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EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 37, Num. 1 (2022), 75 – 90

doi:10.17398/2605-5686.37.1.75

Available online November 17, 2021

On isolated points of the approximate point spectrum
of a closed linear relation

M. Lajnef, M. Mnif

Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax

Route de Soukra Km 3.5, BP 1171, 3000 Sfax, Tunisia

maliklajnaf@gmail.com , maher.mnif@gmail.com

Received June 28, 2021 Presented by M. Mbekhta
Accepted September 30, 2021

Abstract: We investigate in this paper the isolated points of the approximate point spectrum of a
closed linear relation acting on a complex Banach space by using the concepts of quasinilpotent part

and the analytic core of a linear relation.

Key words: Linear relation, isolated point of the approximate point spectrum, analytic core,
quasinilpotent part.

MSC (2020): 47A06, 47A10.

1. Introduction and preliminaries

Throughout this paper, (X,‖.‖) will denote a complex Banach space. In
2008, González et al. [8] have shown that if 0 is isolated in the approximate
point spectrum of a bounded operator T, then the quasinilpotent part H0(T)
and the analytic core K(T) of T are closed, H0(T)∩K(T) = {0}, H0(T)⊕K(T)
is closed and there exists λ0 6= 0 such that

H0(T) ⊕K(T) = K(T −λ0I) =
⋂∞

n=0
Im(T −λ0I)n.

In recent years, the study of isolated spectral points of a multivalued linear
operator (linear relation) has generated a great deal of research attention. It
was proved in [9] that for a closed and bounded linear relation T such that 0
is a point of its spectrum, we have the equivalence:

0 is isolated in the spectrum of T ⇐⇒

{
H0(T) and K(T) are closed

and X = H0(T) ⊕K(T).

The previous studies on isolated spectral points in the two cases of linear op-
erators and relations and their extensions motivate us to focus on establishing

ISSN: 0213-8743 (print), 2605-5686 (online)

c© The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0)

https://doi.org/10.17398/2605-5686.37.1.75
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mailto:maher.mnif@gmail.com
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76 m. lajnef, m. mnif

some necessary conditions for which a point of the approximate point spectrum
of a closed linear relation be isolated. By the way, this work could be consid-
ered as an extension of the study carried out for the case of operators since
it covers the case of closed operators which are not necessary bounded. The
importance of the investigation of linear relations is shown by the examples of
issues in the study of some Cauchy problems associated with parabolic type
equations in Banach spaces [6]. Thus, the generalization of the existing results
for bounded operators to the general setting of closed linear relations seems
to appear quite naturally. We recall now some basic definitions and proper-
ties which are needed in the sequel. A linear relation (or a multivalued linear
operator) in a Banach space X, T : X → X, is a mapping from a subspace
D(T), called the domain of T into the set of nonempty subsets of X verifying
T(α1x + α2y) = α1T(x) + α2T(y) for all non zero scalars α1,α2 and vectors x
and y ∈ D(T). We denote by LR(X) the class of all linear relations in X. A
linear relation T ∈ LR(X) is completely determined by its graph defined by
G(T) := {(x,y) ∈ X ×X : x ∈ D(T),y ∈ Tx}. Let T ∈LR(X). The inverse
of T is the relation T−1 given by G(T−1) := {(u,v) ∈ X×X : (v,u) ∈ G(T)}.
We say that T is closed if its graph is a closed subspace of X × X, open if
γ(T) > 0, where

γ(T) =



∞ if D(T) ⊆ Ker(T),

inf

{
‖Tx‖

d(x, Ker(T))
: x ∈ D(T)\Ker(T)

}
otherwise.

The set of all closed linear relations is denoted by CR(X). We say that
T is continuous if the operator QTT is continuous when QT denoted the
quotient map from X onto X

T(0)
. In such a case the norm of T is defined by

‖T‖ := ‖QTT‖. We say that T is bounded if it is continuous and everywhere
defined. The set of all bounded and closed linear relations acting between
two Banach spaces X and Y is denoted by BCR(X,Y ). If X = Y , we write
BCR(X,X) := BCR(X). The subspaces Ker(T) := T−1(0) and Im(T) :=
T(D(T)) are called respectively the null space and the range space of T. We
say that T is surjective if T(D(T)) = X and injective if Ker(T) = {0}. Note
that T is an operator if and only if T(0) = {0}. In addition, the generalized
range of T is defined by

R∞(T) :=
⋂

n∈N
Im(Tn).



isolated points of the approximate point spectrum 77

For linear relations S, T ∈ LR(X) the relations S + T, ST and S+̂T are
defined respectively by

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

ST := {(x,z) : (x,y) ∈ G(T) and (y,z) ∈ G(S) for some y ∈ X},

S+̂T := {(x + u,y + v) : (x,y) ∈ G(S) and (u,v) ∈ G(T)}.

