S‐essential spectra and application to an example of transport operators

Jeribi, Aref; Moalla, Nedra; Yengui, Sonia

doi:10.1002/mma.1564

Cited by 36 publications

(20 citation statements)

References 16 publications

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“…: According to the fact that K 22 is a non-negative regular operator with Theorem 2.2 in [20], make us to conclude that: , 1 Ä i Ä 6.…”

Section: Walhamentioning

confidence: 89%

“…So, let

λ_{0} MathClass-rel\in ρ_{M_{1}} MathClass-open(A_{1} MathClass-close) MathClass-punc,

then A 1 − λ 0 M 1 ∈ Φ( X ) and i ( A 1 − λ 0 M 1 ) = 0. The fact that i ( A 1 − μM 1 ) is constant on any component of

Φ_{A_{1}, M_{1}}

(see Proposition 2.1 in ) and

ρ_{M_{1}} (A_{1}) \subset ρ_{4, M_{1}} (A_{1})

leads to i ( A 1 − μM 1 ) = 0 for all

μ MathClass-rel\in ρ_{M_{1}} MathClass-open(A_{1} MathClass-close)

, and in this case, it follows from Equation that

i MathClass-open({\overset{true¯}{S}}_{λ} MathClass-bin- μ M_{4} MathClass-close) MathClass-rel= 0

. Hence, the last expression in conjunction with Equation yields:

\begin{matrix} σ_{e 5, M} (L) = σ_{e 5, M_{1}} (A_{1}) \cup^{​} σ_{e 5, M_{4}} ({\overset{S}{true}}_{λ}) . & (7) \end{matrix}

Lemma 2.1 in with the fact that

C_{σ_{e 5, M_{1}} (A_{1})}

is connected and Equation , imply that

σ_{e 6, M} (L) = σ_{e 6, M_{1}} (A_{1}) \cup^{​} σ_{e 6, M_{4}} ({\overset{S}{true}}_{λ}) .

□…”

Section: The M‐essential Spectra Of Lmentioning

confidence: 95%

“…Theorem If the operators H , M 2 , M 3 are Fredholm perturbations, K 12 , K 22 are non‐negative regular operators and if

\frac{κ_{21}}{(x, ., v') | v' |}

is relatively weakly compact, then

σ_{e i, M} (L) : = {λ \in ℂ such that R e λ \leq - \min (\frac{λ_{*}^{1}}{μ_{*}^{1}}, \frac{λ_{*}^{2}}{μ_{*}^{4}})}, 1 \leq i \leq 6.

Proof It is shown in , that

\begin{matrix} σ_{e i, M_{1}} (T_{1}) = {λ \in ℂ such that Re λ \leq - \frac{λ_{1}^{*}}{μ_{1}^{*}}}, 1 \leq i \leq 6. & (9) \end{matrix}

Let

λ MathClass-rel\in ρ_{M_{1}} MathClass-open(T_{1} MathClass-close)

. The operator S λ can be expressed as:

S_{λ} MathClass-punc: MathClass-rel= T_{2}^{H} MathClass-bin+ K_{22} MathClass-bin+ MathClass-open(K_{21} MathClass-bin- λ M_{3} MathClass-close) K_{λ} Γ_{Y} MathClass-bin- F MathClass-open(λ MathClass-close) MathClass-open(K_{12} MathClass-bin- λ M_{2} MathClass-close) MathClass-punc.

Using Lemma 4.2 and the fact that H is Fredholm perturbation, we deduce from Theorem 2.1 in that …”

Section: Application To Two‐group Transport Equationsmentioning

confidence: 99%

“…The operator S λ can be expressed as:

S_{λ} MathClass-punc: MathClass-rel= T_{2}^{H} MathClass-bin+ K_{22} MathClass-bin+ MathClass-open(K_{21} MathClass-bin- λ M_{3} MathClass-close) K_{λ} Γ_{Y} MathClass-bin- F MathClass-open(λ MathClass-close) MathClass-open(K_{12} MathClass-bin- λ M_{2} MathClass-close) MathClass-punc.

