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

TL;DR

This paper characterizes non-negative matrices whose operator norms between sequence spaces are determined solely by their action on decreasing sequences, focusing on cases where p or q is 1 or infinity.

Contribution

It provides necessary and sufficient conditions for such matrices, extending understanding of operator norms in sequence spaces for these special cases.

Findings

Conditions are both necessary and sufficient for finite matrices.

Characterization applies specifically when p or q equals 1 or infinity.

Results clarify when norms are determined by decreasing sequences.

Abstract

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