left–right fredholm and weyl spectra of the sum of two bounded operators and applications

Mediterr. J. Math.

DOI 10.1007/s00009-013-0372-zc© Springer Basel 2013

Left–Right Fredholm and Weyl Spectraof the Sum of Two Bounded Operatorsand Applications

Hatem Baloudi and Aref Jeribi

Abstract. In this paper, we consider the sum of two bounded linearoperators defined on a Banach space and we present some new andquite general conditions to investigate their essential spectra.

Mathematics Subject Classification (2000). 47A53, 47A55, 82D75.

1. Introduction

Let X be a Banach space. We denote by L(X) (resp. C(X)) the set of allbounded (resp. closed, densely defined) linear operators on X and K(X) theideal of compact operators of L(X). For A ∈ C(X), we write D(A) ⊂ X forthe domain, N (A) ⊂ X for the null space and R(A) ⊂ X for the range of A.The nullity, α(A), of A is defined as the dimension of N (A) and the deficiency,β(A), of A is defined as the codimension of R(A) in X. Let σ(A) (resp. ρ(A))denote the spectrum (resp. the resolvent set) of A.

Sets of upper and lower Fredholm operators, respectively, are defined as

Φ+(X) = {A ∈ C(X) such that R(A) is closed in X and α(A) < ∞},and

Φ−(X) = {A ∈ C(X) such that R(A) is closed in X and β(A) < ∞}.Operators in Φ±(X) = Φ+(X)∪Φ−(X) are called semi-Fredholm oper-

ators. For such operators, the index is defined by i(A) = α(A)−β(A). If A ∈Φ+(X)\Φ(X) then i(A) = −∞ and if A ∈ Φ−(X)\Φ(X) then i(A) = +∞.The set of Fredholm operators is defined as Φ(X) = Φ+(X) ∩ Φ−(X).

Sets of left and right Fredholm operators, respectively, are defined as

Φl(X) = {A ∈ Φ+(X) such that R(A) is complemented subspace of X}and

Φr(X) = {A ∈ Φ−(X) such that N (A) is complemented subspace of X}.

H. Baloudi and A. Jeribi MJOM

An operator A ∈ C(X) is left (resp. right) Weyl if A is left (resp. right) Fred-holm operator and i(A) ≤ 0 (resp. i(A) ≥ 0) . We use Wl(X) (resp. Wr(X))to denote the set of all left (resp. right) Weyl operators.

Let

Φ−+(X) = {A ∈ Φ+(X) such that i(A) ≤ 0}

and

Φ+−(X) = {A ∈ Φ−(X) such that i(A) ≥ 0}.

A complex number λ is in ΦA, Φ−+A, Φ+

−A, Φ+A, ΦlA, Φ−A, ΦrA, WlA, WrA

or Φ±A if λ−A is in Φ(X), Φ−+(X), Φ+

−(X), Φ+(X), Φl(X), Φ−(X), Φr(X),Wl(X), Wr(X) or Φ±(X), respectively. A is said to be a Weyl operator ifA is Fredholm operator having index 0. Let Φ0(X) be the class of all Weyloperators.

Corresponding spectra of A ∈ C(X) are defined as

σe1(A) := {λ ∈ C : λ−A ∈ Φ+(X)}, the Gustafson essential spectrum,σe1l

(A) := {λ ∈ C : λ−A ∈ Φl(X)}, the left Fredholm spectrum,σe2(A) := {λ ∈ C : λ−A ∈ Φ−(X)}, the Weidmann essential spectrum,σe2r

(A) := {λ ∈ C : λ−A ∈ Φr(X)}, the right Fredholm spectrum,σe3(A) := {λ ∈ C : λ−A ∈ Φ±(X)}, the Kato essential spectrum,σe4(A) := {λ ∈ C : λ−A ∈ Φ(X)}, the Wolf essential spectrum,σe5(A) := {λ ∈ C : λ−A ∈ Φ0(X)}, the Schechter essential spectrum,σe6(A) := C\ρ6(A), the Browder essential spectrum,σe7(A) := {λ ∈ C : λ−A ∈ Φ−

+(X)}, the essential approximate pointspectrum,

σe7l(A) := {λ ∈ C : λ−A ∈ Wl(X)}, the left Weyl spectrum,

σe8(A) := {λ ∈ C : λ−A ∈ Φ+−(X)}, the essential defect spectrum,

σe8r(A) := {λ ∈ C : λ−A ∈ Wr(X)}, the right Weyl spectrum,

where ρ6(A) the set of those λ ∈ C\σe5(A) such that all scalars nearλ are in ρ(A).

