The S-Jeribi essential spectrum
UDC 517.9We study some properties and results on the S-Jeribi essential spectrum of linear bounded operators on a Banach space. In particular, we give some criteria for coincidence of this spectrum for two linear operators and the relation of this type of spectrum with the well-known S-Schechter ess...
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| author | Belabbaci, C. Belabbaci, C. Belabbaci, C. |
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| description | UDC 517.9We study some properties and results on the S-Jeribi essential spectrum of linear bounded operators on a Banach space. In particular, we give some criteria for coincidence of this spectrum for two linear operators and the relation of this type of spectrum with the well-known S-Schechter essential spectrum. |
| doi_str_mv | 10.37863/umzh.v73i3.163 |
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DOI: 10.37863/umzh.v73i3.163
UDC 517.9
C. Belabbaci (Laboratory Pure and Appl. Math., Dep. Math., Univ. Laghouat, Algeria)
THE S-JERIBI ESSENTIAL SPECTRUM
СУТТЄВИЙ S-СПЕКТР ДЖЕРIБI
We study some properties and results on the S-Jeribi essential spectrum of linear bounded operators on a Banach space. In
particular, we give some criteria for coincidence of this spectrum for two linear operators and the relation of this type of
spectrum with the well-known S-Schechter essential spectrum.
Вивчено властивостi суттєвого S-спектра Джерiбi для лiнiйних операторiв у просторi Банаха та отримано деякi вiд-
повiднi результати. Зокрема, сформульовано критерiї збiгу цих спектрiв для двох лiнiйних операторiв i встановлено
зв’язок мiж цим типом спектра та вiдомим суттєвим S-спектром Шехтера.
1. Introduction. The spectral theory of operator pencils \lambda S - T, \lambda \in \BbbC (operator-valued functions
of a complex argument) play a crucial role in several branches of mathematical physics (see, for
example, [1, 5, 4, 11]), this notion has recently attracted the attention of many mathematicians.
In particular, F. Abdmouleh, A. Ammar and A. Jeribi investigate, in [1], the S-Browder essential
spectrum of bounded linear operators on a Banach space X and extended many results to various
types of S-essential spectra. Moreover, in [6], we have introduced the notion of the S-Jeribi essential
spectrum and we gave some localization of this S-essential spectrum using the notion of measure of
noncompactness. An obvious question to ask, when introducing this notion, is whether the different
results of the S-essential spectra continue to hold for the S-Jeribi essential spectrum?
This work continues research begun in [6], we will give many properties for the S-Jeribi essential
spectrum. In particular, a characterization of the S-Jeribi essential spectrum is given, in Theorem 3.1,
when the operator S is Fredholm with index zero. In Theorem 3.2, we show the existence of a weakly
compact operator W for which the S-Jeribi essential spectrum of T coincides with the S-Schechter
essential spectrum of the operator (T +W ). Furthermore, we state a condition for which the S-Jeribi
essential spectrum coincides with the S-Schechter essential spectrum in Theorem 3.5.
The paper is organized as follows. Section 2 contains an overview of the necessary background.
The main results of the paper are found in Section 3.
2. Preliminaries. The purpose of this section is to recall some results on Fredholm operators
and S-essential spectra. Let X be a Banach space. Denote by \scrL (X) (resp., \scrK (X)) the set of all
bounded linear (resp., compact) operators on X. The nullity, \alpha (T ), of an operator T is defined as
the dimension of \mathrm{k}\mathrm{e}\mathrm{r}(T ) and the deficiency, \beta (T ), of T is defined as the codimension of the range
R(T ) in X.
The sets of upper and lower semi-Fredholm operators in \scrL (X) are defined, respectively, by
\Phi +(X) = \{ T \in \scrL (X) : such that \alpha (T ) < \infty and R(T ) is closed\} ,
\Phi - (X) = \{ T \in \scrL (X) : such that \beta (T ) < \infty \} .
The sets of Fredholm and semi-Fredholm operators are defined, respectively, by
\Phi (X) = \Phi +(X) \cap \Phi - (X) and \Phi \pm (X) = \Phi +(X) \cup \Phi - (X).
