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Title: Lower bound and upper bound of operators on block weighted sequence spaces (English)
Author: Lashkaripour, Rahmatollah
Author: Talebi, Gholomraza
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 62
Issue: 2
Year: 2012
Pages: 293-304
Summary lang: English
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Category: math
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Summary: Let $A=(a_{n,k})_{n,k\geq 1}$ be a non-negative matrix. Denote by $L_{v,p,q,F}(A)$ the supremum of those $L$ that satisfy the inequality $$ \|Ax\|_{v,q,F} \ge L\| x\|_{v,p,F}, $$ where $x\geq 0$ and $x\in l_p(v,F)$ and also $v=(v_n)_{n=1}^\infty $ is an increasing, non-negative sequence of real numbers. If $p=q$, we use $L_{v,p,F}(A)$ instead of $L_{v,p,p,F}(A)$. In this paper we obtain a Hardy type formula for $L_{v,p,q,F}(H_\mu )$, where $H_\mu $ is a Hausdorff matrix and $0<q\leq p\leq 1$. Another purpose of this paper is to establish a lower bound for $\|A_{W}^{NM} \|_{v,p,F}$, where $A_{W}^{NM}$ is the Nörlund matrix associated with the sequence $W=\{w_n\}_{n=1}^\infty $ and $1<p<\infty $. Our results generalize some works of Bennett, Jameson and present authors. (English)
Keyword: lower bound
Keyword: weighted sequence space
Keyword: Hausdorff matrices
Keyword: Euler matrices
Keyword: Cesàro matrices
Keyword: Hölder matrices
Keyword: Gamma matrices
MSC: 26D15
MSC: 40G05
MSC: 46A45
MSC: 47A30
MSC: 54D55
idZBL: Zbl 1265.26074
idMR: MR2990178
DOI: 10.1007/s10587-012-0031-8
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Date available: 2012-06-08T09:34:20Z
Last updated: 2016-04-07
Stable URL: http://hdl.handle.net/10338.dmlcz/142829
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Reference: [10] Lashkaripour, R., Foroutannia, D.: Lower bounds for matrices on block weighted sequence spaces. I.Czech. Math. J. 59 (134) (2009), 81-94. Zbl 1217.47065, MR 2486617, 10.1007/s10587-009-0006-6
Reference: [11] Lashkaripour, R., Foroutannia, D.: Computation of matrix operators bounds with applying new extension of Hardy inequality on weighted sequence spaces. I.Lobachevskii J. Math. 30 (2009), 40-45. Zbl 1177.26039, MR 2506053, 10.1134/S1995080209010065
Reference: [12] Lashkaripour, R., Talebi, G.: Lower bound of Copson type for Hausdorff matrices on weighted sequence spaces.J. Sci., Islam. Repub. Iran 22 (2011), 153-157. MR 2884149
Reference: [13] Lashkaripour, R., Talebi, G.: Lower bound for the norm of lower triangular matrices on block weighted sequence spaces.J. Math. Inequal. 5 (2011), 33-38. Zbl 1211.26018, MR 2799056, 10.7153/jmi-05-04
Reference: [14] Lashkaripour, R., Talebi, G.: Bounds for the operator norms of some Nörlund matrices on weighted sequence spaces.Preprint.
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