This last sum is direct when G(S) ∩G(T) = {(0, 0)}. In such case, we write
S⊕T. We denote by B(X,Y ) the Banach algebra of all bounded operators on
X and Y . If X = Y , we write B(X,X) := B(X). Recall that a linear relation
T is regular if Im(T) is closed and Ker(Tn) ⊆ Im(Tm) for all n, m ∈ N. The
class of all regular linear relations in X will be denoted by R(X). In what
follows we write

reg(T) := {λ ∈ C : T −λI is regular}.

For r > 0 we denote D(0,r) := {λ ∈ C : 0 ≤ |λ| < r} and D∗(0,r) :=
D(0,r)\{0}. Now, we essentially aim to define and study some basic tools of
the spectral theory. Given a closed linear relation T. For λ ∈ C, we denote
by Rλ(T) = (λI −T)−1 the resolvent of T at λ. The resolvent set of T is the
set defined by

ρ(T) = {λ ∈ C : (λI −T)−1 is everywhere defined and single valued}.

We say that T is invertible if 0 ∈ ρ(T). The spectrum of T is the set σ(T) =
C\ρ(T).

Furthermore, we say that T is bounded below if there exists some δ > 0
such that δ‖x‖≤‖Tx‖ for every x ∈ D(T). The approximate point spectrum
of T is defined by

σap(T) = {λ ∈ C : T −λI is not bounded below}.

Let us state now a useful lemma that we use below.

Lemma 1.1. Let T ∈CR(X). Then we have the following assertions.

(i) If T is surjective then there exists � > 0 such that T −λI is surjective
for every |λ| < �.

(ii) If T is bounded below then there exists � > 0 such that T−λI is bounded
below for every |λ| < �.



78 m. lajnef, m. mnif

Proof. The proof is similar to the proof of [10, Lemma 15].

The structure of this paper is as follows. In Section 2, we are mainly in-
terested in studying the quasinilpotent part and the analytic core of a closed
linear relation. Most properties of these latter subspaces are also gathered.
The stated results generalize the concepts of quasinilpotent part and the an-
alytic core recently introduced in [12] to the setting of closed not necessary
bounded linear relations. Section 3 begins by a generalization to the case of
closed linear relations of [9, Theorem 3.1] stated above. After that, we develop
a significant quantity of interesting technical lemmas. In particular, we set out
the concepts of regular linear relations and gap of two subspaces. This leads
us to find some necessary conditions for which a point of the approximate
point spectrum be isolated.

2. Quasinilpotent part and analytic core of a closed linear
relation

Let T ∈CR(X). Consider the graph norm ‖.‖T on D(T) defined by

‖x‖T := ‖x‖ + ‖Tx‖.

In what follows XT denotes the space D(T) endowed with the graph norm.
Observe that XT is a Banach space (since QTT is a closed operator). Consider
the relation T̃ defined by

T̃ : XT −→ X
x 7−→ Tx.

Evidently, T̃ is closed and D(T̃) = D(T). Then, by virtue of [5, II.5.1] we get
that T̃ ∈BCR(XT ,X).

Remark 2.1. Note that QT T̃ : XT → XT(0) is bounded. Moreover, for all
x ∈ D(T), ‖QTiTx‖ = d(x,T(0)) ≤ ‖x‖ ≤ ‖x‖T , then, QTiT is bounded,
where

iT : XT −→ X
x 7−→ x.

Now, let’s further extend the concepts of quasinilpotent part and the an-
alytic core developed in [11, 12] to the case of closed not necessary bounded
linear relations.



isolated points of the approximate point spectrum 79

Definition 2.1. Let T ∈CR(X).

(i) The quasinilpotent part of T, denoted by H0(T), is the set of all x ∈
D(T) for which there exists a sequence (xn)n ⊆ D(T) satisfying

x0 = x, xn+1 ∈ Txn for all n ∈ N and ‖xn‖
1
n
T → 0.

(ii) The analytic core of T, denoted by K(T), is defined as the set of all
x ∈ X for which there exist c > 0 and a sequence (xn)n∈N satisfying
x0 = x and for all n ≥ 0, xn+1 ∈ D(T), xn ∈ Txn+1 and

d(xn, Ker(T) ∩T(0)) ≤ cnd(x, Ker(T) ∩T(0)).