The index of T, denoted \mathrm{i}\mathrm{n}\mathrm{d}(T ), is defined as \mathrm{i}\mathrm{n}\mathrm{d}(T ) = \alpha (T ) - \beta (T ). An operator F is called a
Fredholm perturbation if T+F \in \Phi (X) whenever T \in \Phi (X). Denote by \scrF (X) the set of Fredholm
c\bigcirc C. BELABBACI, 2021
308 ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
THE S-JERIBI ESSENTIAL SPECTRUM 309
perturbations. Note that \scrF (X) is a closed two-sided ideal of \scrL (X) (see [7, 17]). For more details
about the theory of Fredholm operators we refer to [2, 3, 14, 17].
Now, we recall the definitions of strictly singular operators and weakly compact operators. The
concept of strictly singular operators was introduced in the pioneering paper by Kato [12] as a
generalization of the notion of compact operators.
Definition 2.1. Let T \in \scrL (X). T is said to be strictly singular if the restriction of T to any
infinite-dimensional subspace of X is not an homeomorphism.
The set of strictly singular operators on X is denoted by \scrS (X). Note that \scrS (X) is a closed two-
sided ideal of \scrL (X) containing \scrK (X) (see [8, 12]). If X is a Hilbert space, then \scrK (X) = \scrS (X). In
general, we have
\scrK (X) \subset \scrS (X) \subset \scrF (X).
Definition 2.2. An operator T \in \scrL (X) is said to be weakly compact if for every bounded subset
B \subset X, T (B) is relatively weakly compact.
The family of weakly compact operators on X is denoted by \scrW (X). Note that \scrW (X) is a
closed two-sided ideal of \scrL (X) containing \scrK (X) (see [8, 12]). Let us notice that, according to [15]
(Theorem 1), we have \scrW (L1(\Omega , d\mu )) = \scrS (L1(\Omega , d\mu )), where (\Omega ,\Sigma , d\mu ) be an arbitrary positive
measure space. If 1 < p < \infty , Lp(\Omega , d\mu ) is reflexive and then \scrL (Lp(\Omega , d\mu )) = \scrW (Lp(\Omega , d\mu )).
Moreover, from [7] (Theorem 5.2) we deduce that
\scrL (Lp(\Omega , d\mu )) \varsubsetneq \scrS (Lp(\Omega , d\mu )) \varsubsetneq \scrW (Lp(\Omega , d\mu )) with p \not = 2.
For p = 2, we obtain
\scrL (L2(\Omega , d\mu )) = \scrS (L2(\Omega , d\mu )) = \scrW (L2(\Omega , d\mu )).
Let T, S be two bounded linear operators on \scrL (X) such that S is non-zero. We define the S-
spectrum of T as \sigma S(T ) = \BbbC \setminus \rho S(T ), where \rho S(T ) is the S-resolvent set defined by
\rho S(T ) = \{ \lambda \in \BbbC : \lambda S - T has a bounded inverse\} .
There are several definitions of the S-essential spectra of an operator T defined on a Banach space
(see, for example, [1, 4, 5]). In this paper, we are concerned with the S-Jerbi essential spectrum
introduced in [6] and the S-Schechter essential spectrum.
Definition 2.3. Let S and T be two bounded linear operators on a Banach space X. The
S-Jeribi essential spectrum of an operator T \in \scrL (X) is defined by
\sigma j,S(T ) :=
\bigcap
K\in \scrW \ast (X)
\sigma S(T +K),
where \scrW \ast (X) stands for each one of the sets \scrW (X) and \scrS (X). In other words, \scrW \ast (X) is either
the set \scrW (X) or the set \scrS (X).
The S-Schechter essential spectrum of an operator T, denoted \sigma e1,S (T ), is defined as follow:
\sigma e1,S (T ) =
\bigcap
K\in \scrK (X)
\sigma S(T +K).
Since \scrK (X) \subset \scrW \ast (X), then \sigma j,S(T ) \subset \sigma e1,S (T ). Note that if S = I (the identity operator), we
recover the definition of the Jeribi essential spectrum (see [9, 10, 13]) and the Schechter essential
spectrum, respectively,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
310 C. BELABBACI
\sigma j(T ) =
\bigcap
K\in \scrW \ast (X)
\sigma (T +K) and \sigma e1(T ) =
\bigcap
K\in \scrK (X)
\sigma (T +K).
