Compact matrix operators on a new sequence space related to ℓ p $\ell_{p}$ spaces

Alotaibi, Abdullah; Mursaleen, M.; Mohiuddine, S. A.

doi:10.1186/s13660-016-1129-6

Cited by 2 publications

(1 citation statement)

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“…Mursaleen and Noman [12,13] established the compact operators on some difference sequence spaces. Alotaibi et al [14,15] examined the compact operators on some Fibonacci difference sequence spaces and on a new sequence space related to ℓ p spaces. The multiplication maps on Cesàro sequence spaces equipped with the Luxemburg norm investigated by Komal et al [16].…”

Section: Introductionmentioning

confidence: 99%

Kannan Contraction Operator on the Domain of $r (.)$ -Cesàro Matrix in $ℓ<Bakery 1, Dewaik 2 2021 Journal of Function Spaces0000View full text Add to dashboard CiteIn this article, the sequence space

Ξ

r
,
t

υ

has been built by the domain of

r

l

-Cesàro matrix in Nakano sequence space

ℓ

t

l

, where

t
=

t

l

and

r
=

r

l

are sequences of positive reals with

1
≤

t

l

<
∞

, and

υ

f

=

∑

l
=
0

∞

∑

z
=
0

l

r

z

f

z

/

∑

z
=
0

l

r

z

t

l

, with

f
=

f

z

∈
Ξ

r
,
t

. Some topological and geometric behavior of

Ξ

r
,
t

υ

, the multiplication maps acting on

Ξ

r
,
t

υ

, and the eigenvalues distribution of operator ideal constructed by

Ξ

r
,
t

υ

and

s

-numbers have been examined. The existence of a fixed point of Kannan prequasi norm contraction mapping on this sequence space and on its prequasi operator ideal are investigated. Moreover, we indicate our results by some explanative examples and actions to the existence of solutions of nonlinear difference equations.show abstract“…Mursaleen and Noman [12, 13] established the compact operators on some difference sequence spaces. Alotaibi et al [14, 15] examined the compact operators on some Fibonacci difference sequence spaces and on a new sequence space related to ℓ p spaces. The multiplication maps on Cesàro sequence spaces equipped with the Luxemburg norm investigated by Komal et al [16] .…” Section : Introduction mentioning confidence: 99% Kannan Contraction Operator on the Domain of r (.) -Cesàro Matrix in ℓ<Bakery 1, Dewaik 2 2021 Journal of Function Spaces0000View full text Add to dashboard CiteIn this article, the sequence space

