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EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 34, Num. 1 (2019), 29 – 40

doi:10.17398/2605-5686.34.1.29

Available online February 3, 2019

Browder essential approximate pseudospectrum and
defect pseudospectrum on a Banach space

Aymen Ammar, Aref Jeribi, Kamel Mahfoudhi

Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax
Route de Soukra Km 3.5, B.P. 1171, 3000 Sfax, Tunisia

ammar aymen84@yahoo.fr , Aref.Jeribi@fss.rnu.tn , kamelmahfoudhi72@yahoo.com

Received January 18, 2018 Presented by Pietro Aiena
Accepted July 5, 2018

Abstract: In this paper, we introduce and study the Browder essential approximate pseudospectrum

and the Browder essential defect pseudospectrum of bounded linear operators on a Banach space.
Moreover, we characterize these spectra and will give some results concerning the stability of them

under suitable perturbations.

Key words: Pseudospectrum, Browder essential approximate pseudospectrum, Browder essential
defect pseudospectrum.

AMS Subject Class. (2010): 39B82, 44B20, 46C05.

1. Introduction

Let X be an infinite-dimensional Banach space, let L(X) be the set of all
bounded linear operators acting on X, and let K(X) be its ideal of compact
operators on X.

Let T ∈ L(X). Then D(T), N(T), α(T), R(T), β(T), T
′

and σ(T) are,
respectively, used to denote the domain, the kernel, the nullity, the range,
the defect, the adjoint and the spectrum of T. If the range R(T) is closed
and α(T) < ∞ (resp. β(T) < ∞) then T is said to be an upper semi-
Fredholm operator (resp. a lower semi-Fredholm operator). The set of upper
semi-Fredholm operators (resp. lower semi-Fredholm operators) is denoted by
Φ+(X) (resp. Φ−(X)). The set of all semi-Fredholm operators is defined by

Φ±(X) := Φ+(X) ∪ Φ−(X), and

the class Φ(X) of all Fredholm operators is defined by

Φ(X) := Φ+(X) ∩ Φ−(X).

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

https://doi.org/10.17398/2605-5686.34.1.29
mailto:ammar_aymen84@yahoo.fr
mailto:Aref.Jeribi@fss.rnu.tn
mailto:kamelmahfoudhi72@yahoo.com
https://www.eweb.unex.es/eweb/extracta/
https://creativecommons.org/licenses/by-nc/3.0/


30 a. ammar, a. jeribi, k. mahfoudhi

The index of a semi-Fredholm operator T is defined by

i(T) = α(T) −β(T).

An operator F ∈L(X) is called a Fredholm perturbation if T + F ∈ Φ(X)
whenever T ∈ Φ(X). The set of Fredholm perturbations is denoted by F(X).
An operator F ∈L(X) is called an upper semi-Fredholm perturbation (resp. a
lower semi-Fredholm perturbation) if T + F ∈ Φ+(X) (resp. T + F ∈ Φ−(X))
whenever T ∈ Φ+(X) (resp. T ∈ Φ−(X)). The set of upper semi-Fredholm
perturbations (resp. lower semi-Fredholm perturbations) is denoted by F+(X)
(resp. F−(X)). Now, we define the minimum modulus

m(T) := inf
{
‖Tx‖ : x ∈D(X), ‖x‖ = 1

}
,

and the defect modulus

q(T) := sup
{
r > 0 : rBX ⊂ TBX

}
,

where BX is the closed unit ball of X. For more information see [16] and [20].
Note that m(T) > 0 if and only if T is bounded below, i.e. T is injective and
T has closed range and q(T) > 0 if and only if T is surjective. Recall also that
m(T∗) = q(T) and q(T∗) = m(T).

The ascent (resp. descent) of T ∈L(X) is the smallest nonnegative integer
a := asc(T) (resp. d := desc(T)) such that N(Ta) = N(Ta+1) (resp. R(Td) =
R(Td+1)). If such an integer does not exist, then asc(T) = ∞ (resp. desc(T) =
∞). We also introduce some special parts of pseudospectrum having valuable
spectral properties such as

σap(T) :=
{
λ ∈ C : m(λ−T) = 0

}
,

σδ(T) :=
{
λ ∈ C : q(λ−T) = 0

}
.

