the number i(A) :=α(A)−β(A). When X=Y, the sets ᏸ(X,Y), Ꮿ(X,Y), Φ+(X,Y), Φ−(X,Y), andΦ(X,Y) are replaced, respectively, byᏸ(X),Ꮿ(X),Φ+(X),Φ−(X), and Φ(X).

Definition 2.1. LetXandYbe two Banach spaces andT∈ᏸ(X,Y).Tis said to be strictly singular, if its restriction to every closed infinite-dimensional subspace ofX is not an isomorphism.

Let᏿(X,Y) denote the set of strictly singular operators from X intoY. In general, strictly singular operators are not compact (see [12,4]). IfX=Y,᏿(X) :=᏿(X,X) is a closed two-sided ideal ofᏸ(X) containing᏷(X) and ifXis a separable Hilbert space, then

᏿(X) :=᏷(X). For basic properties of strictly singular operators, we refer to [5,8,10,12].

LetNbe a closed subspace of a Banach spaceX. we denote byπN the quotient map X→X/N. The codimension ofN, codim(N), is defined to be the dimension of the vector spaceX/N.

Definition 2.2. LetX,Y be two Banach spaces andT∈ᏸ(X,Y).T is said to be strictly cosingular if there exists no closed subspaceNofY withcodim(N)= ∞such thatπNT: X→Y/Nis surjective.

LetᏯ᏿(X,Y) denote the set of strictly cosingular operators onX. This class of opera- tors was introduced by Pelczynski [13]. The setᏯ᏿(X,Y) is a closed subspace ofᏸ(X,Y).

IfX=Y,Ꮿ᏿(X) :=Ꮿ᏿(X,X) forms a closed two-sided ideal ofᏸ(X) (see [14]).

Definition 2.3. LetXandYbe two Banach spaces and letF∈ᏸ(X,Y).Fis called a Fred- holm perturbation ifU+F∈Φ(X,Y) wheneverU∈Φ(X,Y).Fis called an upper (resp., lower) semi-Fredholm perturbation if U+F ∈Φ+(X,Y) (resp., U+F ∈Φ−(X,Y)) wheneverU∈Φ+(X,Y) (resp.,U∈Φ−(X,Y)).

The sets of Fredholm, upper semi-Fredholm, and lower semi-Fredholm perturbations are denoted byᏲ(X,Y),Ᏺ+(X,Y), andᏲ−(X,Y), respectively. If, inDefinition 2.3, we replaceΦ(X,Y),Φ+(X,Y), andΦ−(X,Y) byΦ^b(X,Y),Φ^b+(X,Y), andΦ^b₋(X,Y), we ob- tain the setsᏲ^b(X,Y),Ᏺ+^b(X,Y), andᏲ₋^b(X,Y). These classes of operators were intro- duced and investigated in [12]. In particular, it is shown thatᏲ^b(X,Y) is a closed subset ofᏸ(X,Y), andᏲ^b(X) :=Ᏺ^b(X,X) is a closed two-sided ideal ofᏸ(X). In general, we have

᏷(X,Y)⊆᏿(X,Y)⊆Ᏺ+^b(X,Y)⊆Ᏺ^b(X,Y), (2.4)

᏷(X,Y)⊆᏿Ꮿ(X,Y)⊆Ᏺ−^b(X,Y)⊆Ᏺ^b(X,Y). (2.5) The inclusion᏿(X,Y)⊆Ᏺ+^b(X,Y) is due to Kato [1], whereas the inclusionᏯ᏿(X,Y)⊆ Ᏺ^b₋(X,Y) was proved by Vladimirski [14].

LetXbe a Banach space andR∈ᏸ(X).Ris said to be a Riesz operator ifRsatisfies Riesz-Schauder theory of compact operators. The set of all Riesz operators will be de- noted by᏾(X). However, we point out that, in general,᏾(X) is not an ideal ofᏸ(X).

Moreover, M. Schechter [7] has proved thatᏲ^b(X) is the largest closed two-sided ideal of

᏾(X). By using (2.4) and (2.5), we deduce that the classes᏷(X),᏿(X),Ꮿ᏿(X),Ᏺ^b+(X) := Ᏺ^b+(X,X), andᏲ^b₋(X) :=Ᏺ^b₋(X,X) are included in᏾(X), therefore, ifSbelongs to one of these sets, 0 is an accumulation point of its spectrum (see [15,7]).

