Difference sequence spaces derived by using generalized means

Abstract

This paper deals with new sequence spaces X(r, s, t; ∆) for X ∈ {l ∞ , c, c 0 } defined by using generalized means and difference operator. It is shown that these spaces are complete normed linear spaces and the spaces X(r, s, t; ∆) for X ∈ {c, c 0 } have Schauder basis. Furthermore, the α-, β-, γ-duals of these sequence spaces are computed and also established necessary and sufficient conditions for matrix transformations from X(r, s, t; ∆) to X.2010 Mathematics Subject Classification: 46A45, 46A35.

Measures of noncompactness in (N¯qΔ)summable difference sequence spaces

“…Proof. Using [8, Corollary 3.6 (a), (c)], the statements in (17) and (18) can be easily shown. (c) Define P m : ∞ → ∞ by P m (x) = x [m] for all x ∈ ∞ and m = 0, 1, .…”

“…The operator L ∈ B(X, Y) is said to be of finite rank if dim R(L) < ∞, where R(L) denotes the range space of L. A finite rank operator is clearly compact. [6,Chapter 2] The concept of difference sequence spaces was first introduced by Kizmaz [13] and later several authors studied new sequence spaces defined by using difference operators like Mursaleen and Noman [19], Mursaleen et al [18], Jalal [10], Manna et al [17], Polat et al [20]. In the past, several authors studied matrix transformations on sequence spaces that are the matrix domains of the difference operator, or of the matrices of the classical methods of summability in spaces such as p , c 0 , c, ∞ or others.…”