The spectrum σap(T) (resp. σδ(T)) is called the approximate spectrum (resp.
defect spectrum). The Browder essential spectrum of T is defined as

σeb(T) :=
⋂
KT (X)

σ(T + K), (1.1)

the Browder essential approximate point spectrum of T is defined as

σeab(T) :=
⋂
KT (X)

σap(T + K), (1.2)



browder essential approximate pseudospectrum 31

and the Browder essential defect spectrum of T is defined as

σeδb(T) :=
⋂
KT (X)

σδ(T + K), (1.3)

where
KT (X) :=

{
K ∈ K(X) : TK = KT

}
.

For more information on the Browder essential approximate spectrum and his
essential defect spectrum one may refer to [1, 9, 17, 18]. It is clear that

σeb(T) = σeab(T) ∪σeδb(T).

The pseudospectrum of bounded linear operators T on a Banach space X
can be split into subsets in many different ways, depending on the purpose
one has in mind. We may refer to [2, 3, 5, 7, 13] as examples.

Definition 1.1. Let T ∈L(X) and ε > 0. We define the following sets:

(i) the pseudospectrum

σε(T) =
⋃
DT (X)

σ(T + D),

(ii) the approximate pseudospectrum

σap,ε(T) =
⋃
DT (X)

σap(T + D),

(iii) the defect pseudospectrum

σδ,ε(T) =
⋃
DT (X)

σδ(T + D),

where
DT (X) =

{
D ∈L(X) : ‖D‖ < ε, TD = DT

}
.

In this paper we study some parts of the pseudospectrum of bounded
linear operators on a Banach space from the viewpoint of Fredholm theory. In
particular, we study the Browder essential approximate pseudospectrum and
the Browder essential defect pseudospectrum. We have already mentioned
that (1.1), (1.2) and (1.3) inherit ε-versions, which are the Browder essential



32 a. ammar, a. jeribi, k. mahfoudhi

pseudospectrum σeb,ε(·), the Browder essential approximate pseudospectrum
σeab,ε(·) and the Browder essential defect pseudospectrum σeδb,ε(·) defined by

σeb,ε(T) =
⋂
KT (X)

σε(T + K),

σeab,ε(T) =
⋂
KT (X)

σap,ε(T + K),

σeδb,ε(T) =
⋂
KT (X)

σδ,ε(T + K).

This paper is divided into three sections. In the second one we recall
some facts which are helpful to prove the main results. Throughout the third
section we characterize Browder essential approximate pseudospectrum and
the Browder essential defect pseudospectrum. Finally, we prove the invariance
of the Browder essential approximate pseudospectrum and his essential defect
pseudospectrum and establish some results of perturbation on the context of
linear operators on a Banach space.

2. Auxiliary results

In order to prove our main results we begin by introducing some well
known perturbation results

Lemma 2.1. ([16, Theorem 9]) Let T,K ∈L(X). We have

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

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

The following result was proved in [11].

Lemma 2.2. Let T ∈ L(X) and K ∈ K(X) such that K commutes with
T. We have

(i) If T ∈ Φ+(X) then asc(T) < ∞ if, and only if, asc(T + K) < ∞.

(ii) If T ∈ Φ−(X) then desc(T) < ∞ if, and only if, desc(T + K) < ∞.

A bounded operator R ∈L(X) on a Banach space X is said to be a Riesz
operator if λ−T ∈ Φ(X) for every λ ∈ C\{0}. The class of all Riesz operators
is denoted by R(X).



browder essential approximate pseudospectrum 33

Lemma 2.3. ([15, Theorem 3.5]) Let R ∈R(X) which commutes with T.
We have

(i) If T ∈ Φ+(X) then asc(T) < ∞ if and only if asc(T + R) < ∞.

(ii) If T ∈ Φ−(X) then desc(T) < ∞ if and only if desc(T + R) < ∞.

Lemma 2.4. ([14, Theorem 3.9]) Let T ∈ Φ+(X). The following state-
ments are equivalent:

(i) i(T) ≤ 0.

(ii) T can be expressed in the form T = S + K where K ∈ K(X) and
S ∈L(X) is an operator with closed range and α(S) = 0.