LetXandY be two Banach spaces andA∈Ꮿ(X,Y). For everyx∈D(A) (the domain ofA), we write

x A:= x + Ax (graph norm). (2.6)

As already observed,D(A) endowed with the norm · Ais a Banach space denoted by X_A, andA, as operator fromX_AintoY, is bounded. IfD(A)⊆D(J), thenJisA-defined.

Furthermore, we have

α(A) =α(A), β(A) =β(A), R(A) =R(A), α(A+J)=α(A+J), (2.7) β(A+J)=β(A+J),R(A+J) =R(A+J). (2.8) It is clear that the relations (2.7) and (2.8) lead to

A∈Φ+(X,Y)⇐⇒A∈Φ+

X_A,Y, (2.9)

A∈Φ−(X,Y)⇐⇒A∈Φ−

X_A,Y, (2.10)

A∈Φ(X,Y)⇐⇒A∈ΦX_A,Y. (2.11)

3. Main results

We start this study by stating the following result which was established in [16].

Proposition 3.1. LetXandY be two Banach spaces, then

Ᏺ^b(X,Y)=Ᏺ(X,Y). (3.1)

Before completing our analysis, let us give some elements which will be useful after- wards. First, let us recall the definition of totally incomparable Banach spaces that has been introduced for the first time by H. Rosenthal [17].

Definition 3.2. Two infinite-dimensional Banach spacesXandYare called totally incom- parable if there exists no infinite-dimensional Banach spaceZwhich is isomorphic to a subspace ofXand to a subspace ofY.

It should be observed that every two different spaces from the set{c0}

{l_p}are totally incomparable ([4]). More precisely, we have the following.

Letp∈[1,∞[, ifp < r(resp.r < p), then

ᏸl_r,l_p=᏷l_r,l_p=᏿l_r,l_p=Ᏺ^bl_r,l_p=Ᏺl_r,l_p

(resp.,ᏸl_r,l_p=᏿l_r,l_p=Ᏺ^bl_r,l_p=Ᏺl_r,l_p=᏷l_r,l_p. (3.2) For more examples satisfying the previous identities, we can quote, for example, [18].

On the other hand, it is easy to observe that if X and Y are totally incomparable, then any bounded operator fromXintoY is strictly singular. Moreover, the definition of Fredholm perturbations allows us to establish the following result.

Lemma 3.3. LetXandYbe two Banach spaces such thatᏸ(X,Y)=Ᏺ^b(X,Y)=Ᏺ(X,Y), thenΦ^b(X,Y)= ∅.

We give now the definition of hereditarily indecomposable Banach spaces which will be used afterwards.

Definition 3.4. LetXbe a Banach space.Xis said to be indecomposable if it can not be written as a direct sum of two closed infinite-dimensional subspaces.

Definition 3.5. LetXbe a Banach space.Xis said to be hereditarily indecomposable (H.I) if all of its closed infinite-dimensional subspaces are indecomposables.

For a detailed study on these spaces, we refer to the famous results established by Gow- ers and Maurey [19,20] in which we can find an example of a separable reflexive hered- itarily indecomposable Banach space denoted byX_GMwhose dual quotients inherit this property.

Theorem 3.6 (see [21, Theorem 2.1]). LetXbe anXGMBanach space, then (a)

ᏸX_GM×X_GM,X_GM=Ᏺ^b+

X_GM×X_GM,X_GM=᏿X_GM×X_GM,X_GM, (3.3)

(b)

ᏸX^∗GM,X^∗GM×X^∗GM

=Ᏺ₋^bX^∗GM,X^∗GM×X^∗GM

=Ꮿ᏿X^∗GM,X^∗GM×X^∗GM

. (3.4)

Remark 3.7. As an immediate consequence of this theorem, we deduce that (a)

ᏸX_GM×X_GM,X_GM=Ᏺ^bX_GM×X_GM,X_GM=᏿X_GM×X_GM,X_GM, (3.5) (b)

ᏸX^∗_GM,X^∗_GM×X^∗_GM=Ᏺ^bX^∗_GM,X^∗_GM×X^∗_GM

=Ꮿ᏿X^∗_GM,X^∗_GM×X^∗_GM. (3.6) We will prove thatTheorem 3.6remains true, respectively, for the perturbation classes Ᏺ+(XGM×XGM,XGM) and Ᏺ−(X^∗GM,X^∗GM×X^∗GM); however, the proofs are more complicated.

