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Boundedness and compactness of a class of matrix operators in weighted sequence spaces
Eurasian National University, Astana.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
2008 (English)In: Journal of Mathematical Inequalities, ISSN 1846-579X, E-ISSN 1848-9575, Vol. 2, no 4, p. 555-570Article in journal (Refereed) Published
Abstract [en]

Characterisations of bounded and compact multiple weighted summation operators from weighted ℓ p into weighted ℓ q spaces are established. Let 1<p,q<∞ , let ℓ p denote the space of all p -summable real sequences, let (ω i,k ) ∞ k=1 for i=1,2,…,n−1 , u=(u i ) ∞ i=1 and v=(v i ) ∞ i=1 be nonnegative sequences, and let ℓ p,v be the space of all sequences f=(f i ) ∞ i=1 such that fv=(f i v i ) ∞ i=1 ∈ℓ p , endowed with the natural norm ∥⋅∥ ℓ p,v defined by (∑ ∞ i=1 |f i v i | p ) 1/p . The n -tuple summation operator S n is defined by (S n f) i =∑ k 1 =1 i ω 1,k 1 ∑ k 2 =1 k 1 ω 2,k 2 ∑ k 3 =1 k 2 ω 3,k 3 ⋯∑ k n−1 =1 k n−2 ω n−1,k n−1 ∑ j=1 k n−1 f j . A necessary and sufficient condition is established for the inequality ∥S n f∥ ℓ q,u ≤C∥f∥ ℓ p,u to hold in the case 1<p≤q<∞ , for all sequences f∈ℓ q,u , where C is an absolute constant. This condition immediately yields a necessary and sufficient condition for S n to be a bounded operator from ℓ q,u into ℓ p,v . This result is a generalisation of a known result by K. F. Andersen and H. P. Heinig in the case n=1 when the operator S n reduces to a discrete Hardy operator of the form (S 1 f) i =∑ i j=1 f j . Finally, a necessary and sufficient condition is established for S n to be a compact operator from ℓ q,u into ℓ p,v when 1<p≤q<∞ . It should be noted that if n=2 then S 2 f can be expressed as a special matrix transformation of the form (Af) i =∑ i j=1 a ij f j .

Place, publisher, year, edition, pages
2008. Vol. 2, no 4, p. 555-570
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:ltu:diva-13733Local ID: d02eff0e-e315-4793-a1bb-9461d958962fOAI: oai:DiVA.org:ltu-13733DiVA, id: diva2:986686
Note

Upprättat; 2008; 20130627 (andbra)

Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2017-11-24Bibliographically approved

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Temirkhanova, Ainur

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