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The absolute Nörlund spaces Nu p k; ≥ 1; have more recently been introduced and studied by Hazar and Sargöl [On absolute Nörlund spaces and matrix operators, Acta Math. Sin. (Engl. Ser.), 34 (5) (2018), 812-826]: In the present paper, we characterize the classes of infinite matrix and compact operators transforming from Nu p k into X and obtain some identities or estimates for the Hausdorff measures of noncompactness, where X is one of the spaces '1, c and c0:. [ABSTRACT FROM AUTHOR]

In a more recent paper, the second author has introduced a space |Cα|k as the set of all series by absolute summable using Cesàro matrix of order α > −1. In the present paper we extend it to the absolute Nörlund space |Npθ|k taking Nörlund matrix in place of Cesàro matrix, and also examine some topological structures, α-β-γ-duals and the Schauder base of this space. Further we characterize certain matrix operators on that space and determine their operator norms, and so extend some well-known results. [ABSTRACT FROM AUTHOR]

In this paper we first introduce N̅qΔ − summable difference sequence spaces and prove some properties of these spaces. We then obtain the necessary and sufficient conditions for infinite matrix A to map these sequence spaces on the spaces c, c0 and l∞. Finally, the Hausdorff measure of noncompactness is then used to obtain the necessary and sufficient conditions for the compactness of the linear operators defined on these spaces. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to establish necessary and sufficient conditions for a matrix A ∈ (( ℓ), ℓ), where T is an arbitrary triangle, 1 ≤ p ≤ ∞, to be a compact operator. In the past, only sufficient conditions were established in almost all of those cases, by using the Hausdorff measure of noncompactness. We improve those results by applying another method for the characterizations of compact linear operators between BK spaces. [ABSTRACT FROM AUTHOR]

According to Hardy [5], Cesàro summability is usually considered for the range α ≥ -1. In a more recent paper [14], the space ¦cα¦k κ is studied for α > -1: In this paper we define ¦C-1¦k using the Cesàro mean (C;-1) of Thorpe [26] , compute its α-, β- and γ-duals, give some algebraic and topological properties, and characterize related matrix operators, and also obtain some identities or estimates for the their operator norms and the Hausdorff measure of noncompactness. Further, by applying the Hausdorff measure of noncompactness, we establish the necessary and sufficient conditions for such operators to be compact. So some results in [14] is also extended to the range α ≥ -1. [ABSTRACT FROM AUTHOR]

In the present paper, some results on matrix mappings and Hausdorff measure of noncompactness of certain generalized Euler difference sequence spaces of fractional order are discussed. Also, the Hausdorff measures of noncompactness of certain matrix operators that map an arbitrary BK-space into the classical sequence spaces are established. Furthermore, by using this measure, the characterization of some classes of Euler mean compact operators are determined in the BK-spaces. [ABSTRACT FROM AUTHOR]

In this paper, for α > –1 and k ≥ 1; we characterize the classes of all infinite matrices , where the absolute spaces are defined by Sarlgöl [22 – 24]. Also we obtain estimates for the norms of bounded linear operators corresponding to matrices in these classes. So not only some problems are solved but also some well known results of Mehdi [12], Mazhar [11], Mohapatra and Das [13] and Sarıgöl [22] are generalized. [ABSTRACT FROM AUTHOR]

In a way different from traditional methods, recently, new series spaces have been established by using absolute summability methods. Among these absolute summability methods, the absolute generalized Nörlund summability method |N, pn, qn, θn|k is extremely important as it includes several summability methods such as absolute Nörlund, weighted, Cesàro and Euler summability methods. Now, it is natural to ask behavior of the space derived by the absolute summability of generalized Nörlund means. The purpose of this study is to derive a more general series space including some well known spaces such as the spaces | N ˜ q θ | k , |Cα|k, |Cα, β|k, | N p θ | k and | E r θ | k , and to investigate its topological structure, α-, β- and γ-duals. Beside this, some matrix transformations related to these spaces have been characterized. [ABSTRACT FROM AUTHOR]

If X and Y are F K spaces, then every infinite matrix A ∊ (X, Y) defines a bounded linear operator LA ∊ B(X, Y) where LA(x) = Ax for each x ∊ X. But the converse is not always true. Indeed, if L is a general bounded linear operator from X to Y, that is, L ∊ B(X, Y), we are interested in the representation of such an operator using some infinite matrices. In this paper we establish the representations of the general bounded linear operators from the space bv into the spaces ℓ∞ and c0. We also prove some estimates for their Hausdorff measures of noncompactness. In this way we show the difference between general bounded linear operators between some sequence spaces and the matrix operators associated with matrix transformations. [ABSTRACT FROM AUTHOR]

A well-known result by Cohen and Dunford ([1], 1937) characterizes the class of all bounded linear operators from the space of all convergent complex sequences into itself. It follows that a regular matrix transformation cannot be compact. We use the theory of BK spaces and the Hausdorff measure of noncompactness to present a new proof for these results and establish their extensions to the spaces of strongly summable and strongly convergent sequences, and of convergent series. [ABSTRACT FROM AUTHOR]