In the next lemma, we collect some elementary properties of H0(T) and
K(T).

Lemma 2.1. Let T ∈CR(X). Then the following statements hold.

(i) If F is a closed subspace of X such that T(F∩D(T)) ⊆ F , then H0(T)∩
F = H0(T|F).

(ii) For λ 6= 0, H0(T) ⊆ (λI −T)H0(T) (the closure of (λI − T)H0(T) in
X).

(iii) T(D(T) ∩K(T)) = K(T).
(iv) If F is a closed subspace of X such that T(D(T) ∩F) = F, then F ⊆

K(T).

(v) If T ∈R(X), then K(T) = R∞(T) and it is closed.

Proof. (i) Let x ∈ H0(T|F). Then, by Definition 2.1, there exists
(xn)n ⊆ F ∩D(T) such that

• x0 = x,
• xn+1 ∈ T|Fxn,

• ‖xn‖
1
n

T |F → 0.

But T|Fxn = Txn and ‖xn‖T |F = ‖xn‖T , then x ∈ H0(T) ∩F.
Conversely, assume that x ∈ H0(T)∩F. Then there exists (xn)n ⊆ D(T)
such that



80 m. lajnef, m. mnif

• x0 = x ∈ F ∩D(T),
• xn+1 ∈ Txn,

• ‖xn‖
1
n
T → 0.

Since T(F ∩ D(T)) ⊆ F, then (xn) ⊆ F and so, xn+1 ∈ T |Fxn and
‖xn‖T |F = ‖xn‖T . Consequently, x ∈ H0(T|F).

(ii) Let x ∈ H0(T). Then, there exists (xn)n such that x0 = x, xn+1 ∈ Txn
and ‖xn‖

1
n
T −−−→n→∞ 0. Let yn =

∑n
k=0

xk
λk+1

. As, for all n ∈ N, xn ∈ H0(T),
then yn ∈ H0(T) and we have

(λI −T)yn = x−
xn+1
λn+1

+ T(0).

Therefore,

x−
xn+1
λn+1

∈ (λI −T)yn.

Whence,

x−
xn+1
λn+1

∈ (λI −T)H0(T).

Using the fact that ‖xn‖
1
n ≤‖xn‖

1
n
T → 0, one can deduce that

xn+1
λn+1

−−−→
n→∞

0,

which implies that x ∈ (λ−T)H0(T). Thus, H0(T) ⊆ (λI −T)H0(T).
(iii) Let prove the first inclusion T(D(T)∩K(T)) ⊆ K(T). Let y ∈ T(D(T)∩

K(T)). Then, there exists x ∈ D(T) ∩ K(T) such that y ∈ Tx. Since
x ∈ K(T) then there exist δ > 0 and a sequence (xn)n such that

• x0 = x,
• for all n ≥ 0, xn+1 ∈ D(T) and xn ∈ Txn+1,
• d(xn, Ker(T) ∩T(0)) ≤ δnd(x, Ker(T) ∩T(0)) for all n ∈ N.

Let (yn)n be the sequence defined by yn+1 = xn for all n ∈ N and
y0 = y. Since xn−1 ∈ Txn, then yn ∈ Tyn+1 for all n ≥ 1. On the other
hand, since y1 = x0 = x and y ∈ Tx, then y0 ∈ Ty1. We need only to
prove that d(yn,T(0)∩Ker(T) ≤ δ′nd(y,T(0)∩Ker(T)) for some δ′ > 0.
Trivially, if y ∈ T(0) ∩ Ker(T) there is nothing to prove. If not we get



isolated points of the approximate point spectrum 81

that d(yn,T(0) ∩ Ker(T) ≤ δ′nd(y,T(0) ∩ Ker(T)) with δ′ > 0 be such
that

δ′ = max

{
δ ,
d(x,T(0) ∩ Ker(T))
d(y,T(0) ∩ Ker(T))

}
.

Thus, y ∈ K(T).
For the reverse inclusion, let x ∈ K(T). Then there exists δ > 0 and a
sequence (un)n such that

• u0 = x,
• for all n ≥ 0, un+1 ∈ D(T) and un ∈ Tun+1,
• d(un, Ker(T) ∩T(0)) ≤ δnd(x, Ker(T) ∩T(0)) for all n ∈ N.