In this study, we investigate the essential spectra of the sum of twobounded linear operators defined on a Banach space by means of the essentialspectra of the two operators where their products are Riesz operators (seeDefinition 2.1). More precisely, let A and B be two bounded linear operatorson a Banach space X such that AB is a Riesz operator. If AB − BA is aFredholm perturbation (see Definition 2.5), then σei

(A+B)\{0} = [σei(A)∪

σei(B)]\{0}, i = 1l, 2r, 4 and σei

(A+B)\{0} ⊂ [σei(A) ∪ σei

(B)]\{0}, i =5, 7l, 8r. If, further, ΦA is connected, then σe5(A + B)\{0} = [σe5(A) ∪σe5(B)]\{0}. Moreover, if ΦB is connected, then σei

(A+B)\{0} = [σei(A) ∪

σei(B)]\{0}, i = 7l, 8r. According to the same previous hypotheses and if

ΦA+B is connected, then σe6(A+B)\{0} = [σe6(A)∪σe6(B)]\{0}. Let Fb+(X)

(resp. Fb−(X)) the ideal of upper semi-Fredholm (resp. lower semi-Fredholm)

perturbations. If AB−BA ∈ Fb+(X) (resp. Fb

−(X)), then σe1(A+B)\{0} =

Left–Right Fredholm and Weyl spectra

[σe1(A) ∪ σe1(B)]\{0} and σe7(A + B)\{0} ⊂ [σe7(A) ∪ σe7(B)]\{0} (resp.σe2(A + B)\{0} = [σe2(A) ∪ σe2(B)]\{0} and σe8(A + B)\{0} ⊂ [σe8(A) ∪σe8(B)]\{0}). If, further, ΦA and ΦB are connected, then σe7(A+B)\{0} =[σe7(A) ∪ σe7(B)]\{0} (resp. σe8(A + B)\{0} = [σe8(A) ∪ σe8(B)]\{0}). IfAB − BA ∈ Fb

+(X) ∩ Fb−(X), then σe3(A + B)\{0} = [(σe3(A) ∪ σe3(B)) ∪

(σe1(A)∩σe2(B))∪ (σe2(A)∩σe1(B))]\{0}. Finally, we will apply the resultsdescribed above to investigate the essential spectra of the following integro-differential operator:

AHψ(x, ξ) = −ξ ∂ψ∂x

(x, ξ) − σ(ξ)ψ(x, ξ) +

1∫

−1

κ(x, ξ, ξ′)ψ(x, ξ

′)dξ

′

= THψ +Kψ

with general boundary conditions where x ∈ [−a, a], a > 0, and ξ ∈ [−1, 1].This operator describes the transport of particles (neutrons, photons, mole-cules of gas, etc.) in a plane parallel domain with a width of 2a mean freepaths. The function ψ(x, ξ) represents the number (or probability) densityof gas particles having the position x and the direction cosine of propaga-tion ξ. The variable ξ may be thought of as the cosine of the angle betweenthe velocity of particles and the x-direction. The function σ(.) and κ(., ., .)are called, respectively, the collision frequency and the scattering kernel. Theboundary conditions are modeled by

ψ|Γ−= Hψ|Γ+

where Γ− (resp. Γ+) is the incoming (resp. outgoing) part of the phase spaceboundary, ψ|Γ−

(resp. ψ|Γ+) is the restriction of ψ to Γ− (resp. Γ+) and H

is a linear bounded operator from a suitable function on Γ+ to a similar oneon Γ−.

This paper is divided into four sections. In Sect. 2, we give some pre-liminary results and notations used in the sequel of the paper. In Sect. 3,we present a new characterization of the essential spectra of the sum of twooperators. In Sect. 4, we apply the results of Sect. 3 to describe the essen-tial spectra of the one-dimensional transport operator with general boundarycondition.

2. Preliminaries Results

In this section, we recall some definitions and we give some lemmas that wewill need in the sequel.

Definition 2.1. Let X be a Banach space and R ∈ L(X). R is said to be Rieszoperator if ΦR = C\{0}.Remark 2.1. The family of Riesz operators is not an ideal of L(X) (see [3]).

Let R(X) denote the set of Riesz operators. For further information onthe family of Riesz operators, we refer to [3,11] and the reference therein.


Definition 2.2. Let X be a Banach space. An operator A ∈ L(X) is said to beweakly compact if A(B) is relatively weakly compact in X for every boundedsubset B ⊂ X.

Definition 2.3. Let X be a Banach space. An operator S ∈ L(X) is calledstrictly singular if, for every infinite-dimensional subspace M of X, therestriction of S to M is not a homeomorphism.

Let S(X) denote the set of strictly singular operators onX. The conceptof strictly singular operators was introduced in the pioneering paper by Kato[12] as a generalization of the notion of compact operators. For a detailedstudy of the properties of strictly singular operators we refer to [7,12]. Notethat S(X) is a closed two-sided ideal of L(X) containing K(X). The class ofweakly compact operators on L1-spaces (resp. C(Ω)-spaces with Ω a compactHaussdorff space) is nothing but the family of strictly singular operators onL1-spaces (resp. C(Ω)-spaces) (see [18, Theorem 1]).

