Operator norms and lower bounds of generalized Hausdorff matrices

Abstract

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 the following inequality: The purpose of this article is to establish a Bennett-type formula for and a Hardy-type formula for and , where is a generalized Hausdorff matrix and 0 < p ≤ 1. Similar results are also established for and for other ranges of p and q. Our results extend [Chen and Wang, Lower bounds of Copson type for Hausdorff matrices, Linear Algebra Appl. 422 (2007), pp. 208–217] and [Chen and Wang, Lower bounds of Copson type for Hausdorff matrices: II, Linear Algebra Appl. 422 (2007) pp. 563–573] from to with α ≥ 0 and completely solve the value problem of , , and for α ∈ ℕ ∪ {0}.