Since x ∈ Tu1 then, in order to show that K(T) ⊆ T(D(T) ∩ K(T)),
it is sufficient to prove that u1 ∈ K(T). If u1 ∈ T(0) ∩ Ker(T), then
there is nothing to prove. If not, let (wn)n be the sequence such that
wn = un+1. We have wn = un+1 ∈ Tun+2 = Twn+1. Furthermore,

d(wn, Ker(T)∩T(0)) = d(un+1, Ker(T)∩T(0)) ≤ δ′nd(u1, Ker(T)∩T(0)),

where δ′ = δ2
d(x,Ker(T)∩T(0))
d(u1,Ker(T)∩T(0))

. Hence, u1 ∈ K(T).

(iv) First, we claim that F ∩ D(T) is closed in XT . Indeed, let (xn)n ⊆
F ∩D(T) be such that xn

XT−−−→
n→∞

x. Trivially, x ∈ D(T). On the other
hand, we have ‖xn − x‖T −−−→

n→∞
0. As F is closed in X, then x ∈ F.

Hence, F ∩D(T) is closed in XT , as claimed. Recall that the relation T̃
is closed. Let us consider T0 : D(T) ∩F → F, the restriction of T̃. We
have G(T0) = G(T̃)∩((D(T̃)∩F)×F) is closed in XT ×X. Then, T0 is
closed. We have, by hypothesis, Im(T0) = F then, by the open mapping
theorem [5, Theorem III.4.2], we deduce that T0 is open. Thus, there
exists a constant γ > 0 such that for all x ∈ D(T0) = D(T) ∩F,

dT (x, Ker(T0)) ≤ γ‖T0x‖,

where dT (x,G) := infα∈G‖x − α‖T . As, for all x ∈ D(T0) and α ∈
Ker(T0), ‖x − α‖ ≤ ‖x − α‖T , then d(x, Ker(T0)) ≤ dT (x, Ker(T0)).
Hence,

d(x, Ker(T0)) ≤ γ‖T0x‖. (2.1)

Now, consider � > 0 and let u ∈ F. Then, there exists x ∈ D(T) ∩ F
such that u ∈ Tx. By (2.1) there exists y ∈ Ker(T0) ⊆ Ker(T) such that



82 m. lajnef, m. mnif

‖x − y‖ ≤ (γ + �)d(u,T(0)). Take u1 = x − y ∈ D(T) ∩ F. We have
u ∈ T(u1) and

d(u1,T(0) ∩ Ker(T)) ≤ (γ + �)d(u,T(0) ∩ Ker(T)).

Continuing in the same manner, we build a sequence (un)n such that
u0 = u, for all n ≥ 0, un+1 ∈ D(T) ∩F, un ∈ Tun+1 and

d(un,T(0) ∩ Ker(T)) ≤ (γ + �)nd(u,T(0) ∩ Ker(T)).

Hence, u ∈ K(T). Thus, F ⊆ K(T).
(v) As T is regular then, by [2, Proposition 2.5] and [1, Lemma 20], we

get that R∞(T) is closed and T(R∞(T) ∩ D(T)) = R∞(T). If follows
from (iv) that R∞(T) ⊆ K(T). On the other hand, it is clear that
K(T) ⊆ R∞(T). So, K(T) = R∞(T) which is closed, as desired.

3. Isolated point of the approximate point spectrum

The main objective of this section is to give necessary conditions to ensure
that the approximate point spectrum of a closed linear relation T does not
cluster at a point λ. To do this, we begin with a generalization to the case of
closed linear relations of a theorem stated in [9] dealing with the characteri-
zation of isolated points of the spectrum of a bounded closed linear relation.
For this, we need the following technical lemma.

Lemma 3.1. Let T ∈CR(X) and x ∈ X. Then,

R̃.(T)x : ρ(T) −→ XT
µ 7−→ R̃µ(T)x = Rµ(T)x := (µI −T)−1x

is analytic.

Proof. Let λ ∈ ρ(T). By virtue of [5, Corollary VI.1.9], we get that if
|λ−µ| < ‖Rλ(T)‖−1, then

Rµ(T) =
∑∞

n=0
Rλ(T)

n+1(µ−λ)n,

which implies that R̃µ(T)x =
∑∞

n=0 R̃λ(T)
n+1x(µ − λ)n. It was like

proving that
∑∞

n=0 R̃λ(T)
n+1x(µ − λ)n is convergent on XT . Observe that



isolated points of the approximate point spectrum 83

‖R̃λ(T)n+1x‖T = ‖R̃λ(T)n+1x‖ + ‖TR̃λ(T)n+1x‖. Moreover, we have

‖TR̃λ(T)n+1x‖ = ‖QTTR̃λ(T)n+1x‖
= ‖QT (T −λI + λI)R̃λ(T)n+1x‖
= ‖QT [(T −λI)R̃λ(T)n+1x + λR̃λ(T)n+1x]‖
= ‖QTRλ(T)nx + λQTRλ(T)n+1x‖
≤‖QTRλ(T)nx‖ + |λ|‖QTRλ(T)n+1x‖
≤‖Rλ(T)nx‖ + |λ|‖Rλ(T)n+1x‖.

