spectral mapping theorem for rakocević and schmoeger essential spectra of a multivalued linear...

Mediterr. J. Math.DOI 10.1007/s00009-014-0437-7c© Springer Basel 2014

Spectral Mapping Theorem for Rakocevicand Schmoeger Essential Spectraof a Multivalued Linear Operator

Faical Abdmouleh, Teresa Alvarez, Aymen Ammarand Aref Jeribi

Abstract. The purpose of this paper is to investigate a detailed treatmentof some subsets of essential spectrum and following we will establish thespectral mapping theorems essential approximate point spectrum anddefect spectrum of multivalued linear operator.

Mathematics Subject Classification. 47A06.

Keywords. Linear relations, essential spectra, Fredholm relations,upper (resp. lower) semi Fredholm relation.

1. Introduction

Let X and Y be two Banach spaces. A linear relation T from X to Y denotedby T ∈ LR(X,Y ) is a mapping from a subspace D(T ) ⊂ X, called the domainof T, into the collection of nonempty subsets of Y such that T (αx1 + βx2) =αT (x1) + βT (x2) for all nonzero α, β scalars and x1, x2 ∈ D(T ). A linearrelation T ∈ LR(X,Y ) is uniquely determined by its graph, G(T ), which isdefined by

G(T ) = {(x, y) ∈ X × Y : x ∈ D(T ), y ∈ Tx}.

The inverse of T is the linear relation T−1 defined by

G(T−1) = {(y, x) ∈ Y × X such that (x, y) ∈ G(T )}.

We say that, T is injective if N(T ) := T−1(0) = {0}, surjective if R(T ) :=T (D(T )) = Y. T is called bijective if it is injective and surjective. The sub-spaces N(T ), R(T ) and T (0) are called the null space, the range and themultivalued part of T , respectively. For T, S ∈ LR(X,Y ), the sum T + S isdefined by

G(T + S) := {(x, y + z) : (x, y) ∈ G(T ), (x, z) ∈ G(S)}.

F. Abdmouleh et al. MJOM

If λ ∈ K and T ∈ LR(X) := LR(X,X), the linear relation T − λ is given by

G(T − λ) := {(x, y − λx) : (x, y) ∈ G(T )}.

Let T ∈ LR(X,Y ). If QT denotes the quotient map from X onto Y/T (0)then QT T is an operator and we can define ‖Tx‖ := ‖QT Tx‖, x ∈ D(T )and ‖T‖ := ‖QT T‖. We say that T is closed if its graph is a closed subspaceequivalently if QT T is closed and T (0) is closed, continuous if ‖T‖ < ∞, openif γ(T ) > 0 where γ(T ) := sup{λ ≥ 0 : λd(x,N(T )) ≤ ‖Tx‖, x ∈ D(T )},bounded below if it is injective and open and T is called compact if QT TBX

is compact where BX is the unit ball of X.For a closed linear relation T ∈ LR(X) where X is a complex Banach

space, we introduce the following essential spectra.

σeap(T ) :=⋂

K∈KT (X)

σap(T + K),

σeδ(T ) :=⋂

K∈KT (X)

σδ(T + K),

where KT (X) := {K ∈ LR(X) : K is compact, D(T ) ⊂ D(K), K(0) ⊂T (0)}, σap(T ) := {λ ∈ C : T − λ not bounded below} and σδ(T ) := {λ ∈C : T − λ is not surjective}. For a bounded operator T , σeap(T ) was intro-duced by V. Rakocevic in [7] and denotes the essential approximate pointspectrum of T and σeδ(T ) is the essential defect spectrum of T which wasintroduced by Schmoeger in [12]. When T is a densely defined closed operator,σeap(T ) and σeδ(T ) are considered in [6], where the authors characterize suchessential spectra in terms of semi-Fredholm operators and they also obtainspectral mapping theorems for σeap(T ) and σeδ(T ) in a special case whichoccurs in applications to singular neutron transport operators. The purposeof this paper is to extend the results of [6] above mentioned to the generalcase of linear relations.

