Matrices Whose Norms Are Determined by Their Actions on Decreasing Sequences

Characterization of the matrix whose norm is determined by its action on decreasing sequences:The exceptional cases

“…In [7], the present authors indicate that the matrix A, defined by a 2,2 = 1 and 0 otherwise, possesses the property: A ℓp,ℓ∞ > A ℓp,ℓ∞,↓ , where 1 ≤ p < ∞. This phenomenon can be interpreted by applying the following result to the case Λ = {2}.…”

“…This problem has been partially solved by [1, page 422], [6,Lemma 2.4], and [12,Theorem 2]. Recently, in [7], the present authors gave a more general setting, which includes these as special cases. They characterized A and proved that E and F can be ℓ p , d(w, p), or ℓ p (w), where d(w, p) is the Lorentz sequence space associated with non-negative decreasing weights w n and ℓ p (w) consists of all sequences x = {x k } ∞ k=1 such that…”

“…However, the case F = ℓ ∞ is excluded in [7]. The main purpose of this paper is to deal with this case.…”

On weighted mean matrices whose l^p norms are determined on decreasing sequences

“…We refer the reader to the articles [12] and [13] for more recent developments in this area. It is our goal in this paper to give a condition on weighted mean matrices in Section 2 so that (1.10) will hold.…”

“…Motivated by the study of inequalities (1.6)-(1.7), we seek for extra inputs that may lead to a resolution of the remaining case of (1.7) for 1 < p < 2, 1 < α < 2. For this, we note the following natural question related to the l p norms of any matrix asked by Bennett [5,Problem 7.23 We refer the reader to the articles [12] and [13] for more recent developments in this area. It is our goal in this paper to give a condition on weighted mean matrices in Section 2 so that (1.10) will hold.…”