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

Aasma, Ants ; Natarajan, Pinnangudi Narayanasubramanian

Serdica Mathematical Journal, Tome 50 (2024) no. 2, pp. 137-150

Résumé

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

Détail

Keywords: matrix transforms, boundedness, convergence and α-absolute convergence with speed, α-absolute λ-conservativity, improvement of α-absolute λ-convergence, 40C05, 40D05, 41A25