We organize the paper in the following way: The Sect. 2 contains theauxiliary properties that will need to prove the results of Sect. 3. We begingiving some results concerning the product and the perturbation of upperand lower semi-Fredholm linear relations. In the second part of Sect. 2, wepresent some properties of polynomials in a linear relation which be used forthe proof of spectral mapping theorems. The main results of this paper areestablished in Sect. 3. The Theorem 3.1 gives a characterization of σeap(T )(resp. σeδ(T )) by means of upper (resp lower) semi-Fredholm a linear rela-tions. This characterization together with some results concerning the indexof polynomial in a linear relation T are applied to obtain spectral mappingtheorems for σeap(T ) and σeδ(T ) (Theorem 3.2 below).

2. Preliminary Results

Throughout the paper, we adhered to the notations and terminology of themonographs [3] and [9]. Let X, Y and Z be Banach spaces, for T ∈ LR(X,Y )and S ∈ LR(Y,Z). The product ST is defined by

Spectral Mapping Theorem for Rakocevic

G(ST ) :={(x, z) ∈ X×Z : (x, u) ∈ G(T ) and (u, z) ∈ G(S) for some u ∈ Y }.

For T ∈ LR(X,Y ), we define α(T ) := dim N(T ), β(T ) := dimY/R(T ) andthe index of T is the quantity i(T ) := α(T ) − β(T ) provided α(T ) and β(T )are not both infinite.

Definition 2.1 [3, Definition V.1.8]. Let T ∈ LR(X,Y ) be closed. We say thatT is a Φ+ (resp. Φ−)-relation, denoted by T ∈ Φ+(X,Y ) (resp. Φ−(X,Y ), ifα(T ) < ∞ (resp. β(T ) < ∞) and R(T ) is closed. If T ∈ Φ+(X,Y )∩Φ−(X,Y ),we say that T is a Φ-relation.

The adjoint T ′ of T ∈ LR(X,Y ) is defined by

G(T ′) := G(−T−1)⊥ ⊂ Y ′ × X ′,

where for a subspace M of a normed space E,

M⊥ := {x′ ∈ E′ : x′(m) = 0 for all m ∈ M}.

Lemma 2.1. Let T ∈ LR(X,Y ) and S ∈ LR(Y,Z) be closed. Then(i) T ∈ Φ+(X,Y ) if and only if QT T ∈ Φ+(X,Y/T (0)) if and only if

T ′ ∈ Φ−(Y ′,X ′).In such case i(T ) = i(QT T ) = −i(T ′).

(ii) T ∈ Φ−(X,Y ) if and only if QT T ∈ Φ−(X,Y/T (0)) if and only ifT ′ ∈ Φ+(Y ′,X ′).In such case i(T ) = i(QT T ) = −i(T ′).

(iii) If T ∈ Φ+(X,Y ) and S ∈ Φ+(Y,Z), then ST ∈ Φ+(X,Z).(iv) If T ∈ Φ−(X,Y ), S ∈ Φ−(Y,Z) and ST is closed, then ST ∈ Φ−(X,Z).(v) If T ∈ Φ(X,Y ) and S ∈ Φ(Y,Z) then ST ∈ Φ(X,Z) and

i(ST ) = i(S) + i(T ) + dimY

R(T ) + D(S)− dim(T (0) ∩ N(S)).

Proof. The statements (i) and (ii) follow immediately from ([1, Lemma 5])combined with the closed graph theorem for linear relations [3, TheoremIII.4.2].

(iii) We first note that ST is closed by [3, Proposition II.5.17] and by [11,Lemma 5.1] we infer that α(ST ) ≤ α(S) + α(T ). It remains to show thatR(ST ) is closed. To do this, we consider the linear relation S0 := S|N(S)+R(T )

which is defined by

G(S0) := {(y, z) : y ∈ N(S) + [R(T ) ∩ D(S)], z ∈ Sy}.

Then, it is clear that S0 is closed and since S0 and S have the same null spacewe have that γ(S) ≤ γ(S0), so that S0 is open equivalently R(S0) is closed.But

R(S0) := S(N(S) + (R(T ) ∩ D(S)))= SN(S) + S(R(T ) ∩ D(S)), (see [3, Proposition I.31])= R(ST ), (see [3,Corollary I.2.10]).

(iv) Since ST is a closed linear relation and by [11, Lemma 5.1], we obtainthat β(ST ) ≤ β(S) + β(T ), then applying [4, Corollary 3.1], we infer thatR(ST ) is closed.