Let X be a Banach space. If N is a closed subspace of X, we denoteby πN the quotient map X → X/N . The codimension of N, codim(N), isdefined to be the dimension of the vector space X/N .

Definition 2.4. Let X be a Banach space. An operator S ∈ L(X) is said to bestrictly cosingular if there exists no closed subspace N of X with codim(N) =∞ such that πNS : X → X/N is surjective.

Let CS(X) denote the set of strictly cosingular operators on X. Thisclass of operators was introduced by Pelczynski [18]. It forms a closed two-sided ideal of L(X) (cf. [20]).

Definition 2.5. Let X be a Banach space and F ∈ L(X).(i) The operator F is called Fredholm perturbation if A+ F ∈ Φ(X) when-

ever A ∈ Φ(X).(ii) F is called an upper semi-Fredholm perturbation if A + F ∈ Φ+(X)

whenever A ∈ Φ+(X).(iii) F is called a lower semi-Fredholm perturbation if A+F ∈ Φ−(X) when-

ever A ∈ Φ−(X).

We denote by F(X) the set of Fredholm perturbations and byF+(X) (resp. F−(X)) the set of upper semi-Fredholm (resp. lower semi-Fredholm) perturbations.

Remark 2.2. Let Φb(X), Φb+(X) and Φb

−(X) denote the sets of Φ(X) ∩L(X), Φ+(X)∩L(X) and Φ−(X)∩L(X), respectively. If in Definition 2.5 wereplace Φ(X), Φ+(X) and Φ−(X) by Φb(X), Φb

+(X) and Φb−(X) we obtain

the sets Fb(X), Fb+(X) and Fb

−(X).

The sets of Fredholm perturbations and semi-Fredholm perturbationswere introduced and investigated in [6]. In particular, it is shown thatFb(X), Fb

+(X) and Fb−(X) are closed two-sided ideals of L(X). In general,

we have

K(X) ⊂ S(X) ⊂ Fb+(X) ⊂ Fb(X),

K(X) ⊂ CS(X) ⊂ Fb−(X) ⊂ Fb(X).


The inclusion S(X) ⊂ Fb+(X) is due to Kato [12], whereas the inclusion

CS(X) ⊂ Fb−(X) was proved by Vladimirskii [20]. We begin with the follow-

ing lemma which is fundamental for our purpose [23, Theorem 7 and 8] and[21, Corollary 2].

Lemma 2.1. Let A ∈ L(X) and let E ∈ R(X).(i) If A ∈ Φ+(X) and AE − EA ∈ Fb

+(X), then A + E ∈ Φ+(X) andi(A+ E) = i(A).

(ii) If A ∈ Φ−(X) and AE − EA ∈ Fb−(X), then A + E ∈ Φ−(X) and

i(A+ E) = i(A).(iii) If A ∈ Φl(X) (resp. Wl(X), Φr(X), Wr(X)) and AE − EA ∈ Fb(X),

then A+E ∈ Φl(X) (resp. Wl(X), Φr(X), Wr(X)).

Remark 2.3. It is easy to see that Φ(X) = Φl(X) ∩ Φr(X) and Φ0(X) =Wl(X) ∩ Wr(X). So, it follows, immediately, from Lemma 2.1 that, ifA ∈ Φ(X) (resp. Φ0(X)) and AE − EA ∈ Fb(X), then A + E ∈Φ(X) (resp. Φ0(X)).

In the next proposition we will recall some well-Known properties of theFredholm-sets (see, for example, [5,6,19]).

Proposition 2.1. (i) ΦA is open.(ii) i(λ−A) is constant on any component of ΦA.(iii) α(λ−A) and β(λ−A) are constant on any component of ΦA except on

a discrete set of points at which they have larger values.

Lemma 2.2. [2, Theorem 1.54, p. 32] Let A ∈ L(X) and let B ∈ L(X).(i) If A ∈ Φl(X) and B ∈ Φl(X), then BA ∈ Φl(X).(ii) If A ∈ Φr(X) and B ∈ Φr(X), then BA ∈ Φr(X).(iii) If BA ∈ Φl(X), then A ∈ Φl(X).(iv) If BA ∈ Φr(X), then B ∈ Φr(X).

3. Main Result

We first prove the following useful stability result.

Theorem 3.1. Let A ∈ C(X) and B ∈ C(X). If there exists λ0 ∈ ρ(A) ∩ ρ(B)such that (A − λ0)−1 − (B − λ0)−1 ∈ R(X), then the following statementsare satisfied.

(i) If (A− λ0)−1 (B − λ0)

−1 − (B − λ0)−1 (A− λ0)

−1 ∈ Fb+(X), then

σei(A) = σei

(B), i = 1, 7.(ii) If (A− λ0)

−1 (B − λ0)−1 − (B − λ0)

−1 (A− λ0)−1 ∈ Fb

−(X), thenσei

(A) = σei(B), i = 2, 8.

(iii) If (A− λ0)−1 (B − λ0)

−1 − (B − λ0)−1 (A− λ0)

−1 ∈ Fb+(X) ∩ Fb

−(X),then σe3(A) = σe3(B).