Using Lemma 4.2 and the fact that H is Fredholm perturbation, we deduce from Theorem 2.1 in that

σ_{e i, M_{4}} MathClass-open(S_{λ} MathClass-close) MathClass-rel= σ_{e i, M_{4}} (() T_{2}^{H} + K_{22}) MathClass-punc.

On the other hand, for

λ MathClass-rel\in ρ_{M_{4}} ((), T_{2}^{H})

such that

r_{σ} ({(λ M_{4} - T_{2}^{H})}^{- 1} K_{22}) < 1

( r σ (.) the spectral radius), implies that

{(λ M_{4} - T_{2}^{H} - K_{22})}^{- 1} - {(λ M_{4} - T_{2}^{H})}^{- 1} = \sum_{n \geq 1} {({(λ M_{4} - T_{2}^{H})}^{- 1} K_{22})}^{n} {(λ M_{4} - T_{2}^{H})}^{- 1} .

According to the fact that K 22 is a non‐negative regular operator with Theorem 2.2 in , make us to conclude that: …”

Section: Application To Two‐group Transport Equationsmentioning

confidence: 99%

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On the M-essential spectra of two-group transport equations

Walha

2013

Math. Meth. Appl. Sci.

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“…: According to the fact that K 22 is a non-negative regular operator with Theorem 2.2 in [20], make us to conclude that: , 1 Ä i Ä 6.…”

Section: Walhamentioning

confidence: 89%

“…So, let

λ_{0} MathClass-rel\in ρ_{M_{1}} MathClass-open(A_{1} MathClass-close) MathClass-punc,

then A 1 − λ 0 M 1 ∈ Φ( X ) and i ( A 1 − λ 0 M 1 ) = 0. The fact that i ( A 1 − μM 1 ) is constant on any component of

Φ_{A_{1}, M_{1}}

(see Proposition 2.1 in ) and

ρ_{M_{1}} (A_{1}) \subset ρ_{4, M_{1}} (A_{1})

leads to i ( A 1 − μM 1 ) = 0 for all

μ MathClass-rel\in ρ_{M_{1}} MathClass-open(A_{1} MathClass-close)

, and in this case, it follows from Equation that

i MathClass-open({\overset{true¯}{S}}_{λ} MathClass-bin- μ M_{4} MathClass-close) MathClass-rel= 0

. Hence, the last expression in conjunction with Equation yields:

\begin{matrix} σ_{e 5, M} (L) = σ_{e 5, M_{1}} (A_{1}) \cup^{​} σ_{e 5, M_{4}} ({\overset{S}{true}}_{λ}) . & (7) \end{matrix}

Lemma 2.1 in with the fact that

C_{σ_{e 5, M_{1}} (A_{1})}

is connected and Equation , imply that

σ_{e 6, M} (L) = σ_{e 6, M_{1}} (A_{1}) \cup^{​} σ_{e 6, M_{4}} ({\overset{S}{true}}_{λ}) .

□…”

Section: The M‐essential Spectra Of Lmentioning

confidence: 95%

“…Theorem If the operators H , M 2 , M 3 are Fredholm perturbations, K 12 , K 22 are non‐negative regular operators and if

\frac{κ_{21}}{(x, ., v') | v' |}

is relatively weakly compact, then

σ_{e i, M} (L) : = {λ \in ℂ such that R e λ \leq - \min (\frac{λ_{*}^{1}}{μ_{*}^{1}}, \frac{λ_{*}^{2}}{μ_{*}^{4}})}, 1 \leq i \leq 6.

Proof It is shown in , that

\begin{matrix} σ_{e i, M_{1}} (T_{1}) = {λ \in ℂ such that Re λ \leq - \frac{λ_{1}^{*}}{μ_{1}^{*}}}, 1 \leq i \leq 6. & (9) \end{matrix}

Let

λ MathClass-rel\in ρ_{M_{1}} MathClass-open(T_{1} MathClass-close)

. The operator S λ can be expressed as:

S_{λ} MathClass-punc: MathClass-rel= T_{2}^{H} MathClass-bin+ K_{22} MathClass-bin+ MathClass-open(K_{21} MathClass-bin- λ M_{3} MathClass-close) K_{λ} Γ_{Y} MathClass-bin- F MathClass-open(λ MathClass-close) MathClass-open(K_{12} MathClass-bin- λ M_{2} MathClass-close) MathClass-punc.

