Matrix transformations of double convergent sequences with powers

Abstract

In 1967, I. J. Maddox generalized the spaces c0, c, ℓ∞ by adding the powers pk (k ∈ ℕ) in the definitions of the spaces to the terms of elements of sequences (xk). Gökhan and Çolak in 2004–2006 defined the corresponding double sequence spaces for the Pringsheim and the bounded Pringsheim convergence. We will additionally define the corresponding double sequence spaces for the regular convergence. In 2009, Gökhan, Çolak and Mursaleen characterized some classes of matrix transformations involving these double sequence spaces with powers. However, many of their results appeared to be wrong. In this paper, we give corresponding counterexamples and prove the correct results. Moreover, we present the conditions for a wider class of matrices.

Keywords

• matrix transformation
• maddox sequence spaces
• double sequence