(iv) If (A− λ0)−1 (B − λ0)

−1 − (B − λ0)−1 (A− λ0)

−1 ∈ Fb(X), thenσei

(A) = σei(B), i = 1l, 2r, 4, 5, 7l, 8r.

(v) If the hypotheses of (iv) is satisfied and if C\σe5(A) and C\σe5(B) areconnected, then σe6(A) = σe6(B).


To prove Theorem 3.1, we first prove the following lemma:

Lemma 3.1. Let A ∈ C(X) such that 0 ∈ ρ(A). Then for λ = 0, we haveλ ∈ σei

(A) if and only if 1λ ∈ σei

(A−1), for i = 1l, 2r, 7l, 8r.

Proof. We can write for λ = 0A− λ = −λ (

A−1 − 1λ

)A.

Since, A is one to one and onto, then

N (A− λ) = N (A−1 − 1λ

) and R(A− λ) = R(A−1 − 1λ

).

This shows that λ ∈ ΦlA (resp.ΦrA) if and only if 1λ ∈ ΦlA−1 (resp. ΦrA−1)

and we have i(A − λ) = i(A−1 − 1λ ). Therefore, we infer that λ ∈ σei

(A) ifand only 1

λ ∈ σei(A−1), for i = 1l, 2r, 7l, 8r. �

Proof of Theorem 3.1 Let R ∈ R(X) such that (A−λ0)−1 = (B−λ0)−1 +R.Then,

R(B−λ0)−1−(B − λ0)−1R=(A− λ0)−1(B−λ0)−1 − (B−λ0)−1(A−λ0)−1.

(i) Since (A−λ0)−1(B−λ0)−1 − (B−λ0)−1(A−λ0)−1 ∈ Fb+(X), applying

Lemma 2.1 (i), we find that Φ+(A−λ0)−1 = Φ+(B−λ0)−1 and Φ−+(A−λ0)−1

= Φ−+(B−λ0)−1 .

Again, applying [1, Theorem 2] we infer that Φ+A = Φ+B andΦ−

+A = Φ−+B .

(ii) A similar reasoning as before.(iii) Since, the equalities σe3(A) = σe1(A) ∩ σe2(A) and σe3(B) = σe1(B) ∩

σe2(B) are known and (A−λ0)−1(B−λ0)−1 − (B−λ0)−1(A−λ0)−1 ∈Fb

+(X) ∩ Fb−(X), then by (i) and (ii) we deduce that

σe3(A) = σe3(B).

(iv) Since (A− λ0)−1(B − λ0)−1 − (B − λ0)−1(A− λ0)−1 ∈ Fb(X), then byLemma 2.1 (iii), Remark 2.3 and Lemma 3.1, we have

σei(A) = σei

(B), i = 1l, 2r, 4, 5, 7l, 8r.

(v) The sets C\σe5(A) and C\σe5(B) are connected. So, using [9, Lemma3.1], we deduce that

σe5(A) = σe6(A) and σe5(B) = σe6(B).

So, (iv) gives

σe6(A) = σe6(B).

Remark 3.1. The Theorem 3.1 generalizes [10, Theorem 3.3] for the Rieszoperators and extends this result for the different essential spectra.


Theorem 3.2. Let A and B be tow bounded linear operators on a Banachspace X such that AB is a Riesz operator.

(i) If AB −BA ∈ Fb+(X), then

σe1(A+B)\{0} = [σe1(A) ∪ σe1(B)] \{0} (3.1)

and

σe7(A+B)\{0} ⊂ [σe7(A) ∪ σe7(B)] \{0}.If, further, ΦA and ΦB are connected, then

σe7(A+B)\{0} = [σe7(A) ∪ σe7(B)] \{0}. (3.2)

(ii) If AB −BA ∈ Fb−(X), then

σe2(A+B)\{0} = [σe2(A) ∪ σe2(B)] \{0}and

σe8(A+B)\{0} ⊂ [σe8(A) ∪ σe8(B)] \{0}.Moreover, if ΦA and ΦB are connected, then

σe8(A+B)\{0} = [σe8(A) ∪ σe8(B)]\{0}.(iii) If AB −BA ∈ Fb

+(X) ∩ Fb−(X), then

σe3(A+B)\{0} = [(σe3(A) ∪ σe3(B)) ∪ (σe1(A) ∩ σe2(B)) ∪ (σe2(A)∩σe1(B))] \{0}.

(iv) If AB −BA ∈ Fb(X), then

σei(A+B)\{0} = [σei

(A) ∪ σei(B)] \{0}, i = 1l, 2r, 4.

and

σei(A+B)\{0} ⊂ [σei

(A) ∪ σei(B)] \{0}, i = 5, 7l, 8r.