We conclude this section with a useful characterization of the S-Schechter essential spectrum by
means of Fredholm operators established in [5].
Proposition 2.1. Let T, S be in \scrL (X). Then
\lambda /\in \sigma e1,S (T ) if and only if (\lambda S - T ) \in \Phi (X) and \mathrm{i}\mathrm{n}\mathrm{d}(\lambda S - T ) = 0.
3. Main results. The purpose of this section is to present our main results on the S-Jeribi
essential spectrum of linear bounded operators on a Banach space X. In this paper, we choose the
definition of the S-Jeribi essential spectrum of T, \sigma j,S(T ), when K belongs to \scrW (X)
\sigma j,S(T ) :=
\bigcap
K\in \scrW (X)
\sigma S(T +K).
From this definition, it is obviously that the S-Jeribi essential spectrum is invariant under weakly
compact perturbations. Now, from the stability of the S-Jeribi essential spectrum under weakly
compact perturbations and Proposition 2.1, we can easily obtain a characterization of the S-Jeribi
essential spectrum by means of Fredholm operators.
Lemma 3.1. Let T, S \in \scrL (X). Then the S-Jeribi essential spectrum is given by
\sigma j,S(T ) = \BbbC \setminus \{ \lambda \in \BbbC : \exists W \in \scrW (X), (\lambda S - T - W ) \in \Phi (X) with \mathrm{i}\mathrm{n}\mathrm{d}(\lambda S - T - W ) = 0\} .
The following theorems give a characteristic of the S-Jeribi essential spectrum, in the particular
case, when S is a Fredholm operator with zero index. Let us consider S \in \Phi (X), then by Atkinson
theorem (see [2], Theorem 4.46) there exist compact operators K1,K2 \in \scrK (X) and an operator
B \in \scrL (X) such that
SB = I +K1 and BS = I +K2.
The statement (iii) of [1] (Proposition 2.2) remains valid for the S-Jeribi essential spectrum. The
following theorem holds.
Theorem 3.1. Let T \in \scrL (X), S \in \Phi (X) such that \mathrm{i}\mathrm{n}\mathrm{d}(S) = 0. Then
\sigma j,S(T ) = \sigma j(TB),
where B is introduced above.
Proof. Let \lambda /\in \sigma j(TB), then there exists a weakly compact operator W on X such that
(\lambda I - TB - W ) \in \Phi (X) with \mathrm{i}\mathrm{n}\mathrm{d}(\lambda I - TB - W ) = 0. Since S \in \Phi (X), the use of [14] (Theorem 5)
implies that (\lambda I - TB - W )S \in \Phi (X) with \mathrm{i}\mathrm{n}\mathrm{d}((\lambda I - TB - W )S) = 0, i.e., (\lambda S - T - TK2 - WS) \in
\in \Phi (X) with \mathrm{i}\mathrm{n}\mathrm{d}(\lambda S - T - TK2 - WS) = 0. Since the operator TK2 is compact, it follows from
the stability of Fredholm operators under compact perturbations that (\lambda S - T - WS) \in \Phi (X)
and \mathrm{i}\mathrm{n}\mathrm{d}(\lambda S - T - WS) = 0. Hence, there exists a weakly compact operator W \prime = WS such
that (\lambda S - T - W \prime ) \in \Phi (X) and \mathrm{i}\mathrm{n}\mathrm{d}(\lambda S - T - W \prime ) = 0. So, \lambda /\in \sigma j,S(T ). This shows the
inclusion \sigma j,S(T ) \subset \sigma j(TB).
By the same argument we get the inverse inclusion.
Theorem 3.1 is proved.
The next main theorem provides conditions for the existence of a weakly compact operator W \prime
on X such that the S-Jeribi essential spectrum of an operator T coincides with the S-Schechter
essential spectrum of the operator (T +W \prime ).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
THE S-JERIBI ESSENTIAL SPECTRUM 311
Theorem 3.2. Let T, S \in \scrL (X), S \in \Phi (X) with \mathrm{i}\mathrm{n}\mathrm{d}(S) = 0. Then there exists a weakly
compact operator W \prime on X such that \sigma e1,S (T +W \prime ) = \sigma j,S(T ).