(v) By (iii) and (iv), we have that ST is a Fredholm relation and the propertyconcerning the index of ST follows immediately from [11, Proposition 5.2].

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Lemma 2.2. Let S, T ∈ LR(X) such that S(0) ⊂ T (0) and D(T ) ⊂ D(S).Then,

(i) QT = QT+S and T = T − S + S.(ii) If S is continuous then S′(0) ⊂ T ′(0), D(T ′) ⊂ D(S′) and further T +S

is closed if so is T .

Proof. (i) From the condition S(0) ⊂ T (0), it is obvious that QT = QT+S .On the other hand, we note that

D(T − S + S) = D(T ) ∩ D(S) = D(T ) and (T − S + S)(0) = T (0)

and if (x, y) ∈ G(T −S +S) then y = u+v for some u, v ∈ X such that(x, u) ∈ G(T ) and (0, v) ∈ G(T ) which implies that G(T −S+S) ⊂ G(T )and thus, we infer from [3, Exercise I.2.14] that T − S + S = T .

(ii) Since D(T ) ⊂ D(S) and by [3, Proposition III.1.4], we have S′(0) =D(S)⊥ then it follows that S′(0) ⊂ T ′(0). Furthermore, we obtain

D(T ′) ⊂ T (0)⊥

= S(0)⊥

, (see [3,Proposition III.1.4 and Exercice II.5.19])= D(S′), (see [3,Proposition III.4.6]).

It only remains to prove that T + S is closed if T is closed. WhenT and S are operators the property is obvious. For the general case,we observe that QT T is closed operator with T (0) closed and by [3,Lemma IV.5.2] we infer that QT S = QT (0)/S(0)QSS is a closed operatorand since (T + S)(0) = T (0) is closed, we obtain that T + S is closed,as required. �

Lemma 2.3. Let T ∈ LR(X) be closed, and let K ∈ KT (X). Then

(i) If T ∈ Φ+(X) then T + K ∈ Φ+(X) and i(T + K) = i(T ).(ii) If T ∈ Φ−(X) then T + K ∈ Φ−(X) and i(T + K) = i(T ).

Proof. By Lemma 2.2, we have that T + K is closed and QT+K = QT T +QT (0)/K(0)QKK.

(i) Assume T ∈ Φ+(X). Then by Lemma 2.1, we have QT T ∈Φ+(X,Y/T (0))and using [5, Theorem V.2.1] we obtain QT+K(T +K) ∈ Φ+(X,Y/T (0))with the same index that QT T and then again by Lemma 2.1, we con-clude that T + K is a Φ+-relation with the same index that T .

(ii) Let T ∈ Φ−(X), then applying Lemma 2.1 we have that T ′ ∈ Φ+(X ′),using [3, Proposition III.1.5], we obtain that (T + K)′ = T ′ + K ′ and itfollows from [3, Corollary V.5.15] together with Lemma 2.2 that K ′ ∈KT (X ′). These facts combined with the part (i) ensure that (T + K)′ ∈Φ+(X ′) and again applying Lemma 2.1 we infer that T + K ∈ Φ−(X)and i(T + K) = i(T ). �


We end this section by giving some properties of polynomials in a linearrelation useful for the proof of spectral mapping theorems. In the rest of thepaper, T will be a linear relation in a complex Banach space X.

Lemma 2.4 [9, Equation (2.9)] and [11, Lemma 6.1 and Proposition 6.2] .

(i) (T − α)n(T − β)p = (T − β)p(T − α)n for all α, β ∈ C and for alln, p ∈ N ∪ {0} and (T − α)n(T − β)p(0) = Tn+p(0).

(ii) If there is η ∈ C such that T − η is bijective, then X = D(Tn) + R(T p)and {0} = Tn(0) ∩ N(T p) for all n, p ∈ N. Further, if T has a finiteindex, then i(Tn) = ni(T ) and i(Tn+p) = i(Tn) + i(T p).

The notion of polynomial in an operator can be naturally generalizedto linear relations as follows:

Definition 2.2. Given a polynomial P (λ) = α0 + α1λ + · · · + αnλn withcoefficients in C, we define the polynomial in T by

P (T ) := α0 + α1T + · · · + αnTn,

where T 0 is the identity operator defined on X.