Using Lemma 4.2 and the fact that H is Fredholm perturbation, we deduce from Theorem 2.1 in that …”

Section: Application To Two‐group Transport Equationsmentioning

confidence: 99%

“…The operator S λ can be expressed as:

S_{λ} MathClass-punc: MathClass-rel= T_{2}^{H} MathClass-bin+ K_{22} MathClass-bin+ MathClass-open(K_{21} MathClass-bin- λ M_{3} MathClass-close) K_{λ} Γ_{Y} MathClass-bin- F MathClass-open(λ MathClass-close) MathClass-open(K_{12} MathClass-bin- λ M_{2} MathClass-close) MathClass-punc.

Using Lemma 4.2 and the fact that H is Fredholm perturbation, we deduce from Theorem 2.1 in that

σ_{e i, M_{4}} MathClass-open(S_{λ} MathClass-close) MathClass-rel= σ_{e i, M_{4}} (() T_{2}^{H} + K_{22}) MathClass-punc.

On the other hand, for

λ MathClass-rel\in ρ_{M_{4}} ((), T_{2}^{H})

such that

r_{σ} ({(λ M_{4} - T_{2}^{H})}^{- 1} K_{22}) < 1

( r σ (.) the spectral radius), implies that

{(λ M_{4} - T_{2}^{H} - K_{22})}^{- 1} - {(λ M_{4} - T_{2}^{H})}^{- 1} = \sum_{n \geq 1} {({(λ M_{4} - T_{2}^{H})}^{- 1} K_{22})}^{n} {(λ M_{4} - T_{2}^{H})}^{- 1} .

According to the fact that K 22 is a non‐negative regular operator with Theorem 2.2 in , make us to conclude that: …”

Section: Application To Two‐group Transport Equationsmentioning

confidence: 99%

See 2 more Smart Citations

On the M-essential spectra of two-group transport equations

Walha

2013

Math. Meth. Appl. Sci.

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“…The map P(λ) := λS − T, λ ∈ C is called a linear bundle. It is known that many problems of mathematical physics (for example, quantum theory, transport theory,...) are reduced to the study of certain reversibility conditions of λS − T, and this study is reduced to the analysis of the essential spectra of the linear relations S −1 T and TS −1 (see, for example [12]).…”

Section: Introductionmentioning

confidence: 99%

Left-right fredholm and left-right browder linear relations

Álvarez

Fakhfakh

Mnif

2017

Filomat

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In this paper we introduce the notions of left (resp. right) Fredholm and left (resp. right) Browder linear relations. We construct a Kato-type decomposition of such linear relations. The results are then applied to give another decomposition of a left (resp. right) Browder linear relation T in a Banach space as an operator-like sum T = A + B, where A is an injective left (resp. a surjective right) Fredholm linear relation and B is a bounded finite rank operator with certain properties of commutativity. The converse results remain valid with certain conditions of commutativity. As a consequence, we infer the characterization of left (resp. right) Browder spectrum under finite rank operator. 2010 Mathematics Subject Classification. Primary 47A06 Keywords. Ascent and descent of a linear relation, left and right Fredholm linear relations, left and right Browder linear relations, left and right Browder spectrum.

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On the essential spectra of coupled operators matrix and application

Belghith

Moalla

Walha

2017

Math Methods in App Sciences

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We define an abstract setting to treat essential spectra of unbounded coupled operator matrix. We prove a well-posedness result and develop a spectral theory which also allows us to prove an amelioration to many earlier works. We point out that a concrete example from integro-differential equation fit into this abstract framework involving a general class of regular operator in L 1 spaces. KEYWORDScoupled operator matrix, essential spectra, Fredholm perturbations theory, perfect periodic boundary conditions, transport operator l, w, ss, b}, 7218

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S‐essential spectra and application to an example of transport operators

Cited by 36 publications

References 16 publications

On the M-essential spectra of two-group transport equations

On the M-essential spectra of two-group transport equations

Left-right fredholm and left-right browder linear relations

On the essential spectra of coupled operators matrix and application

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