Then, ‖R̃λ(T)n+1x‖T ≤ (1 + |λ|)‖Rλ(T)n+1x‖ + ‖R̃λ(T)nx‖. Since∑
n≥0

Rλ(T)
n+1x(µ−λ)n,

∑
n≥0

Rλ(T)
nx(µ−λ)n

are absolutely convergent in X, then
∑

n≥0 ‖R̃λ(T)
nx‖T |µ−λ|n is convergent.

Therefore,
∑

n≥0 R̃λ(T)
n+1x(µ−λ)n is convergent on XT , as required.

Theorem 3.1. Let T ∈CR(X) and let λ0 ∈ σ(T). Then K(λ0I−T) and
H0(λ0I −T) are closed and X = H0(λ0I −T) ⊕K(λ0I −T) if and only if λ0
is an isolated point of σ(T).

Proof. Recall that the quasinilpotent part of a closed linear relation is
a subspace of XT . Then, proceeding as in the proof of [9, Theorem 3.1],
and taking into account Lemma 3.1, we obtain that H0(T) = Im(B1) and
K(T) = Ker(B1) where B1 is the bounded projection defined as follows:

B1 :=
1

2πi

∮
Γλ0

Rλ(T)dλ,

where Γλ0 is a simple closed curve around λ0 such that the closure of the
region bounded by Γλ0 and containing λ0 intersects σ(T) only at λ0.

Now, we develop some supplementary technical lemmas that enable us to
provide some necessary conditions for which a point of the approximate point
spectrum be isolated. The stated notations and terminology are essentially
adhered to [8]. Using the proof of [1, Theorem 23] and Lemma 2.1 (v), we get
the following lemma.

Lemma 3.2. Let T ∈R(X) and λ ∈ K. Then:



84 m. lajnef, m. mnif

(i) γ(λI −T) ≥ γ(T) − 3|λ|.

(ii) K(T) ⊆ K(λI −T) for all 0 < |λ| <
γ(T)

3
.

The following lemma gives an analytic core stability result for a regular
linear relation.

Lemma 3.3. Let T ∈R(X) and λ ∈ K. Then there exists ν > 0 such that
K(T) = K(T −λI) for all |λ| < ν.

Proof. Let |µ0| < 17γ(T). Then, by virtue of [1, Theorem 23], we get that
T −µ0I ∈R(X). According to Lemma 3.2 (ii) we get that

K(T −µ0I) ⊆ K(T −µ0I −λI) whenever |λ| <
γ(T −µ0I)

4
. (3.1)

As |µ0| < 17γ(T) then, by Lemma 3.2 (i), we get that

γ(T −µ0I) > 7|µ0|− 3|µ0| > 4|µ0|.

Hence, for λ = −µ0 in (3.1), we get that K(T − µ0I) ⊆ K(T). Using again
Lemma 3.2, we get that K(T) ⊆ K(T −µ0I). This completes the proof.

Now, we recall some useful tools for the sequel.

Definition 3.1. Let M and N be two closed subspaces of X. Then the
gap between M and N is defined by

g(M,N) = max(δ(M,N)),δ(N,M)),

with δ(M,N) = sup{d(x,N) : x ∈ M and‖x‖ = 1)}.

The following lemma is fundamental to the proof of the main result of this
section.

Lemma 3.4. Let T ∈CR(X), Ω be a connected component of reg(T) and
λ0 ∈ reg(T). Then:

(i) The mapping λ → Ker(T −λI) is continuous at λ0 in the gap metric.
(ii) Im(T −λ0I) is closed, Im(T −λI) is closed in a neighborhood of λ0 and

the mapping λ → Im(T −λI) is continuous at λ0 in the gap metric.



isolated points of the approximate point spectrum 85

(iii)
⋂
i≥0 Im(T−λiI) =

⋂
i>0 Im(T−λiI), with (λi)i>0 a sequence of distinct

points of Ω which converges to λ0.

(iv) K(T−λ0I) =
⋂
i>0 Im(T−λiI) with (λi)i>0 a sequence of distinct points

of Ω which converges to λ0.

