A Note on the Necessary and Sufficient Conditions to Matrix Summability of Infinite Series

Earlier works have established the conditions under which the series \(\sum\)\(a_n\)\(\gamma\)\(_n\) is ϕ − | F, μ; τ |k summable, provided that the series \(\sum\)\(a_n\) is summable by ϕ − | E, μ; τ |. In this study, we present the necessary and sufficient criteria ensuring ϕ−| F, μ; τ | summability of \(\sum\)\(a_n\)\(\gamma\)\(_n\) whenever \(\sum\)\(a_n\) is summable by ϕ−| E, μ; τ |k, where E = (env) and F = (fnv) denote two positive normal matrices, k ≥ 1, τ ≥ 0, and the inequality −μ(τk + k − 1) + k > 0 holds.