The following lemma is essential in provingTheorem 3.10which is regarded as an ex- tension ofTheorem 3.6to the closed densely defined (unbounded) semi-Fredholm per- turbed operators.

Lemma 3.8. LetXbe anXGMBanach space, then

(a)Φ+(XGM×XGM,XGM)= ∅and (b)Φ−(X^∗GM,X^∗GM×X^∗GM)= ∅.

Proof. (a) For the classΦ+(XGM×XGM,XGM), the proof is based on the separability of the spaceX_GM×X_GM (with respect to the topology of its norm) and that of the space X^∗_GMendowed with the∗-weak topology. In fact, [3] ensures the existence of a compact injective operator with a dense range fromXGMtoXGM×XGM, this implies that the oper- atorK⁻¹:R(K)⊆XGM×XGM→XGMis a closed densely defined Fredholm operator and therefore,Φ(X_GM×X_GM,X_GM)= ∅, one sees thatΦ+(X_GM×X_GM,X_GM)= ∅.

(b) A similar approach by duality allows us to establish the result for the class of lower

semi-Fredholm operatorsΦ−(X^∗GM,X^∗GM×X^∗GM).

The next proposition, owing to Weis [9], will play a fundamental role in the proof of Theorem 3.10.

Proposition 3.9. LetY be a Banach space, then

(a)ᏸ(Y,Z)=᏿(Y,Z)Φ+(Y,Z) for every Banach spaceZif and only ifYis an hered- itarily indecomposable Banach space,

(b)ᏸ(X,Y)=Ꮿ᏿(X,Y)Φ−(X,Y) for every Banach spaceX if and only if the quo- tients ofY are hereditarily indecomposable Banach spaces.

We now prove the following theorem.

Theorem 3.10. LetXbe anXGMBanach space, then

(a)ᏸ(XGM×X_GM,X_GM)=Ᏺ+(X_GM×X_GM,X_GM)=᏿(XGM×X_GM,X_GM), (b)ᏸ(X_GM^∗ ,X_GM^∗ ×X_GM^∗ )=Ᏺ−(X_GM^∗ ,X_GM^∗ ×X_GM^∗ )=Ꮿ᏿(X_GM^∗ ,X_GM^∗ ×X_GM^∗ ).

Proof. (a) It suffices to establish the inclusionᏸ(XGM×XGM,XGM)⊆Ᏺ+(XGM×XGM,

XGM).

LetS∈Φ+(XGM×XGM,XGM), and let jbe the embedding operator fromXStoXGM× X_GMdefined by j: (D(S), · _S)=X_S→X_GM×X_GM, j(x)=x, one sees that jis strictly singular. In fact, sinceS∈Φ+(X_GM×X_GM,X_GM), the relation (2.9) shows thatS∈Φ+(X_S, XGM), this implies the existence of finite codimensional subspaceH inXS, which is iso- morphic toR(S) =R(S). Furthermore, as R(S) is a closed hereditarily indecomposable subspace ofXGM, thenH will inherit this property in the Banach spaceXS, which al- lows us to conclude thatXSis a hereditarily indecomposable Banach space. Moreover,j /∈ Φ+(X_S,X_GM×X_GM) because ifj∈Φ+(X_S,X_GM×X_GM), we would haveX_S^∼=X_GM×X_GM, this contradicts the fact thatXGM×XGM is not a hereditarily indecomposable Banach space. Next, by applying theProposition 3.9(a), we deduce that j is a strictly singular operator fromX_SintoX_GM.

Let us take now some bounded operatorT∈ᏸ(XGM×XGM,XGM), we should show first that the spacesXS+T=(D(S+T), · S+T) andXS=(D(S), · S) are isomorphic.

Indeed, letx∈X_S+T=(D(S+T), · _S+T), then x _S+T= x + (S+T)x

≤ x + S(x) + T(x)

≤ x + S(x) +M x

≤(1 +M) x + S(x)

≤(1 +M) x S.

(3.7)

Moreover, ifx∈XS, we can establish the following estimates:

x _S= x + S(x)

≤ x + (S+T)(x)−T(x)

≤ x + (S+T)(x) + T(x)

≤ x + (S+T)(x) +M x

≤(1 +M) x _S+T,

(3.8)

and finally,

x _S

(1 +M)^≤ x S+T≤(1 +M) x S, (3.9) this ensures that the spacesX_S+T andX_Sare isomorphic.