Proof. (i) and (ii) are proved in [3, Theorem 3.3].

(iii) Let 0 6= x0 ∈
⋂
i>0 Im(T −λiI). Then, for all i > 0,

x0
‖x0‖
∈ Im(T −λiI).

First, we note that

d

(
x0
‖x0‖

, Im(T −λ0I)
)

=
1

‖x0‖
d(x0, Im(T −λ0I)).

From the last equality and the definitions mentioned above, we deduce
that

g(Im(T −λiI), Im(T −λ0I)) ≥ δ(Im(T −λiI), Im(T −λ0I))

≥ sup{d(x, Im(T −λ0I) : x ∈ Im(T −λiI),‖x‖ = 1}

≥ d(x, Im(T −λ0I)), ∀x ∈ Im(T −λiI),‖x‖ = 1

≥ d
(

x0
‖x0‖

, Im(T −λ0I)
)

≥
1

‖x0‖
d(x0, Im(T −λ0I)).

Hence,

d(x0, Im(T −λ0I)) ≤ g(Im(T −λiI), Im(T −λ0I))‖x0‖.

This implies, by the use of (ii), that x0 ∈ Im(T −λ0I) = Im(T −λ0I).
Thus, x0 ∈

⋂
i≥0 Im(A−λiI), as required.

(iv) Using Lemma 3.3 and Lemma 2.1 (v), we get that for all i ≥ 0,

K(T −λ0I) = K(T −λiI) =
⋂

n≥0
Im(T −λiI)n ⊆ Im(T −λiI).

Then, the lefthanded side is contained in the righthanded side. So,
it is sufficient to show that

⋂
i>0 Im(T − λiI) ⊆ K(T − λ0I). To do

this, let x ∈
⋂
i>0 Im(T − λiI). Then, by (iii), x ∈

⋂
i≥0 Im(T − λiI).

Whence, x ∈ Im(T − λ0I) and so, there exists y ∈ D(T) such that
x ∈ (T − λ0I)y. Which implies that for all i ≥ 1, x + (λ0 − λi)y ∈



86 m. lajnef, m. mnif

(T − λiI)y. On the other hand, we have for all i ≥ 1, x ∈ Im(T −
λiI). Therefore, (λ0 − λi)y ∈

⋂
i>0 Im(T − λiI). As λi 6= λ0, then

y ∈
⋂
i>0 Im(T −λiI). Consequently, for all x ∈

⋂
i>0 Im(T −λiI), there

exists y ∈
⋂
i>0 Im(T −λiI)∩D(T) such that x ∈ (T −λ0I)y. Whence,

(T−λ0I)
(⋂

i>0 Im(T −λiI) ∩D(T)
)
⊇
⋂
i>0 Im(T−λiI). On the other

hand, it is clear to see that (T − λ0I)
(⋂

i>0 Im(T −λiI) ∩D(T)
)
⊆⋂

i>0 Im(T −λiI). Thus,

(T −λ0I)
(⋂

i>0
Im(T −λiI) ∩D(T)

)
=
⋂

i>0
Im(T −λiI).

Using Lemma 2.1 (iv), we get the desired inclusion.

With all these auxiliary results behind us, we can now state our main result
of this section. We establish a number of important necessary conditions for
a point in the approximate point spectrum to be isolated.

Theorem 3.2. Let T ∈ CR(X) and let 0 be an isolated point of σap(T).
Then:

(i) H0(T) and K(T) are closed.

(ii) H0(T) ∩K(T) = {0}.
(iii) H0(T) ⊕K(T) is closed and there exists λ0 such that

H0(T) ⊕K(T) = K(T −λ0I) = R∞(T −λ0I).

Proof. The proof follows the approach taken in [8, Proposition 9] estab-
lished in the setting of bounded operators. We divide the proof into two
cases.

First case: Assume that T is surjective. Then, by Lemma 1.1, there exists
θ > 0 such that T −λI is bijective for all λ ∈ D∗(0,θ). Therefore, by virtue
of Theorem 3.1, we get H0(T) and K(T) are closed and X = H0(T) ⊕K(T).

Second case: Assume that T is not surjective. Since 0 is isolated in σap(T),
then there exists µ > 0 such that T−λI is bounded below for each 0 < |λ| < µ.
Using Lemma 3.3, the map λ → K(T − λI) is locally constant on D∗(0,µ).
Which implies that the map λ → K(T − λI) is constant on D∗(0,µ). Now,
fix λ0 ∈ D∗(0,µ). Then, by virtue of Lemma 2.1 (v), we get that

K(T −λ0I) = R∞(T −λ0I) := X0 and it is closed.