Ξ

r
,
t

υ

has been built by the domain of

r

l

-Cesàro matrix in Nakano sequence space

ℓ

t

l

, where

t
=

t

l

and

r
=

r

l

are sequences of positive reals with

1
≤

t

l

<
∞

, and

υ

f

=

∑

l
=
0

∞

∑

z
=
0

l

r

z

f

z

/

∑

z
=
0

l

r

z

t

l

, with

f
=

f

z

∈
Ξ

r
,
t

. Some topological and geometric behavior of

Ξ

r
,
t

υ

, the multiplication maps acting on

Ξ

r
,
t

υ

, and the eigenvalues distribution of operator ideal constructed by

Ξ

r
,
t

υ

and

s

-numbers have been examined. The existence of a fixed point of Kannan prequasi norm contraction mapping on this sequence space and on its prequasi operator ideal are investigated. Moreover, we indicate our results by some explanative examples and actions to the existence of solutions of nonlinear difference equations.show abstractHausdorff measure of noncompactness of matrix operators on some new difference sequence spaces Abyar 1, Ghaemi 2 2016 J Inequal Appl 0000 View full text Add to dashboard Cite The new sequence spaces X(r, s, t; ) for X ∈ {l ∞ , c, c 0 } have been defined by using generalized means and difference operator. In this work, we establish identities or estimates for the operator norms and the Hausdorff measure of noncompactness of certain matrix operators on some new difference sequence spaces X(r, s, t; ) where X ∈ {l ∞ , c, c 0 , l p } (1 ≤ p < ∞), as derived by using generalized means. Further, we find the necessary and sufficient conditions for such operators to be compact by applying the Hausdorff measure of noncompactness. Finally, as applications we characterize some classes of compact operators between these new difference sequence spaces and some other BK-spaces.Keywords: sequence space; difference operators; matrix transformation; generalized means; compact operators; Hausdorff measure of noncompactness
Preliminaries and backgroundThe study of sequence spaces has been very useful in many branches of analysis. Recently, some new sequence spaces have been defined by using matrix domain of a suitable matrix. Beside this, the Hausdorff measure of noncompactness is very useful in the classification of compact operators between Banach spaces.The difference sequence spaces were introduced for the first time by Kizmaz in []. Afterwards, many authors have introduced and studied some new sequence spaces defined by using the difference operator. In this paper we obtain some identities or estimates for the operator norms and the Hausdorff measure of noncompactness of certain matrix operators on new difference sequence spaces defined by Manna et al. Further, we find the necessary and sufficient condi- show abstractscite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health. Contact Info customersupport@researchsolutions.com 10624 S. Eastern Ave., Ste. 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Some topological and geometric behavior of \n \n \n \n \n \n Ξ\n \n \n r\n ,\n t\n \n \n \n \n \n \n υ\n \n \n \n , the multiplication maps acting on \n \n \n \n \n \n Ξ\n \n \n r\n ,\n t\n \n \n \n \n \n \n υ\n \n \n \n , and the eigenvalues distribution of operator ideal constructed by \n \n \n \n \n \n Ξ\n \n \n r\n ,\n t\n \n \n \n \n \n \n υ\n \n \n \n and \n \n s\n \n -numbers have been examined. The existence of a fixed point of Kannan prequasi norm contraction mapping on this sequence space and on its prequasi operator ideal are investigated. Moreover, we indicate our results by some explanative examples and actions to the existence of solutions of nonlinear difference equations.","authors":[{"family":"Bakery","given":"Awad A.","affiliation":"Jeddah University","authorSlug":"awad-a-bakery-6MX5v9","authorName":"Awad A. 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In this work, we establish identities or estimates for the operator norms and the Hausdorff measure of noncompactness of certain matrix operators on some new difference sequence spaces X(r, s, t; ) where X ∈ {l ∞ , c, c 0 , l p } (1 ≤ p \u003C ∞), as derived by using generalized means. Further, we find the necessary and sufficient conditions for such operators to be compact by applying the Hausdorff measure of noncompactness. Finally, as applications we characterize some classes of compact operators between these new difference sequence spaces and some other BK-spaces.Keywords: sequence space; difference operators; matrix transformation; generalized means; compact operators; Hausdorff measure of noncompactness\nPreliminaries and backgroundThe study of sequence spaces has been very useful in many branches of analysis. Recently, some new sequence spaces have been defined by using matrix domain of a suitable matrix. 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By applying the Hausdorff measure of noncompactness, we obtain the necessary and sufficient conditions for such operators to be compact. Further, we study some geometric properties of this space.\nMSC: 46B15; 46B45; 46B50","authors":[{"family":"Alotaibi","given":"Abdullah","affiliation":"King Abdulaziz University","authorSlug":"abdullah-alotaibi-OVdWdj","authorName":"Abdullah Alotaibi","authorID":"5134978","authorLastKnownAffiliationId":284990,"authorSequenceNumber":1,"affiliationSlug":"king-abdulaziz-university-kQnX","affiliationID":"14168"},{"family":"Mursaleen","given":"M.","affiliation":"Aligarh Muslim University","authorSlug":"m-mursaleen-r6N4kA","authorName":"M. Mursaleen","authorID":"5617373","authorLastKnownAffiliationId":267581,"authorSequenceNumber":2,"affiliationSlug":"aligarh-muslim-university-ApVL","affiliationID":"21849"},{"family":"Mohiuddine","given":"S. A.","affiliation":"King Abdulaziz University","authorSlug":"s-a-mohiuddine-XxaAW9","authorName":"S. A. 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