Summability Matrices That Preserve Asymptotic Equivalence for Ideal Convergence - Connor, Jeff; Gumus, Hafize (Sarajevo Journal of Mathematics, 2024)
A characterization of summability matrices that preserve the asymptotic equivalence of two sequences is given, where asymptotic equivalence is defined with respect to an ideal $I$ of subsets of $\mathbb{N}$. Pobyvanets (1980) gave necessary and sufficient conditions for a nonnegative summability matrix $A$ to have the property that, for two nonnegative sequences $x$ and $y$ bounded away from 0, the ratio $Ax/Ay$ tends to 1 whenever $x/y$ tends to 1; an analogous characterization is given where the convergence of $x/y$ is with respect to an ideal $I$ as well as a new proof of Pobyvanets’ theorem. Similar extensions are given of theorems of Marouf and Li, in particular characterizations of summabilty matrices that map sequences $x$ and $y$ such that $x/y$ is $I$-convergent to 1 to sequences with the property that the ratio of the sums of their tails or the ratio of the supremum of their tails also tend to 1 with respect to an ideal J.