On compact operators on some spaces related to matrix B(r, s)

Abstract

Many sequence spaces arise from different concepts of summability. Recent results obtained by Altay, Başar and Malkowsky [2] are related to strong Cesàro summability and boundedness. They determined β−duals of the new sequence spaces and characterized some classes of matrix transformations on them. Here, we will present new results supplementing their research with the characterization of classes of compact operators on those spaces.

Compactness of quadruple band matrix operator and geometric properties

“…Cesàro, Norlund, Borel, Riesz and others have pioneered the development of the theory of summability. Many authors have used the matrix domain of difference matrices for constructing new sequence spaces and have investigated some properties of those spaces in c 0 (∆), c(∆) and ∞ (∆) in [19], ∆c 0 (p), ∆c(p) and ∆ ∞ (p) in [1], c 0 (u, ∆, p), c(u, ∆, p) and ∞ (u, ∆, p) in [2], c 0 (∆ 2 ), c(∆ 2 ) and ∞ (∆ 2 ) in [13], c 0 (u, ∆ 2 ), c(u, ∆ 2 ) and ∞ (u, ∆ 2 ) in [26], c 0 (u, ∆ 2 , p), c(u, ∆ 2 , p) and ∞ (u, ∆ 2 , p) in [5], c 0 (∆ m ), c(∆ m ) and ∞ (∆ m ) in [14], ˆ ∞ , ĉ0 , ĉ and ˆ p in [18], c 0 (B), c(B), ∞ (B) and p (B) in [29], w p 0 (r, s), w p (r, s) and w p ∞ (r, s) in [10], c 0 (B), ∞ (B) and p (B) in [9], c 0 (Q), c(Q), ∞ (Q) and p (Q) in [6], f (Q(r, s, t, u)), f 0 (Q(r, s, t, u)) and f s(Q(r, s, t, u)) in [7]. Recently, studies on the matrix domain of generalized difference matrix ∆ 3 i have also been done some authors in [30], [31], [24], [25].…”

Applications of Measure of Noncompactness in the Series Spaces of Generalized Absolute Cesàro Means

“…The Hausdorff measure of noncompactness was defined by Goldens �tein et al (1957). Using the Hausdorff measure of noncompactness, several authors have characterized some classes of compact operators on certain sequence spaces (Başarır and Kara, 2013;Djolović, 2010;Malkowsky et al, 2002;Malkowsky and Rakočević, 2000;Mursaleen and Noman, 2010, 2014Rakočević, 1998).…”

Some notes on matrix mappings and their Hausdorff measure of noncompactness

“…The domain of the matrix B(r, s) in the classical spaces ∞ , c 0 and c has recently been studied by Kirişçi and Başar in [6]. The characterizations of compact matrix operators between some of those spaces were given by Djolović in [2]. Recently difference sequence spaces have extensively been studied, for instance in [2], [7], [6], [1], [11] and [10].…”