Absolute convergence with speed and matrix transforms

Abstract

We define the notion of absolute convergence with speed, where the speed is defined by a monotonically increasing positive sequence λ. Also we present the notion of absolute λ-conservativity of a matrix, and the notion of improvement of λ-convergence by a matrix. Let X, Y be two sequence spaces defined by speeds of convergence. In this paper, we give necessary and sufficient conditions for a matrix A (with real or complex entries) to map X into Y , if X or Y is the set of absolutely λ-convergent sequences. We also present some examples of matrices being absolutely λ-conservative or improving the 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.