Generalized Kothe-Toeplitz Duals of a New Class of Sequence Spaces

BSTRACT. In this paper, we determine the pα-duals of certain newly introduced difference se- quence spaces, namely ℓ∞(u, △m, q), c0(u, △m, q), and c(u, △m, q). Furthermore, we investigate v v v their topological properties and establish the conditions under which these spaces are perfect. 1. Introduction and preliminaries Assume that ω denotes the space of all sequences. Throughout this chapter, we adopt the notation q = (qk) to denote a sequence of positive real numbers. Some generalized sequence spaces, such as ℓ∞(q), c(q), and c0(q), have been studied by various authors [9–11]. Ahmad and Mursaleen [1] introduced the concept of some new generalized difference sequence spaces, defined as follows: △ℓ∞(q) = {x ∈ ω : △x ∈ ℓ∞(q)}. △c(q) = {x ∈ ω : △x ∈ c(q)}. △c0(q) = {x ∈ ω : △x ∈ c0(q)}. Where △xk = xk − xk+1 . After that, Et and Bas¸arer [8] defined these difference sequence spaces in a more general form by introducing the operator △mon these spaces.