Proof. Let \lambda /\in \sigma j,S(T ). From Theorem 3.1, it follows that \lambda /\in \sigma j(TB). So, there exists a
weakly compact operator W on X such that (\lambda I - TB - W ) \in \Phi (X) with \mathrm{i}\mathrm{n}\mathrm{d}(\lambda I - TB - W ) = 0.
Since S \in \Phi (X), the use of [14] (Theorem 5) shows that (\lambda I - TB - W )S \in \Phi (X) with \mathrm{i}\mathrm{n}\mathrm{d}((\lambda I -
- TB - W )S) = 0, i.e., (\lambda S - T - TK2 - WS) \in \Phi (X) with \mathrm{i}\mathrm{n}\mathrm{d}(\lambda S - T - TK2 - WS) = 0. By
using the fact that TK2 \in \scrK (X) and the stability of Fredholm operators under compact perturbations,
we obtain (\lambda S - T - WS) \in \Phi (X) and \mathrm{i}\mathrm{n}\mathrm{d}(\lambda S - T - WS) = 0. Consequently, there exists a weakly
compact operator W \prime = WS such that \lambda /\in \sigma e1,S (T + W \prime ). Hence, \sigma e1,S (T + W \prime ) \subset \sigma j(TB) =
= \sigma j,S(T ). The reverse inclusion is obvious since \sigma j,S(T ) = \sigma j,S(T + W ) \subset \sigma e1,S (T + W )
for all W \in \scrW (X).
Theorem 3.2 is proved.
The other main result is the following theorem.
Theorem 3.3. Let T, S \in \scrL (X) such that 0 \in \rho (T ) \cap \rho (S). Then, for all \lambda \not = 0, we have
\lambda \in \sigma j,S(T ) if and only if \lambda - 1 \in \sigma j,S - 1(T - 1).
Proof. Assume that \lambda - 1 /\in \sigma j,S - 1(T - 1). Then there exists a weakly compact operator W on
X such that (\lambda - 1S - 1 - T - 1 - W ) \in \Phi (X) with \mathrm{i}\mathrm{n}\mathrm{d}(\lambda - 1S - 1 - T - 1 - W ) = 0. We obtain (using
the fact that T and S are Fredholm operators) S(\lambda - 1S - 1 - T - 1 - W )T \in \Phi (X). Consequently,
- \lambda (\lambda - 1T - S - SWT ) \in \Phi (X) with \mathrm{i}\mathrm{n}\mathrm{d}( - \lambda (\lambda - 1T - S - SWT )) = 0. So, (\lambda S - T - W \prime \prime ) \in \Phi (X)
with \mathrm{i}\mathrm{n}\mathrm{d}(\lambda S - T - W \prime \prime ) = 0, where W \prime \prime = - \lambda SWT \in \scrW (X). By using Lemma 3.1, we can see
that \lambda /\in \sigma j,S(T ). This proves that
\lambda \in \sigma j,S(T ) \Rightarrow \lambda - 1 \in \sigma j,S - 1(T - 1).
We now prove that \lambda - 1 \in \sigma j,S - 1(T - 1) implies \lambda \in \sigma j,S(T ). To see this, let \lambda /\in \sigma j,S(T ). Then
there exists W \in \scrW (X) such that (\lambda S - T - W ) \in \Phi (X) with \mathrm{i}\mathrm{n}\mathrm{d}(\lambda S - T - W ) = 0. The operator
(\lambda S - T - W ) can be written in the form
(\lambda S - T - W ) = - \lambda S
\bigl(
\lambda - 1S - 1 - T - 1 + \lambda - 1S - 1WT - 1
\bigr)
T. (3.1)
We see that the second member of equation (3.1) is in \Phi (X) with zero index. Applying the index
theorem [2] (Theorem 4.43), we get\bigl(
\lambda - 1S - 1 - T - 1 + \lambda - 1S - 1WT - 1
\bigr)
\in \Phi (X)
with \mathrm{i}\mathrm{n}\mathrm{d}
\bigl(
\lambda - 1S - 1 - T - 1 + \lambda - 1S - 1WT - 1
\bigr)
= 0. We put W \prime = \lambda - 1S - 1WT - 1, then W \prime \in \scrW (X).