This isomorphism will be denoted byh, which is defined byh(x)=x. On the other hand, the operatorT+Sdefined byT+S:XS+T→XGM, (T+S)(x)=(T jh)(x) + (S jh)(x)

∀x∈X_S+T is an element ofΦ+(X_S+T,X_GM), this follows immediately from the fact that T jhandS jhbelong, respectively, to the classes᏿(XS+T,XGM) andΦ+(XS+T,XGM); next, by the use of the relation (2.9), we obtainT+S∈Φ+(XGM×XGM,XGM) and, therefore,

T∈Ᏺ+(XGM×XGM,XGM). Let us now consider the projection operator Pr :XGM×XGM→ XGMdefined by Pr(x,y)=x. Obviously, Pr∈ᏸ(XGM×XGM,XGM), but this operator is not strictly singular because its restriction to the subspaceX× {0}is an isomorphism;

consequently, ᏸ(XGM × XGM,XGM) = Ᏺ+(XGM × XGM,XGM) = ᏿(XGM × XGM, XGM), which achieves the proof.

(b) LetS∈Φ−(X_GM^∗ ,X_GM^∗ ×X_GM^∗ ) and letX_S=(D(S), . _S), then the operatorj^∗de- fined fromXStoX_GM^∗ byj^∗(x)=xis not an element ofΦ−(XS,X_GM^∗ ) because if we sup- pose that this is not the case, we obtainXSX_GM^∗ and, therefore,X_GM^∗ /N(S)XS/N(S) R(S), which is a finite codimensional decomposable subspace ofX_GM^∗ ×X_GM^∗ , this contra- dicts the fact that the quotients ofX_GM^∗ are indecomposable Banach spaces. Moreover, the Proposition 3.9(b) ensures that j^∗is a strictly cosingular operator.

Let us take T∈ᏸ(X_GM^∗ ,X_GM^∗ ×X_GM^∗ ); as in the proof of (a), the operatorT+Sde- fined fromX_S+T toX_GM^∗ ×X_GM^∗ can be written under the formT+S(x)=(T j^∗h)(x) + (S j^∗h)(x)∀x∈XS+T, which gives that this operator is an element of the set ofΦ−(XS+T, X_GM^∗ ×X_GM^∗ ), this follows immediately from the fact that the operatorsT j^∗handS j^∗hbe- long, respectively, to the classesᏯ᏿(X_S+T,X_GM^∗ ×X_GM^∗ ) andΦ−(X_S+T,X_GM^∗ ×X_GM^∗ ); next, by the use of the relation (2.10), we infer thatT+S∈Φ−(X_GM^∗ ,X_GM^∗ ×X_GM^∗ ) and, there- fore,T∈Ᏺ−(X_GM^∗ ,X_GM^∗ ×X_GM^∗ ), this gives thatᏸ(X_GM^∗ ,X_GM^∗ ×X_GM^∗ )=Ᏺ−(X_GM^∗ ,X_GM^∗ × X_GM^∗ ). Now consider the operatoridefined fromX_GM^∗ toX_GM^∗ ×X_GM^∗ byi:X_GM^∗ →X_GM^∗ × X_GM^∗ ,i(x)=(x, 0), this operator is not strictly cosingular. In fact, sincei∈ᏸ(X_GM^∗ ,X_GM^∗ × X_GM^∗ ), then i∈Ᏺ−(X_GM^∗ ,X_GM^∗ ×X_GM^∗ ). LetH be the closed subspace H= {(0,y),y∈ X_GM^∗ }and denote, byπ_H, the quotient mapπ_H:X_GM^∗ ×X_GM^∗ →(X_GM^∗ ×X_GM^∗ )/H. Clearly, the operatorπHoi:X_GM^∗ →(X_GM^∗ ×X_GM^∗ )/His surjective. Sincecodim(H)= ∞, we infer thatiis not strictly cosingular fromX_GM^∗ toX_GM^∗ ×X_GM^∗ . Consequently,Ᏺ−(X_GM^∗ ,X_GM^∗ × X_GM^∗ )=Ꮿ᏿(X_GM^∗ ,X_GM^∗ ×X_GM^∗ ), which ends the proof.