Lemma 2.5 [2, Theorem 3.3]. Let P and Q be two polynomials in C. Then

(QP )(T ) = Q(T )(P )(T ).

Lemma 2.6. Let β ∈ C. Then

(i) Tn =∑n

k=0

(nk

)βk(T − β)n−k, n ∈ N ∪ {0}.

(ii) Every polynomial in T , P (T ) of degree n can be described as P (T ) =(T − β)Q(T ) + δ where Q(T ) is a polynomial in T of degree n − 1 andδ is a scalar.

Proof. (i) We proceed by induction. For n = 0 it is trivial and since T :=(T − β) + β, we have that the property is true for n = 1. Assume (i)holds for n = m. Then

Tm+1 = A(B + C), (2.1)

where, A :=∑m

k=0

(mk

)βk(T − β)m−k, B := T − β and C := β := βI.

Indeed, Tm+1 = TmT =(∑m

k=0

(mk

)βk(T − β)m−k

)((T − β) + β).

Moreover, we have that

A(B + C) = AB + AC. (2.2)

We first prove that D(A(B + C)) = D(AB + AC).


In effect, we have

D(A(B + C)) := {x ∈ D(B + C) : (B + C)x ∩ D(A) = ∅}:= {x ∈ D(T ) : Tx ∩ D(Tm) = ∅}:= D(Tm+1)

D(AB) := {x ∈ D(B) : Bx ∩ D(A) = ∅}:= {x ∈ D(T − β) : (T − β)x ∩ D(Tm) = ∅}:= D(Tm(T − β)):= D(Tm+1)

D(AC) := {x ∈ D(C) = X : βx ∩ D(A) = ∅}:= D(Tm).

Therefore, A(B+C) and AB+AC have the same domain. Furthermore,we infer from [3, Proposition I.4.2] that A(B + C) is a extension ofAB + AC, so that

G(AB + AC) ⊂ G(A(B + C)) and A(B + C)(0) = AB(0) + AC(0).

In this situation, the property (2.2) follows immediately from [3, Exer-cise I.2.14].

Tm+1 =m+1∑

k=0

(m + 1

k

)βk(T − β)m+1−k. (2.3)

From (2.2), we have that Tm+1 := E + F where E :=( ∑m

k=0

(mk

)βk

(T − β)m−k)(

T − β)

and F := β∑m

k=0

(mk

)βk(T − β)m−k. Then, if in

Lemma 2.5, we take S := T − β, Q(S) :=∑m

k=0

(mk

)βk(T − β)m−k

and P (S) := T − β, we obtain that QP (S) = Q(S)P (S), that is E :=∑m

k=0

(mk

)βk(T − β)m+1−k. Therefore, (2.3) holds.

The proof of the statement (i) is completed.(ii) Let P (T ) = α0 + α1T + · · · + αnTn. The use of (i) leads to P (T ) =

a0 + a1(T − β) + · · · + an(T − β)n, for some scalars a0, a1, ..., an.On the other hand, if in Lemma 2.5, we take S := T − β, P (S) :=a1S + · · · + anSn−1 and Q(S) = S, we deduce that P (T ) − a0 = (T −β)(a1 +a2(T −β)+ · · ·+an(T −β)n−1) = (T −β)Q(T ), where, Q(T ) :=a1 + a2(T − β) + · · · + an(T − β)n−1.

Again, applying the part (i), we have that Q(T ) := b1 + b2T + · · · +bnTn−1, for some scalars b1, b2, ..., bn, so that Q is a polynomial in T ofdegree n − 1, as desired. �

Lemma 2.7. Suppose that T is closed and that there exists β ∈ C such thatT − β is bijective. Let P (T ) = α0 + α1T + · · · + αnTn be a polynomial in Tof degree n. Then P (T ) is a closed linear relation.