Moreover, we have T(R∞(T − λ0I) ∩ D(T)) = (T − λ0I + λ0I)(R∞(T −
λ0I) ∩ D(T)) ⊆ R∞(T − λ0I) + λ0R∞(T − λ0I) ⊆ R∞(T − λ0I). Let T0 :



isolated points of the approximate point spectrum 87

X0 ∩D(T) → X0 be the restriction of T̃, where T̃ is the relation defined by
T̃ : XT → X,x 7→ Tx. Now, we divide the remaining proof into four steps.

First step: Show that K(T) = K(T0). Let x ∈ K(T). We claim that there
exist δ > 0 and a sequence (xn)n such that

• x0 = x,

• xn+1 ∈ D(T) and xn ∈ Txn+1 for all n ≥ 0,

• ‖xn‖T ≤ δn‖x‖ for all n ≥ 1.

Indeed, if x ∈ K(T), then there exist c > 0 and a sequence (yn)n such that

• y0 = x,

• for all n ≥ 0, yn+1 ∈ D(T) and yn ∈ Tyn+1,

• d(yn, Ker(T) ∩T(0)) ≤ cnd(x, Ker(T) ∩T(0)).

Let d > c. Then, for all n ≥ 1 there exists αn ∈ T(0) ∩ Ker(T) ⊆ D(T)
such that ‖yn −αn‖ ≤ dn‖x‖. Let (xn)n be the sequence defined by xn+1 =
yn+1 − αn+1 for all n ≥ 0 and x0 = x. Then, for all n ≥ 0, xn+1 ∈ D(T),
xn ∈ Txn+1 and ‖xn‖ ≤ dn‖x‖. On the other hand, we have ‖xn‖T =
‖xn‖+‖QTTxn‖ = ‖xn‖+‖QT (xn−1)‖. Then, ‖xn‖T = ‖xn‖+d(xn−1,T(0)),
which implies that

‖xn‖T ≤ dn‖x‖ + ‖xn−1‖≤ (dn + dn−1)‖x‖.

Consequently, there exists δ > 0 such that ‖xn‖T ≤ δn‖x‖, as claimed. Let g
be the analytic function g : D(λ0,

1
d
) → XT defined by

g(λ) =
∑∞

n=0
xn+1(λ−λ0)n.

Using Remark 2.1, we get for all λ ∈ D(λ0, 1d),

QTT
(∑

n≥0
(λ−λ0)nxn+1

)
= QT

∑
n≥0

(λ−λ0)nxn.

Which implies that

T
(∑

n≥0
(λ−λ0)nxn+1

)
−
∑

n≥0
(λ−λ0)nxn ⊆ T(0).



88 m. lajnef, m. mnif

Thus, T
(∑

n≥0(λ−λ0)
nxn+1

)
=
∑

n≥0(λ−λ0)
nxn + T(0). Whence,

(T − (λ−λ0)I)
∑

n≥0
(λ−λ0)nxn+1 =

∑
n≥0

(λ−λ0)nxn + T(0)

−
∑

n≥1
xn(λ−λ0)n.

Therefore,

(T − (λ−λ0)I)g(λ) = x + T(0), for each |λ−λ0| <
1

d
. (3.2)

Particularly, x ∈
⋂
λ∈D(λ0, 1d )

Im(T − λI). Hence, it follows from Lemma 3.4
that x ∈ K(T −λ0I), which means that K(T) ⊆ X0. Moreover, let � > 0 be
such that � < inf( 1

d
,µ). Then, by (3.2), we have for each 0 < |λ − λ0| < �,

g(λ) + Ker(T −(λ−λ0)I) = (T −(λ−λ0)I)−1x + (T −(λ−λ0)I)−1(T −(λ−
λ0)I)(0). Then, g(λ) = (T − (λ − λ0)I)−1x ∈ X0 for each 0 < |λ − λ0| < �
and, by continuity, we get that x1 = g(λ0) ∈ X0. Now, let h be the analytic
function h : D(λ0,

1
d
) → XT defined by

h(λ) =
∑∞

n=0
xn+2(λ−λ0)n.

Arguing as in (3.2), we get that

(T − (λ−λ0)I)h(λ) = x1 + T(0), for each |λ−λ0| <
1

d
.