Therefore, \lambda - 1 /\in \sigma j,S - 1(T - 1).
Theorem 3.3 is proved.
From Theorem 3.3 with S = I we obtain the following corollary.
Corollary 3.1. Let T \in \scrL (X) such that 0 \in \rho (T ). Then, for all \lambda \not = 0, we have
\lambda \in \sigma j(T ) if and only if \lambda - 1 \in \sigma j(T
- 1).
In next theorem, we will give a relation between the S-Jeribi essential spectrum and the S-
Schechter essential spectrum.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
312 C. BELABBACI
Theorem 3.4. Let T, S \in \scrL (X), W \in \scrW (X) and \xi \in \rho S(T +W ). We put T\xi = (\xi S - T -
- W ) - 1. Then, for \lambda \not = \xi , we have
\lambda \in \sigma j,S(T ) if and only if ( - \lambda + \xi ) - 1 \in \sigma e1(T\xi S).
Proof. Let W \in \scrW (X). The operator (\lambda S - T - W ) can be written in the form
(\lambda S - T - W ) = ( - \lambda + \xi )(\xi S - T - W )(( - \lambda + \xi ) - 1 - T\xi S). (3.2)
Suppose that ( - \lambda + \xi ) - 1 /\in \sigma e1(T\xi S). Then we get (( - \lambda + \xi ) - 1I - T\xi S) \in \Phi (X) and \mathrm{i}\mathrm{n}\mathrm{d}(( - \lambda +
+ \xi ) - 1I - T\xi S) = 0. Since (\xi S - T - W ) \in \Phi (X) with \mathrm{i}\mathrm{n}\mathrm{d}(\xi S - T - W ) = 0, then, by using
[14] (Theorem 5), we see that the second member of equation (3.2) is a Fredholm operator with
zero index. So, we have (\lambda S - T - W ) \in \Phi (X) with \mathrm{i}\mathrm{n}\mathrm{d}(\lambda S - T - W ) = 0, which means that
\lambda /\in \sigma e1,S (T +W ), i.e., \lambda /\in \sigma j,S(T +W ). By using the stability of S-Jeribi essential spectrum under
weakly compact perturbations, we conclude that \lambda /\in \sigma j,S(T ).
Conversely, assume that \lambda /\in \sigma j,S(T ), then there exists W \in \scrW (X) which satisfies (\lambda S - T -
- W ) \in \Phi (X) and ind (\lambda S - T - W ) = 0. By equation (3.2), we have (\xi S - T - W )(( - \lambda +\xi ) - 1 -
- T\xi S) \in \Phi (X). Since (\xi S - T - W ) \in \Phi (X), then by [2] (Theorem 4.43) we get (( - \lambda + \xi ) - 1 -
- T\xi S) \in \Phi (X) and ind (( - \lambda + \xi ) - 1 - T\xi S) = 0. Hence, ( - \lambda + \xi ) - 1 /\in \sigma e1(T\xi S).
Theorem 3.4 is proved.
In the next theorem, we state a condition under which the S-Jeribi essential spectrum coincides
with the S-Schechter essential spectrum.
Theorem 3.5. Let X be a Banach space, T, S \in \scrL (X) and W \in \scrW (X). If there exists
\lambda \in \rho S(T +W ) such that W (\lambda S - T - W ) - 1 \in \scrF (X), then \sigma e1,S (T ) = \sigma j,S(T ).
Proof. It suffices to show the following inclusion: \sigma e1,S (T ) \subset \sigma j,S(T ). Writing the operator
(\lambda S - T ) as follows:
(\lambda S - T ) =
\bigl(
I +W (\lambda S - T - W ) - 1
\bigr)
(\lambda S - T - W ) for any W \in \scrW (X). (3.3)
Let \lambda \in \BbbC such that \lambda /\in \sigma j,S(T ). Then there exists a weakly compact operator W on X satisfies
(\lambda S - T - W ) \in \Phi (X) with ind(\lambda S - T - W ) = 0. Using the hypothesis, Theorem 5 cited in
[14] and together with equation (3.3), we get (\lambda S - T ) \in \Phi (X) with \mathrm{i}\mathrm{n}\mathrm{d}(\lambda S - T ) = 0. Then
\lambda /\in \sigma e1,S (T ).
