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 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.