On almost B-summable double sequence spaces

Orhan Tuğ

doi:10.1186/s13660-017-1606-6

Abstract

The concept of a four-dimensional generalized difference matrix and its domain on some double sequence spaces was recently introduced and studied by Tuğ and Başar (AIP Conference Proceedings, vol. 1759, 2016) and Tuğ (J. Inequal. Appl. 2017(1):149, 2017). In this present paper, as a natural continuation of (J. Inequal. Appl. 2017(1):149, 2017), we introduce new almost null and almost convergent double sequence spaces B(mathcal{C}_{f}) and B(mathcal{C}_{f_{0}}) as the four-dimensional generalized difference matrix B(r,s,t,u) domain in the spaces mathcal{C}_{f} and mathcal{C}_{f_{0}}, respectively. Firstly, we prove that the spaces B(mathcal{C}_{f}) and B(mathcal{C}_{f_{0}}) of double sequences are Banach spaces under some certain conditions. Then we give an inclusion relation of these new almost convergent double sequence spaces. Moreover, we identify the α-dual, beta(bp)-dual and γ-dual of the space B(mathcal{C}_{f}). Finally, we characterize some new matrix classes (B(mathcal{M}_{u}):mathcal{C}_{f}), (mathcal{M}_{u}:B(mathcal {C}_{f})), and we complete this work with some significant results.

Highlights

It is clearly seen that p – limm,n→∞ xmn = 0 but x ∞ = supm,n∈N |xmn| = ∞, so x ∈ Cp – Mu
Appl. 2017(1):149, 2017), we introduce new almost null and almost convergent double sequence spaces B(Cf ) and B(Cf0 ) as the four-dimensional generalized difference matrix B(r, s, t, u) domain in the spaces Cf and Cf0, respectively
6 Conclusion The concept of almost convergence of single sequence was introduced by Lorentz [12]

Summary

Introduction

As natural continuation of [2] and [11], we introduce new almost null and almost convergent double sequence spaces B(Cf ) and B(Cf0 ) as the domain of fourdimensional generalized difference matrix B(r, s, t, u) in the spaces Cf and Cf0 , respectively. Lemma 4.7 ([21]) A four-dimensional matrix A = (amnkl) is almost regular, i.e., A ∈ (Cbp : Cf )reg, iff the condition (4.1) and the following conditions hold: lim a i, j, q, q , m, n = 0, q,q →∞

Results

Conclusion