A characterization of $(\mathcal{I}, \mathcal{J})$-regular matrices

TL;DR

This paper characterizes matrices that transform sequences converging under one ideal into sequences converging under another ideal, extending classical results and applying to permutation and diagonal matrices.

Contribution

It provides necessary and sufficient conditions for $( ext{I}, ext{J})$-regular matrices, generalizing Silverman--Toeplitz's classical characterization.

Findings

Characterization of $( ext{I}, ext{J})$-regular matrices

Conditions for matrices to preserve ideal limits

Applications to permutation and diagonal matrices

Abstract

Let $\mathcal{I},\mathcal{J}$ be two ideals on $\mathbf{N}$ which contain the family $\mathrm{Fin}$ of finite sets. We provide necessary and sufficient conditions on the entries of an infinite real matrix $A=(a_{n,k})$ which maps $\mathcal{I}$-convergent bounded sequences into $\mathcal{J}$-convergent bounded sequences and preserves the corresponding ideal limits. The well-known characterization of regular matrices due to Silverman--Toeplitz corresponds to the case $\mathcal{I}=\mathcal{J}=\mathrm{Fin}$. Lastly, we provide some applications to permutation and diagonal matrices, which extend several known results in the literature.