Lower bounds of Copson type for Nörlund matrices

✍ Scribed by Chang-Pao Chen; Meng-Kuang Kuo; Kuo-Zhong Wang

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 153 KB
• Volume: 428
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2007.10.037

✦ Synopsis

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