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

Abstract

We firstly summarize the related literature about ( , , , )-summability of double sequence spaces and almost ( , , , )-summable double sequence spaces. Then we characterize some new matrix classes of (L : C ), ( (L ) : C ), and (L : (C )) of four-dimensional matrices in both cases of 0 < ≤ 1 and 1 < < ∞, and we complete this work with some significant results. Preliminaries, Background, and NotationsWe denote the set of all complex valued double sequence by Ω which is a vector space with coordinatewise addition and scalar multiplication. Any subspace of Ω is called a double sequence space. A double sequence = ( ) of complex numbers is called bounded if ‖ ‖ ∞ = sup , ∈N | | < ∞, where N = {0, 1, 2, . . .}. The space of all bounded double sequences is denoted by M which is a Banach space with the norm ‖ ⋅ ‖ ∞ . Consider the double sequence = ( ) ∈ Ω. If for every > 0 there exists a natural number 0 = 0 ( ) and ∈ C such that | − | < for all , > 0 , then the double sequence is said to be convergent in Pringsheim's sense to the limit point ; say that − lim , →∞ = , where C indicates the complex field. The space C denotes the set of all convergent double sequences in Pringsheim's sense. Although every convergent single sequence is bounded, this is not hold for double sequences in general. That is, there are such double sequences which are convergent in Pringsheim's sense but not bounded. Actually, Boos [1, p. 16] defined the sequence = ( ) byThen it is clearly seen that − lim , →∞ = 0 but ‖ ‖ ∞ = sup , ∈N | | = ∞, so ∈ C −M . The set C denotes the space of both bounded and convergent double sequences; that is, C = C ∩ M . Hardy [2] showed that a double sequence = ( ) is said to converge regularly to if ∈ C and the limits := lim , ( ∈ N) and := lim , ( ∈ N) exist, and the limits lim lim and lim lim exist and are equal to the -limit of . Moreover, by C 0 and C 0 , we may denote the spaces of all null double sequences contained in the sequence spaces C and C , respectively. Móricz [3] proved that the double sequence spaces C , C 0 , C , and C 0 are Banach spaces with the norm ‖ ⋅ ‖ ∞ . The space L of all absolutely −summable double sequences corresponding to the space ℓ of −summable single sequences was defined by Başar and Sever [4]; that is,which is a Banach space with the norm ‖ ⋅ ‖ . Then, the space L which is a special case of the space L with = 1 is introduced by Zeltser [5].Let be a double sequence space and converging with respect to some linear convergence rule is − lim : → C. Then, the sum of a double series ∑ , relating to this rule is defined by − ∑ , = − lim , →∞ ∑ , , =0. Throughout the paper, the summations from 0 to ∞ without limits, that is, ∑ , , mean that ∑ ∞ , =0. Here and after, unless otherwise stated, we consider that denotes any of the symbols , , or .

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…”

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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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