Lower bounds of Copson type for the transposes of lower triangular matrices

Chang Pao Chen*, Kuo-Zhong Wang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

Let A = (an, k)n, k ≥ 0 be a non-negative matrix. Denote by Lp, q (A) the supremum of those L satisfying the following inequality:(underover(∑, n = 0, ∞) (underover(∑, k = 0, ∞) an, k xk)q)1 / q ≥ L (underover(∑, k = 0, ∞) xkp)1 / p (X ∈ ℓp, X ≥ 0) . In this paper, we focus on the evaluation of Lp, p (At) for a lower triangular matrix A, where 0 < p < 1. A Borwein-type result is established. We also derive the corresponding result for the case Lp, p (A) with - ∞ < p < 0. In particular, we apply them to summability matrices, the weighted mean matrices, and Nörlund matrices. Our results not only generalize the work of Bennett, but also provide several analogues of those given in [Chang-Pao Chen, Dah-Chin Lour, Zong-Yin Ou, Extensions of Hardy inequality, J. Math. Anal. Appl. 273 (1) (2002) 160-171] and [P.D. Johnson Jr., R.N. Mohapatra, D. Ross, Bounds for the operator norms of some Nörlund matrices, Proc. Amer. Math. Soc. 124 (2) (1996), Corollary on p. 544]. Our results also improve Bennett's results for some cases.

Original languageEnglish
Pages (from-to)1284-1294
Number of pages11
JournalJournal of Mathematical Analysis and Applications
Volume341
Issue number2
DOIs
StatePublished - 15 May 2008

Keywords

  • Cesàro matrices
  • Lower bounds
  • Nörlund matrices
  • Summability matrices
  • Weighted mean matrices

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