Theorem 3.5 is proved.
We next show that Theorems 4.7 and 4.8 of [16] are also valid for the S-Jeribi essential spectrum.
Theorem 3.6. Let T, S, L \in \scrL (X) and W \in \scrW (X). If for some \xi \in \rho S(T +W )\cap \rho S(L+W )
the operator (\xi S - T - W ) - 1 - (\xi S - L - W ) - 1 is compact on X, then
\sigma j,S(T ) = \sigma j,S(L).
Proof. Let \lambda \in \sigma j,S(T ). Then, according to Theorem 3.4, we have ( - \lambda + \xi ) - 1 \in \sigma e1(T\xi S)
for all \lambda \not = \xi . We see that this is equivalent to ( - \lambda + \xi ) - 1 \in \sigma e1((\xi S - L - W ) - 1S) since
((\xi S - T - W ) - 1 - (\xi S - L - W ) - 1)S \in \scrK (X) and the Schechter essential spectrum is invariant
under compact perturbations. This means that ( - \lambda +\xi ) - 1 \in \sigma e1(L\xi S) where L\xi = (\xi S - L - W ) - 1.
Hence, by Theorem 3.4 again, we have \lambda \in \sigma j,S(L). This proves the claim.
Theorem 3.6 is proved.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
THE S-JERIBI ESSENTIAL SPECTRUM 313
Theorem 3.7. Let T, L operators in \Phi (X) such that \mathrm{i}\mathrm{n}\mathrm{d}(T ) = \mathrm{i}\mathrm{n}\mathrm{d}(L) = 0 and S \in \scrL (X). If
the operator (A - B) is compact, then
\sigma j,S(T ) = \sigma j,S(L).
Operators A, B are defined in \scrL (X) such that TA = I - K1 and LB = I - K2, K1, K2 \in \scrK (X).
Proof. For any scalar \lambda \in \BbbC and W \in \scrW (X), we can write
(\lambda S - T - W )A - (\lambda S - L - W )B = K1 - K2 + (\lambda S - W )(A - B). (3.4)
Let \lambda /\in \sigma j,S(T ). Then there exists a weakly compact operator W on X such that (\lambda S - T - W ) \in
\in \Phi (X) with \mathrm{i}\mathrm{n}\mathrm{d}(\lambda S - T - W ) = 0. So, (\lambda S - T - W )A \in \Phi (X) since A \in \Phi (X). The second
member of equation (3.4) is compact, then, by the stability of Fredholm operators under compact
perturbations, we see that (\lambda S - L - W )B \in \Phi (X). This implies, by the use of index theorem, that
(\lambda S - L - W ) \in \Phi (X) and \mathrm{i}\mathrm{n}\mathrm{d}(\lambda S - L - W ) = 0. We deduce from Lemma 3.1 that \lambda /\in \sigma j,S(L). So,
\sigma j,S(L) \subset \sigma j,S(T ).
Similarly we prove the opposite inclusion.
Theorem 3.7 is proved.
References
1. F. Abdmouleh, A. Ammar, A. Jeribi, Stability of the s-essential spectra on a Banach space, Math. Slovaca, 63, № 2,
299 – 320 (2013).
2. Y. A. Abramovich, C. D. Aliprantis, An invitation to operator theory, Grad. Stud. Math., vol. 50, Amer. Math. Soc.,
Providence (2002).
3. P. Aiena, Fredholm and local spectral theory, with applications to multipliers, Springer Sci. & Business Media (2004).
4. A. Ammar, B. Boukettaya, A. Jeribi, Stability of the S-left and S-right essential spectra of a linear operator, Acta
Math. Sci. Ser. A (Chin. Ed.), 34, № 6, 1922 – 1934 (2014).
5. A. Ammar, M. Zerai Dhahri, A. Jeribi, A characterization of S- essential spectrum by means of measure of non-strict-
singularity and application, Azerb. J. Math., 5, № 1 (2015).