Proof. We shall proceed by induction. For n = 1, it is clear since P (T ) =α0 + α1T is closed by Lemma 2.2. Assume that the property is true forn = m. Then if P (T ) is a polynomial in T of degree m + 1, we infer from thecondition (ii) in Lemma 2.6 that P (T ) = (T − β)Q(T ) + δ where Q(T ) is apolynomial in T of degree m and δ is a scalar. Then Q(T ) is closed by theinduction hypothesis and since T − β is closed and bijective, we deduce from[3, Proposition II.5.17] that (T − β)Q(T ) is closed, so that P (T ) is closed, asrequired. �

Definition 2.3. Fix λ ∈ C and let P (λ) − μ = c∏n

k=1(λ − λk)mk . Then, byLemma 2.4, the polynomial P (T ) in T given by P (T )−μ = c

∏nk=1(T −λk)mk

is a linear relation in X.

The following properties concerning the behavior of the domain, therange, the null space and the multivalued part of P (T ) were proved in[9, Theorems 3.2, 3.3, 3.4 and 3.6] by Sandovici.

Lemma 2.8. Let P (T ) as in Definition 2.3. Then

D(P (T )) = D(P (T ) − μ) = D(T∑n

k=1 mk).

R(P (T )) =n⋂

k=1

R(T − λk)mk .

N(P (T )) =n∑

k=1

N(T − λk)mk .

P (T )(0) = T∑n

k=1 mk(0).

3. Main Results

In the rest of the paper, X will denote a complex Banach space and T we willdenote a closed linear from X to X. The following result was established in [8,Theorems 2.1] and [12, Proposition 7] for bounded operators and generalizedto densely defined closed operators in [6, Proposition 3.1].

Theorem 3.1. Let λ ∈ C. Then

(i) λ /∈ σeap(T ) if and only if T − λ ∈ Φ+(X) and i(T − λ) ≤ 0.(ii) λ /∈ σeδ(T ) if and only if T − λ ∈ Φ−(X) and i(T − λ) ≥ 0.

Proof. (i) Let λ /∈ σeap(T ). Then there exists K ∈ KT (X) for which T −λ − K is bounded below, so that T − λ − K ∈ Φ+(X) and i(T − λ −K) ≤ 0. This together with Lemmas 2.2 and 2.3 leads to T − λ ∈Φ+(X) and i(T − λ) ≤ 0. Conversely, let λ ∈ C such that T − λ ∈Φ+(X) and i(T − λ) ≤ 0. Choose a basis {x1, x2, ..., xn} for N(T − λ)and let y1, y2, ..., yn be linearly independent elements of X such that[y1], [y2], ..., [yn] ∈ X/R(T − λ) are linearly independent (such elementsexist, since dim N(T − λ) ≤ dim X/R(T − λ)). Let {x′

1, x′2, ..., x

′n} be


chosen in X ′ such that x′i(xj) = δij for 1 ≤ i, j ≤ n. Define

K : x ∈ X −→ Kx :=n∑

i=1

x′i(x)yi ∈ X.

Then, it is clear that K is an everywhere-defined continuous operatorwith dimR(K) ≤ dim N(T −λ), so that K ∈ KT (X) and thus we deducefrom Lemma 2.3 that T−λ−K ∈ Φ+(X) and i(T−λ−K) = i(T−λ) ≤ 0.Furthermore, reasoning exactly as in the proof of Theorem 9.1 (i) in [10],we obtain that T − λ − K is injective. Therefore (i) holds.

(ii) Assume that λ ∈ σeδ(T ) and let K ∈ KT (X) such that T − λ − K issurjective, in particular T −λ−K ∈ Φ−(X) and i(T −λ−K) ≥ 0. Thisimplies by the use of Lemmas 2.2 and 2.3 that T − λ is a Φ−-relationwith i(T − λ) ≥ 0.

Let T − λ ∈ Φ−(X) with i(T − λ) ≥ 0, so that β(T − λ) := m < ∞ andm ≤ α(T − λ). Let x1, x2, ..., xm be a set of m linearly independent elementsof N(T −λ). Choose linear functionals x′

1, x′2, ..., x

′m such that x′

i(xj) = δij for1 ≤ i, j ≤ m and choose elements y1, y2, ..., yn in X such that the correspond-ing cosets [y1], [y2], ..., [ym] ∈ X/R(T − λ) determine a basis of X/R(T − λ).Define

K : x ∈ X −→ Kx :=m∑

i=1

x′i(x)yi ∈ X.

