On the Characterization of Some Classes of Four-Dimensional Matrices and AlmostB-Summable Double Sequences

Orhan Tug

doi:10.1155/2018/1826485

Abstract

We firstly summarize the related literature aboutBr,s,t,u-summability of double sequence spaces and almostBr,s,t,u-summable double sequence spaces. Then we characterize some new matrix classes ofLs′:Cf,BLs′:Cf, andLs′:BCfof four-dimensional matrices in both cases of0<s′≤1and1<s′<∞, and we complete this work with some significant results.

Highlights

We denote the set of all complex valued double sequence by Ω which is a vector space with coordinatewise addition and scalar multiplication
We firstly summarize the related literature about B(r, s, t, u)-summability of double sequence spaces and almost B(r, s, t, u)summable double sequence spaces
It is easy to see for any two spaces λ and μ of double sequences that μα ⊂ λα whenever λ ⊂ μ and λα ⊂ λγ

Summary

Introduction

We denote the set of all complex valued double sequence by Ω which is a vector space with coordinatewise addition and scalar multiplication. Given any four-dimensional infinite matrix A = (amnkl), where m, n, k, l ∈ N, any double sequence x = (xkl), we write Ax= {(Ax)mn}m,n∈N, the A−transform of x, exists for every sequence x = (xkl) ∈ λ and it is in μ, where (Ax)mn = θ − ∑amnklxkl for each m, n ∈ N. The four-dimensional generalized difference matrix B(r, s, t, u) = {bmnkl(r, s, t, u)} and matrix domain of it on some double sequence spaces were recently defined by Tugand Basar [9] and studied by Tug [10,11,12].

Results

Conclusion