3. Main results

In this section we establish an useful result for the Browder essential ap-
proximate pseudospectrum and the Browder essential defect pseudospectrum.
We start our characterization with the following theorem:

Theorem 3.1. Let T ∈L(X) and ε > 0. Then

(i) λ /∈ σeab,ε(T) if and only if, for all D ∈ L(X) such that ‖D‖ < ε, we
have λ−T −D ∈ Φ+(X), i(λ−T −D) ≤ 0 and asc(λ−T −D) < ∞.

(ii) λ /∈ σeδb,ε(T) if and only if, for all D ∈ L(X) such that ‖D‖ < ε, we
have λ−T −D ∈ Φ−(X), i(λ−T −D) ≥ 0 and desc(λ−T −D) < ∞.

Proof. (i) Let λ /∈ σeap,ε(T). Then there exists a compact operator K on
X such that TK = KT and λ /∈ σap,ε(T +K). According to the Definition 1.1,
we obtain that λ /∈ σap(T + D + K) for all D ∈L(X) such that ‖D‖ < ε and
D commutes with T + K. Therefore,

λ−T−D−K ∈ Φ+(X), i(λ−T−D−K) ≤ 0 and asc(λ−T−D−K) = 0

for all D ∈L(X) such that ‖D‖ < ε. Since K commutes with λ−T −D−K,
from Lemma 2.2 we obtain asc(λ−T−D) < ∞. Using Lemma 2.1, we deduce
that

λ−T −D ∈ Φ+(X) and i(λ−T −D) ≤ 0.



34 a. ammar, a. jeribi, k. mahfoudhi

To prove the converse, suppose that for all D ∈ L(X) such that ‖D‖ < ε we
have

λ−T −D ∈ Φ+(X), i(λ−T −D) ≤ 0 and asc(λ−T −D) < ∞.

There are two possible cases:

1stcase : If λ /∈ σap,ε(T) then λ /∈ σap,ε(T + K), so the proof is completed.

2ndcase : If λ ∈ σap,ε(T) then from [16, Theorem 10] we infer that the space
X is decomposed into a direct sum of two closed subspaces X0 and X1 such
that dim X0 < ∞, (λ − T − D)(Xi) ⊆ Xi for i ∈ {1, 2}, (λ − T − D)\X0 is
nilpotent operator and (λ − T − D)\X1 is injective operator. Let K be the
finite rank operator defined by{

K = I on X0,

K = 0 on X1.

It is clear that K is a compact operator commuting with T and D such that
λ−T −D −K is an injective operator (i.e. α(λ−T −D −K) = 0). Then,
from Lemma 2.4 there exists a constant c > 0 such that

‖(λ−T −D −K)x‖≥ c‖x‖, for all x ∈D(T).

This proves that infx∈X, ‖x‖=1 ‖(λ − T − D − K)x‖ ≥ c > 0. Thus λ /∈
σap(T +D+K). Moreover, (T +D)K = K(T +D) and by using Definition 1.1
we infer that λ /∈ σap,ε(T + K). Hence λ /∈ σeab,ε(T).

(ii) Reasoning in the same way as (i), it suffices to replace Φ+(·), σeab,ε(·),
σap,ε(·), i(·) ≤ 0 and (λ − T − D)\X1 , which is injective, by Φ−(·), σeδb,ε(·),
σδ,ε(·), i(·) ≥ 0 and (λ−T −D)\X1 , which is surjective, respectively.

Remark 3.1. It follows immediately from Theorem 3.1 (i) that

σeab,ε(T) =
⋃
‖D‖<ε

σeab(T + D).

Moreover, it follows from Theorem 3.1 (ii) that

σeδb,ε(T) =
⋃
‖D‖<ε

σeδb(T + D).



browder essential approximate pseudospectrum 35

Next, the Browder essential approximate pseudospectrum and the Brow-
der essential defect pseudospectrum will be characterized by means of semi-
Fredholm perturbation. We set

F+T (X) =
{
F ∈F+(X) : TF = FT

}
,

and
F−T (X) =

{
F ∈F−(X) : TF = FT

}
.

Theorem 3.2. Let T ∈L(X) and ε > 0. Then

(i) σeab,ε(T) =
⋂

F∈F+
T
(X)

σap,ε(T + F).