6. C. Belabbaci, M. Aissani, M. Terbeche, S-essential spectra and measure of noncompactness, Math. Slovaca, 67, № 5,
1203 – 1212 (2017).
7. I. C. Gohberg, A. S. Markus, I. A. Feldman, Normally solvable operators and ideals associated with them, Amer.
Math. Soc. Transl. Ser., 2, № 61, 63 – 84 (1967).
8. S. Goldberg, Unbounded linear operators, McGraw-Hill, New York (1966).
9. A. Jeribi, Spectral theory and applications of linear operators and block operator matrices, Springer (2015).
10. A. Jeribi, K. Latrach, Quelques remarques sur le spectre essentiel et application ‘a l’?equation de transport, Compt.
Rend. Acad. Sci. Sér. 1, Math., 323, № 5 (1996).
11. A. Jeribi, N. Moalla, S. Yengui, S-essential spectra and application to an example of transport operators, Math.
Methods Appl. Sci., 37, 2341 – 2353 (2014).
12. T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Anal. Math., 6, № 1,
261 – 322 (1958).
13. K. Latrach, A. Jeribi, On the essential spectrum of transport operators on L1 -spaces, J. Math. Phys., 37, № 12,
6486 – 6494 (1996).
14. V. Muller, Spectral theory of linear operators and spectral systems in Banach algebras, 2nd ed., Oper. Theory: Adv.
and Appl., vol. 139, Birkhäuser, Basel (2007).
15. A. Pelczynski, On strictly singular and strictly cosingular operators. I. Strictly singular and strictly cosingular
operators in C(X)-spaces. II. Strictly singular and strictly cosingular operators in L(?)-spaces, Bull. Acad. Polon.
Sci., 13, № 1, 31 – 36, 37 – 41 (1965).
16. M. Schechter, Spectra of partial differential operators, vol. 14, North-Holland, Amsterdam (1971).
17. M. Schechter, Principles of functional analysis, Amer. Math. Soc. (2002).
Received 22.06.18,
after revision — 13.06.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
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| id | umjimathkievua-article-163 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:03Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2e/58d6ac40c2c8118180cc12d9742eb12e.pdf |
| spelling | umjimathkievua-article-1632025-03-31T08:48:21Z The S-Jeribi essential spectrum The S-Jeribi essential spectrum The S-Jeribi essential spectrum Belabbaci, C. Belabbaci, C. Belabbaci, C. S-essential spectra Jeribi essential spectrum Fredholm operators S-essential spectra Jeribi essential spectrum Fredholm operators UDC 517.9We study some properties and results on the S-Jeribi essential spectrum of linear bounded operators on a Banach space. In particular, we give some criteria for coincidence of this spectrum for two linear operators and the relation of this type of spectrum with the well-known S-Schechter essential spectrum. УДК 517.9 Суттєвий $S$-спектр Джерiбi Вивчено властивостi суттєвого S-спектра Джерiбi для лiнiйних операторiв у просторi Банаха та отримано деякi вiдповiднi результати. Зокрема, сформульовано критерiї збiгу цих спектрiв для двох лiнiйних операторiв та встановлено зв’язок мiж цим типом спектру та вiдомим суттєвим S-спектром Шехтера. Institute of Mathematics, NAS of Ukraine 2021-03-11 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/163 10.37863/umzh.v73i3.163 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 3 (2021); 308 - 313 Український математичний журнал; Том 73 № 3 (2021); 308 - 313 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/163/8975 Copyright (c) 2021 Chafika Belabbaci |
| spellingShingle | Belabbaci, C. Belabbaci, C. Belabbaci, C. The S-Jeribi essential spectrum |
| title | The S-Jeribi essential spectrum |
| title_alt | The S-Jeribi essential spectrum The S-Jeribi essential spectrum |
| title_full | The S-Jeribi essential spectrum |
| title_fullStr | The S-Jeribi essential spectrum |
| title_full_unstemmed | The S-Jeribi essential spectrum |
| title_short | The S-Jeribi essential spectrum |
| title_sort | s-jeribi essential spectrum |
| topic_facet | S-essential spectra Jeribi essential spectrum Fredholm operators S-essential spectra Jeribi essential spectrum Fredholm operators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/163 |
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