On almost B-summable double sequence spaces

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 and as the four-dimensional generalized difference matrix domain in the spaces and , respectively. Firstly, we prove that the spaces and 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, -dual and γ-dual of the space . Finally, we characterize some new matrix classes , , and we complete this work with some significant results.

Four-Dimensional Generalized Difference Matrix and Almost Convergent Double Sequence Spaces

“…The matrix domain on some almost convergent single sequence spaces have been done by several mathematicians (see [31,32,33,34]). The idea of calculating four-dimensional matrix domain on almost convergent double sequences spaces is a new subject and studied by a few mathematicians (see [3,4,19]).…”

“…In this paper, as a natural continuation of the papers [3,4,19] we calculated the four-dimensional generalized difference matrix B…”

“…Almost B−summable double sequence spaces B(C f ) and B(C f0 ) were defined and studied by Tuǧ [3] which was derived by the domain of four-dimensional generalized difference matrix B(r, s, t, u) in the spaces of all almost convergent and almost null double sequences C f and C f0 , respectively.…”

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Most recently, some new double sequence spaces B(Mu), B(C ϑ ) where ϑ = {b, bp, r, f, f 0 } and B(Lq) for 0 < q < ∞ have been introduced as four-dimensional generalized difference matrix B(r, s, t, u) domain on the double sequence spaces Mu, C ϑ where ϑ = {b, bp, r, f, f 0 } and Lq for 0 < q < ∞, and some topological properties, dual spaces, some new four-dimensional matrix classes and matrix transformations related to these spaces have also been studied by Tuğ and Başar and Tuğ (see [1,2,3,4]). In this present paper, we introduce new strongly almost null and strongly almost convergent double sequence spaces B[C f ] and B[C f 0 ] as domain of four-dimensional generalized difference matrix B(r, s, t, u) in the spaces [C f ] and [C f 0 ], respectively.

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