Lower bounds of Copson type for Nörlund matrices

Chang Pao Chen*, Meng Kuang Kuo, Kuo-Zhong Wang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Let A = (an, k)n, k ≥ 0 be a non-negative matrix. Denote by Lp, q (A) the supremum of those L satisfying {norm of matrix} AX {norm of matrix}q ≥ L {norm of matrix} X {norm of matrix}p(X ∈ ℓp, X ≥ 0), and define L(p), q (A) = Lp, q (A)(p > 0) . We derive a range for the value of Lp, q (AWNM), where 0 < q ≤ p < 1 and AWNM denotes the Nörlund matrix associated with the weight function W. By the continuity of L(·), q (AWNM), we show that this range is best possible. It is also proved that there exists a unique ξ ∈ (q, 1] such that L(·), q (AWNM) maps [q, ξ] onto [1, {norm of matrix} W {norm of matrix}q / {norm of matrix} W {norm of matrix}1] and this mapping is continuous and strictly increasing. The case Lp, q ((AWNM)t) with - ∞ < p, q < 0 is also investigated.

Original languageEnglish
Pages (from-to)1939-1948
Number of pages10
JournalLinear Algebra and Its Applications
Volume428
Issue number8-9
DOIs
StatePublished - 15 Apr 2008

Keywords

  • Cesàro matrices
  • Lower bound
  • Nörlund matrices

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