If, further, ΦA is connected, then

σe5(A+B)\{0} = [σe5(A) ∪ σe5(B)] \{0}. (3.3)

Moreover, if ΦB is connected, then

σei(A+B)\{0} = [σei

(A) ∪ σei(B)] \{0}, i = 7l, 8r.

(v) If the hypotheses of (iv) are satisfied and if C\σe5(A+B) is connected,then

σe6(A+B)\{0} = [σe6(A) ∪ σe6(B)] \{0}.To prove Theorem 3.2, we first prove the following lemma:

Lemma 3.2. Let A ∈ C(X). If ΦA is connected and ρ(A) is not empty, then

σe1l(A) = σe7l

(A) and σe2r(A) = σe8r

(A).


Proof. Since the inclusion σe1l(A) ⊂ σe7l

(A) (resp. σe2r(A) ⊂ σe8r

(A)) isknown, it suffices to show that σe7l

(A) ⊂ σe1l(A) (resp. σe8r

(A) ⊂ σe2r(A))

which is equivalent to Cσe1l(A)∩σe7l

(A) = ∅ (resp. Cσe2r(A)∩σe8r

(A) = ∅).Let λ0 ∈ Cσe1l

(A) (resp. Cσe2r(A)). We discuss two cases.

Case 1: If λ0 ∈ ΦlA\ΦA (resp. ΦrA\ΦA), then i(A − λ0) = −∞ < 0 (resp.+ ∞ > 0). In this way we see that λ0 ∈ σe7l

(A) (resp. σe8r(A)).

Case 2: λ0 ∈ ΦA. Since ρ(A) is not empty, then there exists λ1 ∈ C such thatλ1 ∈ ρ(A) and consequently A − λ1 ∈ Φ(X) and i(A − λ1) = 0. Moreover,ΦA is connected, it follows from Proposition 2.1 that i(A− λ) is constant onany component of ΦA. Therefore i(A − λ1) = i(A − λ0). In this way we seethat λ0 ∈ σe7l

(A) (resp. σe8r(A)). �

Remark 3.2. If A ∈ L(X) and ΦA is connected, then

σe1l(A) = σe7l

(A) and σe2r(A) = σe8r

(A).

Remark 3.3. It follows, immediately, from Lemma 2.1 and Remark 2.3 that,if AB ∈ R(X) and AB−BA ∈ Fb

+(X)∪Fb−(X)∪Fb(X), then BA ∈ R(X).

Proof of Theorem 3.2 For λ ∈ C, we can write

(A− λ) (B − λ) = AB − λ (A+B − λ) , (3.4)(B − λ) (A− λ) = BA− λ (A+B − λ) , (3.5)AB (A+B − λ) − (A+B − λ)AB = A (BA−AB) + (AB −BA)B

(3.6)

and

BA (A+B − λ)−(A+B − λ)BA=(BA−AB)A+B (AB −BA) . (3.7)

(i) Let λ ∈ σe1(A) ∪ σe1(B) ∪ {0}, then (A − λ) ∈ Φ+(X) and (B −λ) ∈ Φ+(X). Using [19, Theorem 6.6, p. 129], we have (A− λ) (B − λ) ∈Φ+(X). Since AB − BA ∈ Fb

+(X), we can apply Eq. (3.6), we infer thatλAB (A+B − λ) − λ (A+B − λ)AB ∈ Fb

+(X). Also, since AB ∈ R(X),then by Lemma 2.1 (i) and Eq. (3.4), (A+B−λ) ∈ Φ+(X). So, λ ∈ σe1(A+B).Therefore

σe1(A+B)\{0} ⊂ [σe1(A) ∪ σe1(B)] \{0}. (3.8)

Now, suppose that λ ∈ σe7(A) ∪ σe7(B) ∪ {0},then

(A− λ) ∈ Φ+(X), i(A− λ) ≤ 0, (B − λ) ∈ Φ+(X) and i(B − λ) ≤ 0.

Using [19, Theorem 6.6, p. 129] and [17, Theorem 12, p.159], we have

(A− λ) (B − λ) ∈ Φ+(X) and i [(A− λ) (B − λ)] ≤ 0.

SinceAB−BA ∈ Fb+(X), then by Eq. (3.6), it is clear that λAB (A+B − λ)−

λ (A+B − λ)AB ∈ Fb+(X). Also, since AB ∈ R(X), then by Lemma 2.1 (i)

and Eq. (3.4),

(A+B − λ) ∈ Φ+(X) and i (A+B − λ) ≤ 0.


In this way we see that λ ∈ σe7(A+B) whence

σe7(A+B)\{0} ⊂ [σe7(A) ∪ σe7(B)] \{0}. (3.9)

To prove the inverse inclusion of Eq. (3.8).Suppose λ ∈ σe1(A + B) ∪ {0} then (A + B − λ) ∈ Φ+(X). Since

AB−BA ∈ Fb+(X), then by Eqs. (3.6) and (3.7), we have λAB (A+B − λ)−

λ (A+B − λ)AB ∈ Fb+(X) and λBA (A+B − λ) − λ (A+B − λ)BA ∈

Fb+(X). Also, since AB ∈ R(X) and BA ∈ R(X), then by Eqs. (3.4), (3.5)

and Lemma 2.1 (i), we have

(A− λ) (B − λ) ∈ Φ+(X) and (B − λ) (A− λ) ∈ Φ+(X).