Given a complex Banach spaceXand an operatorT∈ᏸ(X), we define σe(T)=

λ∈C:λI−T /∈Φ^b(X), σ+(T)=

λ∈C:λI−T /∈Φ^b+(X), σ−(T)=

λ∈C:λI−T /∈Φ^b₋(X).

(3.10)

It is well known thatσ_e(T) is a nonempty compact set of the fieldCbecause it coincides with the spectrum of the image ofTin the Calkin algebraᏸ(X)/᏷(X) (see [22]). On the other hand, it is clear that

σ+(T)σ−(T)⊆σ_e(T). (3.11)

Moreover, the stability of the index of a semi-Fredholm operator under small perturba- tions [4, Proposition 2.c.9] provides the inclusions

Frσe(T)⊆σ+(T),

Frσe(T)⊆σ−(T), (3.12)

whereFr(σe(T)) denotes the boundary of the setσe(T).

The following result shows the fact that the setsσe(T) possess empty interiors in the fieldCenables us to derive some nice properties of the classes of semi-Fredholm pertur- bations. More precisely, we have the following proposition.

Proposition 3.11. LetXbe a Banach space such that

0

σe(T)= ∅for everyT∈ᏸ(X), then Ᏺ^b+(X)=Ᏺ^b−(X)=Ᏺ^b(X). (3.13) Proof. In this case, we obtain thatΦ^b(X)=Φ^b+(X)=Φ^b₋(X). In fact, letT∈ᏸ(X), from the inclusions (3.11) and (3.12), we conclude thatF_r(σ_e(T))=σ_e(T)\σ_e(T)⁰ =σ_e(T)= σ+(T)=σ−(T). To prove (3.13), it suffices to establish the identitiesΦ^b(X)=Φ^b+(X)= Φ^b₋(X). We will restrict our proof to the inclusionΦ^b+(X)⊆Φ^b(X) (the inclusionΦ^b₋(X)⊆ Φ^b(X) may be checked in the same way). This is equivalent to show thatC(Φ^b(X))⊆ C(Φ^b+(X)), whereC(Φ^b(X)) andC(Φ^b+(X)) denote, respectively, the setsᏸ(X)\Φ^b(X) andᏸ(X)\Φ^b+(X). Let us considerA∈CΦ^b(X), thenA /∈Φ^b(X), which implies that 0∈ σe(A)=σ+(A) and, consequently, A /∈Φ^b+(X), this allows us to getA∈CΦ^b+(X) and it

ends the proof.

Finally, our last result in this work is stated by the following theorem.

Theorem 3.12. Let Z be an XGM Banach space and letX=XGM×XGM··· ×XGM (n times,n≥2). Denote byY=X×Z=XGM× ··· ×XGM(n+ 1times), then

(a)

Ᏺ^b+(Y)=Ᏺ^b(Y)=᏿(Y), (3.14)

(b)

Ᏺ^b₋(Y^∗)=Ᏺ^b(Y^∗)=Ꮿ᏿(Y^∗). (3.15) Proof. First, we observe that every operatorA∈ᏸ(Y) can be written under the form

A=

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

A11 . . . A1n

. . . . . . . . . . . . . . . . . . . . An1. . . . Ann

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠ B

C D

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠

, (3.16)

whereAi j∈ᏸ(XGM)∀i,j=1,...,n, B∈ᏸ(XGM,X)=Ᏺ^b(XGM,X), C∈ᏸ(X,XGM)= Ᏺ^b(X,XGM) (becauseᏲ^b(H,M)=ᏸ(H,M) if and only ifᏲ^b(M,H)=ᏸ(M,H), see [22]), andD∈ᏸ(XGM).

Let us denote

A0=

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

A11 . . . A1n

. . . . . . . . . . . . . . . . . . . . An1. . . . Ann

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠ 0

0 D

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠

. (3.17)

We haveσ_e(A)=σ_e(A0). Since cardσ_e(A_{i j})=cardσ_e(D)=1,∀i,j=1,...,n(see [21, Proposition 3.1]), we infer that the setσe(A) consists of an isolated point with finite number inC, thus its interior is empty. Moreover, according to Proposition 3.3, we get Ᏺ^b(Y)=Ᏺ^b+(Y). ThusP_j=_0JZ

0 0

, where j_Z :X_GM→X,j_Z(x)=(x, 0,..., 0) gives us an operator inᏲ^b(Y)=Ᏺ^b+(Y), which is not strictly singular.

