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: [GRAPHICS] The purpose of this article is to establish a Bennett-type formula for parallel to H(mu)(0)parallel to(down arrow)(p,p) and a Hardy-type formula for L(p,p)(down arrow)(H(mu)(alpha)) and L(p,p)H((alpha)(mu)), where H(mu)(alpha) is a generalized Hausdorff matrix and 0 < p <= 1. Similar results are also established for L(p,p)(H(mu)(alpha)) and L(p,p)H(((alpha)(mu))(t)) 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 H(mu)(0) to H(mu)(alpha) with alpha >= 0 and completely solve the value problem of parallel to H(mu)(0)parallel to(down arrow)(p,p), L(p,p)(down arrow)(H(mu)(alpha)), L(p,p)H((alpha)(mu)) and L(p,p)H(((alpha)(mu))(t)) for alpha is an element of N boolean OR {0}.