Again, using [17, Theorem 6, p. 157], we have (A−λ) ∈ Φ+(X) and (B−λ) ∈Φ+(X). Hence λ ∈ σe1(A) ∪ σe1(B). Therefore

[σe1(A) ∪ σe1(B)] \{0} ⊂ σe1(A+B)\{0}.This proves that Eq. (3.1). Now, it remains to prove that

[σe7(A) ∪ σe7(B)] \{0} ⊂ σe7(A+B)\{0}.Let λ ∈ σe7(A+B) ∪ {0} then (A+B − λ) ∈ Φ+(X) and i(A+B − λ) ≤ 0.Since AB−BA ∈ Fb

+(X), AB ∈ R(X) and BA ∈ R(X), a similar reasoningas before, it is clear that (A− λ) ∈ Φ+(X) and (B − λ) ∈ Φ+(X).

Let λ0 ∈ ρ(A) and λ1 ∈ ρ(B), then (A − λ0) ∈ Φ(X), (B − λ1) ∈Φ(X), i(A−λ0) = 0 and i(B−λ1) = 0. Since ΦA and ΦB are connected, byProposition 2.1, we have i(A − λ) is constant on any component of ΦA andi(B−λ) is constant on any component of ΦB , then i(A−λ) = i(A−λ0) = 0for all λ ∈ ΦA and i(B − λ) = i(B − λ1) = 0 for all λ ∈ ΦB . On the otherhand, for λ2 ∈ Φ+A\ΦA and λ3 ∈ Φ+B\ΦB ,

i(A− λ2) = −∞ and i(B − λ3) = −∞.

So, i(A − λ) ≤ 0 and i(B − λ) ≤ 0. This proved that λ ∈ σe7(A) ∪ σe7(B)whence

[σe7(A) ∪ σe7(B)] \{0} ⊂ σe7(A+B)\{0}.(ii) The proof of (ii) may be checked in the same way as the proof

of (i).(iii) Since, the equalities σe3(A) = σe1(A) ∩ σe2(A), σe3(B) = σe1(B) ∩

σe2(B) and σe3(A + B) = σe1(A + B) ∩ σe2(A + B) are known, AB ∈R(X), BA ∈ R(X) and AB − BA ∈ Fb

+(X) ∩ Fb−(X) then, by (i) and (ii)

we deduce that

σe3(A+B)\{0} = [(σe3(A) ∪ σe3(B)) ∪ (σe1(A) ∩ σe2(B)) ∪ (σe2(A)∩σe1(B))] \{0}.

(iv) Let λ ∈ σei(A) ∪ σei

(B) ∪ {0}, i = 1l, 2r, 4. Then, (A − λ) ∈Φl(X) (resp. Φr(X), Φ(X)) and (B − λ) ∈ Φl(X) (resp. Φr(X), Φ(X)).Therefore Lemma 2.2 and [17, Theorem 5, p. 156] gives

(A− λ) (B − λ) ∈ Φl(X) (resp. Φr(X), Φ(X)) .


Since AB − BA ∈ Fb(X). Then by Eq. (3.6), we have λAB(A + B − λ) −λ(A + B − λ)AB ∈ Fb(X). By Eq. (3.4), Lemma 2.1 (iii) and Remark 2.3,it is clear that

(A+B − λ) ∈ Φl(X) (resp. Φr(X), Φ(X)).

So, λ ∈ σei(A+B), i = 1l, 2r, 4. This proved that


(A) ∪ σei(B)] \{0}, i = 1l, 2r, 4. (3.10)

Let λ ∈ σei(A) ∪ σei

(B) ∪ {0}, i = 5, 7l, 8r. Then

(A− λ) ∈ Φ(X) (resp. Φl(X), Φr(X)), i(A− λ) = 0 (resp. ≤ 0, ≥ 0),(B−λ) ∈ Φ(X) (resp. Φl(X), Φr(X))andi(B − λ)=0 (resp. ≤0, ≥0).

Using [19, Theorem 2.3, p.111], Lemma 2.2 and [19, Theorem 12, p.153] , wehave (A− λ) (B − λ) ∈ Φ(X) (resp. Φl(X), Φr(X)) andi [(A− λ) (B − λ)] = 0 (resp. ≤ 0, ≥ 0). Moreover, since AB−BA ∈ Fb(X).Then by Eq. (3.6), we have λAB (A+B − λ) − λ (A+B − λ)AB ∈ Fb(X).Also, since AB ∈ R(X), then by Eq. (3.4), Remark 2.3 and Lemma 2.1, wehave

(A+B − λ) ∈ Φ(X) (resp. Φl(X), Φr(X)) and i(A+B − λ) = 0.