Moreover, there exists � > 0 such that for each 0 < |λ − λ0| < �, h(λ) =
(T −(λ−λ0)I)−1x1 ∈ X0 and, by continuity, we get that x2 = h(λ0) ∈ X0. In
a similar way, we prove that xn ∈ X0 for all n ≥ 1. Consequently, x ∈ K(T0)
and hence, K(T) ⊆ K(T0). Observe that K(T0) ⊆ K(T). Then, we obtain
K(T) = K(T0).

Second step: Show that H0(T) = H0(T0). We claim that H0(T) ⊆ X0.
Indeed, according to Lemma 2.1 (ii), we get that H0(T) ⊆ Im(λI −T) =
Im(λI −T) for each 0 < |λ| < µ. Which implies that

H0(T) ⊆ (λI −T)Im(λI −T) for each 0 < |λ| < µ.

Therefore, H0(T) ⊆ Im(λI −T)2. As the power of a bounded below linear
relation is also a bounded below linear relation then, for each λ ∈ D∗(0,µ),
Im(λI−T)2 is closed and hence, H0(T) ⊆ Im(λI−T)2 for each λ ∈ D∗(0,µ).
By repeating this process we get that

H0(T) ⊆ R∞(λI −T) = X0.



isolated points of the approximate point spectrum 89

Therefore, it follows from Lemma 2.1 (i) that H0(T) = H0(T0).
Third step: Show that 0 is isolated in σ(T0). It is easy to see that T0−λI is

injective for each 0 < |λ| < µ. Furthermore, by virtue of Lemma 2.1 (iii) we get
(T0−λI)(X0∩D(T)) = (T−λI)(X0∩D(T)) = (T−λI)(K(T−λ0I))∩D(T)) =
(T −λI)(K(T −λI)∩D(T)) = K(T −λI) = K(T −λ0I) = X0. Then, T0−λI
is surjective whenever 0 < |λ| < µ and so, T0 − λI is bijective for each
0 < |λ| < µ. Hence, 0 is isolated in σ(T0).

Last step: Show that H0(T) and K(T) are closed and X0 = H0(T)⊕K(T).
Using the third step and Theorem 3.1, we get that K(T0) and H0(T0) are
closed in X0 and so in X and X0 = H0(T0) ⊕ K(T0). But, we have, by the
first step and the second step, that K(T) = K(T0) and H0(T) = H0(T0) then
we get the desired result.

Remark 3.1. At this point, a natural question arises: Are necessary condi-
tions given in Theorem 3.2 also sufficient? The answer to this question remains
open. However, it is known that, in the particular case of bounded operators,
the conditions (i) and (ii) of Theorem 3.2 are not sufficient to conclude that
0 is isolated in σap(T) (see the remark in [8, page 4]).

Remark 3.2. It is worthy to point out that the investigation of the quasinil-
potent part H0(T) and the analytic core K(T) of a linear relation T is con-
venient in the study of the spectral properties of relations. However, it is
sometimes difficult to find them explicitly. Theorem 3.2 gives an alternative
way to study the properties of these two subspaces without computing them.

Example 3.1. Let consider the separable Hilbert space l2(N) and let then
(en)n≥0 be her canonical basis. For k ∈ N∗ fixed, we define the bounded and
closed linear relation T in l2(N) by:

T((x0,x1, . . .)) =(x1 + . . . + xk,x2 + . . . + xk, . . . ,xk, 0, 0, 0,xk+1,xk+2, . . .)

+ 〈ek+1〉.

We claim that 0 is an isolated point of σap(T). Indeed, if we set N = 〈(en)kn=0〉
and M = 〈(en)n≥k+1〉, then we have M⊕N = l2(N) and T = TN ⊕TM , where
TN is the bounded nilpotent operator on N of degree k + 1 represented by the
matrix 


0 1 . . . 1
... 0

. . .
...

...
...

. . . 1
0 0 . . . 0






90 m. lajnef, m. mnif

and TM is the linear relation defined on M by TM = S
−1
g Sd, whether Sg

and Sd are the left and right shift operators on M. It is clear that TM is a
bounded below linear relation and hence, by [7, Theorem 3.10], we deduce
that T is a left Drazin invertible linear relation. Therefore, it follows from [4,
Theorem 4.1] that 0 is isolated in σap(T). Thus, by virtue of Theorem 3.2, we
get thatH0(T) and K(T) are closed, H0(T) ∩K(T) = {0}, H0(T) ⊕K(T) is
closed and there exists λ0 such that

H0(T) ⊕K(T) = K(T −λ0I) = R∞(T −λ0I).

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	Introduction and preliminaries
	Quasinilpotent part and analytic core of a closed linear relation
	Isolated point of the approximate point spectrum