(ii) σeδb,ε(T) =
⋂

F∈F−
T
(X)

σδ,ε(T + F).

Proof. (i) For the first inclusion, it is clear that KT (X) ⊂F+T (X). Then,⋂
F∈F+

T
(X)

σap,ε(T + F) ⊂
⋂

F∈KT (X)

σap,ε(T + F) := σeab,ε(T).

For the second inclusion, let λ /∈
⋂

F∈F+
T
(X)

σap,ε(T + F), then there exists

F ∈F+(X) such that TF = FT and λ /∈ σap,ε(T + F). Using Definition 1.1,
we have λ /∈ σap(T + F + D) for all D ∈ L(X) such that ‖D‖ < ε and D
commutes with T and F . Hence,

λ−T−D−F ∈ Φ+(X), i(λ−T−D−F) ≤ 0 and asc(λ−T−D−F) = 0.

Since F commutes with λ−T −D −F, from Lemma 2.3, it follows that

asc(λ−T −D) < ∞

and from Lemma 2.1 we deduce that

λ−T −D ∈ Φ+(X) and i(λ−T −D) ≤ 0.

Hence λ /∈ σeab,ε(T).

(ii) The proof is similar to that of the first part.



36 a. ammar, a. jeribi, k. mahfoudhi

If we set

RT (X) =
{
R ∈R(X) : TR = RT

}
,

then Theorem 3.2 remains true if F+T (X) and F
−
T (X) are replaced by RT (X).

We then have

σeab,ε(T) =
⋂
RT (X)

σap,ε(T + R) and σeδb,ε(T) =
⋂
RT (X)

σδ,ε(T + R).

Definition 3.1. An operator T ∈ L(X) is said to be quasi-compact op-
erator (T ∈ QK(X)) if there exists a compact operator K and an integer m
such that

‖Tm −K‖ < 1.

If T ∈L(X), we define the set

QKT (X) =
{
K ∈QK(X) : TK = KT

}
We invite the reader to [6] for more information about the quasi-compactness
operators. We have the following inclusions

KT (X) ⊂RT (X) ⊂QKT (X).

If T ∈L(X) we define the sets

SεT (X) =
{
K ∈L(X) : K commutes with T + D and

(λ−T −D −K)−1K ∈QKT (X) for all D ∈L(X)

such that ‖D‖ < ε and λ ∈ ρ(T + D + K)
}
,

and

LεT (X) =
{
K ∈L(X) : K commutes with T + D and

K(λ−T −D −K)−1 ∈QKT (X) for all D ∈L(X)

such that ‖D‖ < ε and λ ∈ ρ(T + D + K)
}
.

Theorem 3.3. Let T ∈L(X) with nonempty resolvent set. Then,

σeab,ε(T) =
⋂

K∈ Sε
T
(X)

σap,ε(T + K).



browder essential approximate pseudospectrum 37

Proof. Let λ /∈
⋂
K∈Sε

T
(X) σap,ε(T + K), then there exists K ∈S

ε
T (X) such

that for every ‖D‖ < ε and λ ∈ ρ(T + D + K), we have

(λ−T −D −K)−1K ∈QKT (X) and λ /∈ σap,ε(T + K).

Using [6, Theorem 1.6] we obtain that

I + (λ−T −D −K)−1K ∈ Φ(X) and i
(
I + (λ−T −D −K)−1K

)
= 0.

Since we can write

λ−T −D = (λ−T −D −K)
(
I + (λ−T −D −K)−1K

)
.

According to Definition 1.1, we have for all D ∈L(X) such that ‖D‖ < ε,

(T + K)D = D(T + K) and λ /∈ σap(T + D + K).

We conclude for all D ∈ L(X) such that ‖D‖ < ε that λ−T −D ∈ Φ+(X).
Also, we have

i(λ−T −D) = i(λ−T −D −K) ≤ 0.