In this way we see that λ ∈ σei(A+B), i = 5, 7l, 8r, whence


(A) ∪ σei(B)] \{0}, i = 5, 7l, 8r. (3.11)

To prove the inverse inclusions of Eqs. (3.10) and (3.11).Suppose λ ∈ σei

(A + B) ∪ {0}, i = 1l, 2r, 4, then (A + B − λ) ∈Φl(X) (resp. Φr(X), Φ(X)). Since AB − BA ∈ Fb(X), then by Eqs. (3.6)and (3.7), it is clear that λAB(A + B − λ) − λ(A + B − λ)AB ∈ Fb(X)and λBA(A+B − λ) − λ(A+B − λ)BA ∈ Fb(X). Also, since AB ∈ R(X)and BA ∈ R(X), then by Eqs. (3.4), (3.5), Lemma 2.1 and Remark 2.3, wehave (A − λ)(B − λ) ∈ Φl(X) (resp. Φr(X), Φ(X)) and (B − λ)(A − λ) ∈Φl(X) (resp. Φr(X), Φ(X)). So, by Lemma 2.2 and [17, Theorem 6 (iii), p.157], it is clear that (A − λ) ∈ Φl(X) (resp. Φr(X), Φ(X)) and (B − λ) ∈Φl(X) (resp. Φr(X), Φ(X)). Therefore λ ∈ σei

(A) ∪ σei(B), i = 1l, 2r, 4.

This proved that

[σei(A) ∪ σei

(B)] \{0} ⊂ σei(A+B)\{0}, i = 1l, 2r, 4.

Now, it remain to prove that

[σei(A) ∪ σei

(B)] \{0} ⊂ σei(A+B)\{0}, i = 5, 7l, 8r.

Suppose λ ∈ σe5(A+B)∪{0}, then (A+B−λ) ∈ Φ(X) and i(A+B−λ) = 0.Since AB−BA ∈ Fb(X), AB ∈ R(X) and BA ∈ R(X), a similar reasoningas before it is clear that (A−λ) ∈ Φ(X) and (B−λ) ∈ Φ(X). Again, we canapply Eq (3.4) , [19, Theorem 12, p. 159] and Remark 2.3 we have

i [(A− λ) (B − λ)] = i (A− λ) + i (B − λ) = 0.

Since A is bounded linear operator, we get ρ(A) = ∅. As, ΦA we have i(A−λI) = 0. Again, it is clear that i(B − λI) = 0 . We conclude λ ∈ σe5(A) ∪


σe5(B) whence

[σe5(A) ∪ σe5(B)] \{0} ⊂ σe5(A+B)\{0}.So, we prove Eq. (3.3).

Since ΦA and ΦB are connected. Then, by Lemma 3.2, we haveσe1l

(A) = σe7l(A), σe2r

(A) = σe8r(A), σe1l

(B) = σe7l(B) and σe2r

(B)= σe8r

(B). Therefore

[σe8r(A) ∪ σe8r

(B)] \{0} ⊆ σe2r(A+B)\{0} ⊆ σe8r

(A+B)\{0}.and

[σe7l(A) ∪ σe7l

(B)] \{0} ⊆ σe1l(A+B)\{0} ⊆ σe7l

(A+B)\{0}.(v) This assertion follows immediately from [1, Theorem 2.1], [9, Lemma

3.1] and Eq. (3.3).

4. Application

The aim of this section is to apply Theorems 3.1 and 3.2 to study the essentialspectra of the one-dimensional transport operator on L1-space.

Let

X1 := L1 [(−a, a) × (−1, 1) ; dxdξ] (0 < a < ∞)

and

X01 := L1 [{−a} × (−1, 0); |ξ|dξ] × L1 [{a} × (0, 1); |ξ|dξ]

:= X01,1 ×X0

2,1.

Moreover, we introduce

Xi1 := L1 [{−a} × (0, 1); |ξ|dξ] × L1[{a} × (−1, 0); |ξ|dξ]

:= Xi1,1 ×Xi

2,1.

We define the partial Sobolev space W1 by

W1 ={ψ ∈ X1 such that ξ

∂ψ

∂x∈ X1

}.

It is well known that any function ψ ∈ W1 has traces on {−a} and {a}in X0

1 and Xi1 (see, for instance, [4] or [8]). They are denoted, respectively,

by ψ0 and ψi, and represent the outgoing and the incoming fluxes.We define the advection operator TH by⎧⎪⎪⎨

⎪⎪⎩

TH : D(TH) ⊂ X1 −→ X1

ψ −→ THψ(x, ξ) = −ξ ∂ψ∂x

(x, ξ) − σ(ξ)ψ(x, ξ),

D(TH) ={ψ ∈ W1 such that Hψ0 = ψi

},

where σ(.) ∈ L∞(−1, 1) and H is the boundary operator defined by{H : X0

1 −→ Xi1,

H ∈ L(X01 , X

i1).


