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

Abstract

Let A = (a j,k ) j,k≥1 be a non-negative matrix. In this paper, we characterize those A for which A ℓp,ℓq are determined by their actions on non-negative decreasing sequences, where one of p and q is 1 or ∞. The conditions forcing on A are sufficient and they are also necessary for nonnegative finite matrices.