Then K is a bounded finite rank operator and then it follows from Lemma2.3 that T −λ−K ∈ Φ−(X) with i(T −λ−K) ≥ 0 and proceeding as in theproof of Theorem 9.1 (ii) in [10], we get T − λ − K is surjective. �

As a direct consequence of Theorem 3.1, we obtain

Corollary 3.1. We have

σeap(T ′) = σeδ(T ) and σeap(T ) = σeδ(T ′).

Proof. Combine Lemma 2.1 with Theorem 3.1. �

Proposition 3.1. The subsets σeap(T ) and σeδ(T ) are closed.

Proof. From the definitions and Corollary 3.1, it suffices to prove that σeap(T )is closed. To do this, let λ /∈ σeap(T ) and let η ∈ C such that |λ−η| < γ(T−λ).Then, by [3, Proposition III.7.4 and Corollary III.7.6], we infer that T − λ isinjective and open. Therefore, σeap(T ) is closed, as desired. �

Recall that the resolvent set of T is the set ρ(T ) defined by

ρ(T ) := {λ ∈ C : T − λ is bijective}.

In the rest of this Section, we are concerned with the study of the spectralmapping theorems for σeap(T ) and σeδ(T ) where T is a closed linear relationin X with nonempty resolvent set.


Proposition 3.2. Let P (T ) as in Definition 2.3. If each factor T − λk hasfinite index, then

i(P (T ) − μ) =n∑

k=1

i((T − λk)mk).

Proof. We first show that

Each T − λk has a nonempty resolvent set. (3.1)

Since T −α = (T −λk)+(λk−α) for all α ∈ C, we have that η−λk ∈ ρ(T −λk)whenever η ∈ ρ(T ).

i((T − λk)mk) = mki(T − λk). (3.2)

Combine (3.1) with Lemma 2.4.We prove the result by induction. For n = 1 it is clear. Assume that it

is true for n = r, that is

i

(r∏

k=1

(T − λk)mk

)=

r∑

k=1

i((T − λk)mk).

Let A :=∏r

k=1(T −λk)mk and B := (T −λr+1)mr+1 . Then, by the inductionhypothesis we infer that A and B have finite index and thus it follows fromLemma 2.1 that

i(AB) = i(A) + i(B) + dimX/(D(A) + R(B)) − dim(N(A) ∩ B(0)),

where

D(A) + R(B) = X. (3.3)

Indeed, applying Lemma 2.8, we have

D(A) = D(Tm1+m2+···+mr) = D((T − λr+1)m1+m2+···+mr),

and

R(B) = R((T − λr+1)mr+1).

Hence D(A)+R(B) = D((T −λr+1)m1+m2+···+mr)+R((T −λr+1)mr+1) = X,where the last equality is obtained by combining (3.1) with the condition (ii)in Lemma 2.4.

dim(N(A) ∩ B(0)) < ∞. (3.4)

It is enough to observe that by [11, Lemma 5.1]

α(A) ≤ α((T − λ1)m1) + α((T − λ2)m2) + · · · + α((T − λr)mr )≤ m1α(T − λ1) + m2α(T − λ2) + · · · + mrα(T − λr)< ∞.

Now, it follows from (3.3) and (3.4) that i(AB) = i(A) + i(B) − δ wheredim(N(A)∩B(0)) := δ < ∞ and hence i(AB) ≤ i(A)+ i(B). It only remainsto verify that i(A) + i(B) ≤ i(AB). We first note that AB = BA by virtueof the part (i) in Lemma 2.4, so that by Lemma 2.1, we get

i(AB) = i(A) + i(B) + dimX/(R(A) + D(B)) − dim(N(B) ∩ A(0)),


where

dim X/(R(A) + D(B)) < ∞. (3.5)

In effect, the use of Lemma 2.1 make us conclude that A is a Fredholmrelation, in particular dimX/R(A) < ∞ which implies that dimX/(R(A) +D(B)) < ∞.

A(0) ∩ N(B) = {0}. (3.6)

Indeed, note that applying Lemmas 2.4 and 2.8, we get

A(0) = Tm1+m2+···+mr(0) = (T − λr+1)m1+m2+···+mr (0).

