Hardy-type inequalities for matrix operators

Abstract

We establish necessary and sufficient conditions the validity of the discrete Hardy - type inequality (∑^∞_i=1(∑^∞_j=1a_i,jf_j)^qu_i^q)^1/q ≤ (∑^∞_i=1 f_i^pv_i^p)^1/p, f={f_i}^∞_i=1≥0, with 0 < p ≤ q < ∞ and 0 < p ≤ 1, where the matrices (a_i;j) is an arbitrary matrix and the entries of the matrix (a_i;j) ≥ 0 such that a_i;j is non - increasing in the second index. Also some further results are pointed out on the cone of monotone sequences. Moreover, we give that the applications of the main results for the non - negative and triangular matrices (a_i;j ≥ 0 for 1 ≤ j ≤ i and a_i;j = 0 for i < j).