It remains to show that asc(λ − T − D) < 0 for all D ∈ L(X) such that
‖D‖ < ε. Let K commutes with T + D, then K commutes with λ−T −D for
every λ ∈ C. Then

(λ−T −D)n = (λ−T −D −K)n
(
I + (λ−T −D −K)−1K

)n
=
(
I + (λ−T −D −K)−1K

)n
(λ−T −D −K)n

for every n ∈ N. Use the fact that (λ−T −D)n is injective
(
i.e., 0 belongs to

N((λ−T −D)n)
)
, This implies that λ−T −D is injective

(
N(λ−T −D) ⊂

N((λ − T − D)n) for every n
)
. Consequently, the ascent of λ − T − D is

0. Then asc(λ − T − D) < ∞. This prove that λ /∈ σeab,ε(T). The opposite
inclusion follows from KT (X) ⊆SεT (X). Then⋂

K∈ Sε
T
(X)

σap,ε(T + K) ⊆
⋂

K∈KT (X)

σap,ε(T + K).

Corollary 3.1. Let T ∈L(X) with nonempty resolvent set. Then,

σeab,ε(T) =
⋂

K∈ Lε
T
(X)

σap,ε(T + K).



38 a. ammar, a. jeribi, k. mahfoudhi

Proposition 3.1. Let T ∈L(X) with nonempty resolvent set. Then,

σeδ,ε(T) =
⋂

K∈ Sε
T
(X)

σδ,ε(T + K) =
⋂

K∈ Lε
T
(X)

σδ,ε(T + K).

Remark 3.2. Let T ∈L(X) and ε > 0.

(i) Let UT (X), (resp. VT (X)) be a subset of L(X). If KT (X) ⊂ UT (X) ⊂
SεT (X), (resp. KT (X) ⊂VT (X) ⊂L

ε
T (X) ) then

σeab,ε(T) =
⋂

K∈UT (X)

σap,ε(T + K) =
⋂

K∈VT (X)

σap,ε(T + K).

(
resp. σeδ,ε(T) =

⋂
K∈UT (X)

σδ,ε(T + K) =
⋂

K∈VT (X)

σδ,ε(T + K)
)
.

(ii) If for all J,J2 ∈ UT (X) (resp. VT (X)) we have J ± J2 ∈ UT (X) (resp.
VT (X)) then for each J ∈UT (X) (resp. VT (X)) we have

σeab,ε(T + J) = σeab,ε(T) and σeδ,ε(T + J) = σeδ,ε(T).

In the next theorem we will give a fine characterization of σeab,ε(·) and
σeδ,ε(·) by means of T + D-bounded perturbations.

Definition 3.2. An operator T ∈L(X) is called T-bounded if there exist
c > 0 such that

‖Bx‖≤ c(‖x‖ + ‖Tx‖) for all x ∈D(T) ⊂D(B).

We define for all T ∈L(X) the set

HεT (X) =
{
K ∈SεT (X) : K is (T + D)-bounded

}
.

Theorem 3.4. Let T ∈L(X) and ε > 0. Then,

σeab,ε(T) =
⋂

K∈Hε
T
(X)

σap,ε(T + K).

Proof. Because KT (X) ⊆HεT (X), then⋂
K∈Hε

T
(X)

σap,ε(T + K) ⊆
⋂

K∈KT (X)

σap,ε(T + K) := σeab,ε(T).



browder essential approximate pseudospectrum 39

Conversely, let λ /∈
⋂
K∈Hε

T
(X) σap,ε(T + K), then there exists K ∈ H

ε
T (X)

such that
λ /∈ σap,ε(T + K),

which means that for all D ∈L(X) such that ‖D‖ < ε we have λ−T −D−K
is injective. Using [6, Theorem 1.6] we obtain that

I + (λ−T −D −K)−1K ∈ Φ(X) and i
(
I + (λ−T −D −K)−1K

)
= 0.

We can write

λ−T −D = (λ−T −D −K)
(
I + (λ−T −D −K)−1K

)
.

The proof of our statement is then obtained by using the same argument of
the proof of Theorem 3.3.

By using analogous arguments to those of the proof of Theorem 3.4 we
obtain:

Theorem 3.5. Let T ∈L(X) and ε > 0. Then,

σeδ,ε(T) =
⋂

K∈Hε
T
(X)

σδ,ε(T + K).

Acknowledgements

The authors wish to warmly thank the referee for all his suggestions.

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[22] M.P.H. Wolff, Discrete approximation of unbounded operators and approx-
imation of their spectra, J. Approx. Theory 113 (2) (2001), 229 – 244.


	Introduction
	Auxiliary results
	Main results