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

Let $A=(a_{j,k}{)}_{j,k\ge 1}$ be a non-negative matrix. In this paper, we characterize those $A$ for which $\mathrm{\parallel }A{\mathrm{\parallel }}_{{\mathrm{\ell }}_p,{\mathrm{\ell }}_q}$ are determined by their actions on non-negative decreasing sequences, where one of $p$ and $q$ is 1 or $\mathrm{\infty }$. The conditions forcing on $A$ are sufficient and they are also necessary for non-negative finite matrices.