So that,

A(0) ∩ N(B) = (T − λr+1)m1+m2+···+mr (0) ∩ N((T − λr+1)mr+1).

This equality together with (3.1) and Lemma 2.4 ensures that A(0)∩N(B) ={0}. We conclude from (3.5) and (3.6) that i(A) + i(B) ≤ i(AB). The proofis completed. �

As an immediate consequence, we get

Proposition 3.3. Let P (T ) as in Definition 2.3.

(i) If each T −λk ∈ Φ+(X) and i(T −λk) ∈]−∞; 0], for some k ∈ N, then

P (T ) − μ ∈ Φ+(X) and i(P (T ) − μ

) ∈] − ∞; 0].

(ii) If each T − λk ∈ Φ−(X) and i(T − λk) ∈ [0;+∞[, for some k ∈ N, then

P (T ) − μ ∈ Φ−(X) and i(P (T ) − μ) ∈ [0;+∞[.

Proposition 3.4. Let P (T ) as in Definition 2.3.

(i) If each T − λk ∈ Φ+(X) and i(T − λk) = −∞, for some k ∈ N. Then

P (T ) − μ ∈ Φ+(X) and i(P (T ) − μ) = −∞.

(ii) If each T − λk ∈ Φ−(X) and i(T − λk) = +∞, for some k ∈ N. Then

P (T ) − μ ∈ Φ−(X) and i(P (T ) − μ) = +∞.

Proof. (i) By Lemma 2.1, (T −λk)mk and P (T ) − μ are Φ+-relations. Fur-thermore, by Lemma 2.8, we get

R(P (T ) − μ) =n⋂

k=1

R((T − λk)mk),

and thus it follows from [3, Lemma IV.5.2] that

(X/R(P (T ) − μ))/(R(T − λk)/R(P (T ) − μ)) X/R(T − λk).

This property combined with α(T −λk) < ∞ and i(T −λk) = −∞ leadsto β(P (T ) − μ) = +∞. Therefore (i) holds.


(ii) We note that ρ(T − λk) = ∅, by (3.1) and since T − λk is closed wededuce from Lemma 2.7 that (T −λk)mk and P (T )−μ are closed linearrelations and since (T − λk)mk ∈ Φ−(X) it follows from the part (iv) inLemma 2.1 that P (T ) − μ is a Φ−-relation. It only remains to see thati(P (T ) − μ) = +∞. To this end, we shall verify that

α((T − λk)2) + 2β(T − λk) = β((T − λk)2) + 2α(T − λk). (3.7)

Indeed, from [11, Theorem 4.1], it follows that

α((T − λk)2) + 2β(T − λk) + dim((T − λk)(0) ∩ N(T − λk))= β((T − λk)2) + 2α(T − λk) + dimX/(D(T − λk) + R(T − λk)).

This implies by the use of (3.1) and Lemma 2.4 the validity of (3.7).

α((T − λk)mk) = +∞. (3.8)

Since, T −λk is a lower semi-Fredholm relation with nonempty resolventset, then we have (T − λk)2 ∈ Φ−(X), by Lemmas 2.1 and 2.7, sothat β(T − λk) and β((T − λk)2) are both finite. Furthermore, sincei(T − λk) = +∞, by hypothesis and β(T − λk) < ∞, we have thatα(T − λk) = +∞. In this situation, we infer from (3.7) that α((T −λk)2) = +∞ and thus continuing in this way, we obtain the property(3.8).

α(P (T ) − μ) = +∞. (3.9)

By Lemma 2.8, we have that

N(P (T ) − μ) =n∑

i=1

N((T − λi)mi)

=n∑

i=1i�=k

N((T − λi)mi) + N((T − λk)mk)

and thus by (3.8), it follows that α(P (T ) − μ) = +∞. This propertytogether with the fact P (T )−μ ∈ Φ−(X) proves that i(P (T )−μ) = +∞,as desired. �

We now are ready to state the following spectral mapping theorem whichwas established in [8, Theorem 3.3] and [12, Theorem 3] for bounded operatorsand it was generalized to densely defined closed operators in [6, Theorem 4.1].

