Matrix transforms of the set of α-absolutely convergent sequences with speed

We define the notion of \(α\)-absolute convergence with speed, where the speed is defined by a monotonically increasing positive sequence \(λ\) and \(0< α ≤ 1\). Also we present the notion of \(α\)-absolute \(λ\)-conservativity of a matrix, and the notion of improvement of \(α\)-absolute \(λ\)-convergence by a matrix. Let \(lαλ\) be the set of all \(α\)-absolutely \(λ\)-convergent sequences and \(Y\) a sequence space defined by another speed \(μ\). In this paper, we give necessary and sufficient conditions for a matrix \(A\) (with real or complex entries) to map \(lαλ\) into \(Y\). We also present some examples of matrices being \(α\)-absolutely \(λ\)-conservative or improving the \(α\)-absolute \(λ\)-convergence, and consider these problems in the special cases if \(A\) is the Riesz matrix \((R,pₙ)\) or the Zweier matrix \(Z1/2\).