Second, we show that ifH is a reflexive Banach space, we obtain that [Ᏺ^b+(H)]^∗= Ᏺ^b₋(H^∗). ThusPj^∗∈Ᏺ^b₋(Y^∗). However,Pj^∗is not strictly singular because jZ^∗is sur-

jective.

References

[1] T. Kato, “Perturbation theory for nullity, deficiency and other quantities of linear operators,”

Journal d’Analyse Math´ematique, vol. 6, pp. 261–322, 1958.

[2] P. Aiena and M. Gonz´alez, “Essentially incomparable Banach spaces and Fredholm theory,” Pro- ceedings of the Royal Irish Academy. Section A, vol. 93, no. 1, pp. 49–59, 1993.

[3] S. Goldberg and A. H. Kruse, “The existence of compact linear maps between Banach spaces,”

Proceedings of the American Mathematical Society, vol. 13, pp. 808–811, 1962.

[4] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, vol. 1, Springer, Berlin, Germany, 1997.

[5] V. D. Mil’man, “Some properties of strictly singular operators,” Functional Analysis and Its Ap- plications, vol. 3, no. 1, pp. 77–78, 1969.

[6] A. Pietsch, Operator Ideals, vol. 20 of North-Holland Mathematical Library, North-Holland, Am- sterdam, The Netherlands, 1980.

[7] M. Schechter, “Riesz operators and Fredholm perturbations,” Bulletin of the American Mathe- matical Society, vol. 74, pp. 1139–1144, 1968.

[8] L. Weis, “On perturbations of Fredholm operators inLp(μ)-spaces,” Proceedings of the American Mathematical Society, vol. 67, no. 2, pp. 287–292, 1977.

[9] L. Weis, “Perturbation classes of semi-Fredholm operators,” Mathematische Zeitschrift, vol. 178, no. 3, pp. 429–442, 1981.

[10] R. J. Whitley, “Strictly singular operators and their conjugates,” Transactions of the American Mathematical Society, vol. 113, no. 2, pp. 252–261, 1964.

[11] K. Latrach and A. Dehici, “Fredholm, semi-Fredholm perturbations, and essential spectra,” Jour- nal of Mathematical Analysis and Applications, vol. 259, no. 1, pp. 277–301, 2001.

[12] I. C. Gohberg, A. S. Markus, and I. A. Fel’dman, “Normally solvable operators and ideals asso- ciated with them,” American Mathematical Society Translations, vol. 61, pp. 63–84, 1967.

[13] A. Pelczynski, “On strictly singular and strictly cosingular operators—II,” Bulletin L’Acad´emie Polonaise des Science, vol. 13, pp. 37–41, 1965.

[14] J. I. Vladimirskii, “Strictly cosingular operators,” Soviet Mathematics Doklady, vol. 8, pp. 739–

740, 1967.

[15] S. R. Caradus, W. E. Pfaffenberger, and B. Yood, Calkin Algebras and Algebras of Operators on Banach Spaces, vol. 9 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1974.

[16] A. Ben Amar, “Spectral and fixed point theories and applications to problems arising in kinetic theory of gas and in growing cell population,” Ph. D. Thesis in Mathematics, University of Sfax, Sfax, Tunisia, 2007.

[17] H. P. Rosenthal, “On totally incomparable Banach spaces,” Journal of Functional Analysis, vol. 4, pp. 167–175, 1969.

[18] M. Gonz´alez, “On essentially incomparable Banach spaces,” Mathematische Zeitschrift, vol. 215, no. 1, pp. 621–629, 1994.

[19] W. T. Gowers and B. Maurey, “The unconditional basic sequence problem,” Journal of the Amer- ican Mathematical Society, vol. 6, no. 4, pp. 851–874, 1993.

[20] W. T. Gowers and B. Maurey, “Banach spaces with small spaces of operators,” Mathematische Annalen, vol. 307, no. 4, pp. 543–568, 1997.

[21] A. Dehici and K. Saoudi, “Some remarks on perturbation classes of linear operators,” preprint, 2006.

[22] H. G. Heuser, Functional Analysis, John Wiley & Sons, Chichester, UK, 1982.

Abdelkader Dehici: D´epartement des Sciences Exactes, Universit´e 08 Mai 1945, Guelma 24000, Algeria

Email address:[email protected]

Khaled Saoudi: Centre Universitaire de Khenchela, Route de Batna, El houria, 40004 Khenchela, Algeria

Email address:[email protected]