Theorem 3.2. Assume that T has a nonempty resolvent set. Then, for anycomplex polynomial P we have that

(i) σeap(P (T )) ⊂ P (σeap(T )).(ii) σeδ(P (T )) ⊂ P (σeδ(T )).

Proof. Let P (T ) as in Definition 2.3.(i) Assume that μ ∈ σeap

(P (T )

). If λk /∈ σeap(T ) for all k = 1, 2, ..., n, then

it follows from the characterization of σeap(T ) established in Theorem3.1 that T − λk ∈ Φ+(X) with i(T − λk) ≤ 0, for all k = 1, 2, ..., n


and thus we deduce from Lemma 2.1 that P (T ) − μ is an upper semi-Fredholm relation. Let us consider two cases for the index:Case 1: i(T − λk) ∈] − ∞; 0], for all k = 1, 2, ..., n. Then i(P (T ) −μ) ∈] − ∞; 0], by Proposition 3.3 (i) and hence μ /∈ σeap(P (T )) whichcontradicts that μ ∈ σeap(P (T )).Case 2: i(T − λk) = −∞, for some k ∈ {1, 2, ..., n}. Then i(P (T ) − μ)would be −∞, applying Proposition 3.4 (i), so that also μ /∈ σeap(P (T )).Consequently, there exists j ∈ {1, 2, ..., n} for which λj ∈ σeap(T ) andsince μ = P (λj) we conclude that σeap(P (T )) ⊂ P (σeap(T )).

(ii) This assertion may be proved with a similar scheme using Propositions3.3 (ii) and 3.4 (ii). �

Remark 3.1. An example given in [6] shows that in general the inclusions ofTheorems 3.2 are strict even in the particular case of operators.

Acknowledgements

The work of T. Alvarez was supported by MICINN (Spain) Grant MTM2010-20190.

References

[1] Alvarez, T.: Perturbation theorems for upper and lower semi-Fredholm linearrelations. Publ. Math. Debrecen 65(1–2), 179–191 (2004)

[2] Arens, R.: Operational calculus of linear relations. Pac. J.Math. Soc. 11, 9–23(1961)

[3] Cross, R.W.: Multivalued Linear Operators. Marcel Dekker, New York (1998)

[4] Fakhfakh, F., Mnif, M.: Perturbation theory of lower semi-Browder multivaluedlinear operators. Publ. Math. Debrecen 78, 595–606 (2011)

[5] Goldberg, S.: Unbounded Linear Operators, Theory and Applications.MvGraw-Hill, New York (1966)

[6] Jeribi, A., Moalla, N.: A characterization of some subsets of Schechter’s essen-tial spectrum and application to singular transport equation. J. Math. Anal.Appl. 358(2), 434–444 (2009)

[7] Rakocevic, V.: On one subset of M. Schechter’s essential spectrum. Math. Ves-nik 5(18)(33), 389-391 (1981)

[8] Rakocevic, V.: Approximate point spectrum and commuting compact pertur-bations. Glasg. Math. J. 28, 193–198 (1986)

[9] Sandovici, A.: Some basic properties of polynomials in a linear relation in linearspaces. Oper. Theory Adv. Appl. 175, 231–240 (2007)

[10] Sandovici, A., de Snoo, H., Winkler, H.: Ascent, descent, nullity, defect andrelated notions for linear relations in linear spaces. Linear Algebra Appl.423, 456–497 (2007)

[11] Sandovici, A., Snoo, H.de : An index formula for the product of linear rela-tions. Linear Algebra Appl. 431, 2160–2171 (2009)

[12] Schmoeger, C.: The spectral mapping theorem for the essential approximatepoint spectrum. Coll. Math. 74(2), 167–176 (1997)


Faical Abdmouleh, Aymen Ammar and Aref JeribiDepartment of Mathematics, Faculty of Sciences of SfaxUniversity of SfaxRoute de soukra Km 3.5B. P. 1171, 3000 Sfax, Tunisiae-mail: ammar [email protected]

Aref Jeribi

e-mail: [email protected]

Teresa AlvarezDepartment of MathematicsUniversity of Oviedo33007 Oviedo, Asturias, Spaine-mail: [email protected]

Received: December 4, 2013.

Accepted: July 10, 2014.

spectral mapping theorem for rakocević and schmoeger essential spectra of a multivalued linear...

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