Let

λ∗ := lim inf|ξ|→0

σ(ξ).

Remark 4.1. The essential spectra of the operator T0 (T0 designates thestreaming operator with vacuum boundary conditions, i.e., H = 0), wereanalyzed in detail in [10] and [15, Remark 4.1]. In particular it is shown that

σei(T0) = σ(T0) = {λ ∈ C such that Reλ ≤ −λ∗}, for i = 1, . . . , 8.

So, applying Lemma 3.2, we infer that

σei(T0) = σ(T0) = {λ ∈ C such that Reλ ≤ −λ∗}, for i = 1l, 2r, 7l, 8r.

Next, we consider the collision operator K defined by{K : X1 −→ X1

ψ −→ ∫ 1

−1κ(x, ξ, ξ′)ψ(x, ξ′)dξ′,

where κ(., ., .) is a measurable function from [−a, a] × [−1, 1] × [−1, 1] to R.Observe that the operator K acts only on the variable ξ, so x may

be viewed merely as a parameter in [−a, a]. Hence we may consider K as afunction K : x ∈ [−a, a] −→ K(x) ∈ Z where Z := L(L1([−1, 1],dξ)).

In the following we will make the assumptions

(H)

⎧⎪⎪⎨⎪⎪⎩

K is a measurable, i.e.,{x ∈ [−a, a] such that K(x) ∈ O} is measurable if O ⊂ Z is open,there exists a compact subset T ⊂ Z such that K(x) ∈ T a.e. ,and finally K(x) ∈ K(L1([−1, 1],dξ)) a.e.,

where K(L1([−1, 1],dξ)) denotes the set of all compact operators onL1([−1, 1],dξ).

Lemma 4.1 [16, Lemma 2.1]. If K satisfies (H) then, for any λ ∈ C such thatReλ > −λ∗, the operator (λ− TH)−1K is weakly compact on X1.

Theorem 4.1. Suppose that the collision operator K satisfies the hypothesis(H) and the boundary operator H is weakly compact. Then

σei(AH) = {λ ∈ C such that Reλ ≤ −λ∗}, for i = 1, . . . , 8, 1l, 2r, 7l, 8r.

Proof. Let

B =∑n≥0

BλH(MλH)nGλ,

where Bλ, Mλ and Gλ are three bounded operators (for the details we referto [13]).

We have for Reλ > λ∗0, (λ−TH)−1 − (λ−T0)−1 = B. Since H is weakly

compact, so B is a Riesz operator and B(λ− T0)−1 − (λ− T0)−1B is weaklycompact. Therefore, the use of Theorem 3.1 implies that

σei(TH) = σei(T0) = {λ ∈ C such thatReλ ≤ −λ∗},for i = 1, . . . , 8, 1l, 2r, 7l, 8r.


Let λ ∈ ρ(TH) such that rσ(K(λ − TH)−1) < 1 (rσ(.) the spectral radius).Without loss of generality, we suppose that λ = 0. Therefore, we have 0 ∈ρ(TH) ∩ ρ(AH) and

A−1H = C + T−1

H ,

where

C =∑n≥1

T−1H [−T−1

H K]n.

Then, by Lemma 4.1, CT−1H is a Riesz operator and CT−1

H −T−1H C is weakly

compact. Using Theorem 3.2 we have

σei(A−1

H )\{0} = [σei(T−1

H ) ∪ σei(C)]\{0}.

= [σei(T−1

H ) ∪ {0}]\{0}= σei

(T−1H )\{0}, for i = 1, 2, 4, . . . , 8, 1l, 2r, 7l, 8r.

In the same way, using Theorem 3.2 (iii) we have

σe3(A−1H )\{0} = [(σe3(T

−1H ) ∪ σe3(C)) ∪ (σe1(T

−1H ) ∩ σe2(C)) ∪ (σe2(T

−1H )

∩σe1(B))]\{0}.= [(σe3(T

−1H )∪{0}) ∪ (σe1(T

−1H )∩{0}) ∪ (σe2(T

−1H )∩{0}]\{0}.

= σei(T−1

H )\{0}.Consequently, we get

σei(A−1

H )\{0} = σei(T−1

H )\{0}, for i = 1, . . . , 8, 1l, 2r, 7l, 8r.

Applying [1, Theorem 2] and Lemma 3.1 we obtain

σei(AH) = {λ ∈ C such that Reλ ≤ −λ∗},

for i = 1, . . . , 8, 1l, 2r, 7l, 8r.Q.E.D.

�

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Hatem Baloudi and Aref JeribiDepartement de MathematiquesUniversite de SfaxFaculte des sciences de SfaxRoute de soukra Km 3.5B.P. 11713000 SfaxTunisiae-mail: [email protected]


Received: February 14, 2013.

Revised: October 10, 2013.

Accepted: November 20, 2013.

left–right fredholm and weyl spectra of the sum of two bounded operators and applications

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