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For a nonnegative integer k, let T k denote the kth telephone number. Consider the matrix T = (T k r) given by T k r = { r T r − 1 T k + 1 − 1 , 0 ≤ r ≤ k , 0 , r > k , where k , r = 0 , 1 , 2 , ... Using the matrix T , we define matrix domains T p : = (ℓ p) T for 0 < p < ∞ and T ∞ : = (ℓ ∞) T . In this context, we construct a Schauder basis for T p and identify their α-, β-, and γ-duals. We also derive various results pertaining to matrix transformations from the spaces T p and T ∞ to traditional spaces such as ℓ ∞ , c, c 0 , and ℓ 1 . Additionally, we explore several geometric attributes of the spaces T p and T ∞ , including the approximation property, D-P property, Hahn–Banach extension property, and rotundity. [ABSTRACT FROM AUTHOR]

In this paper, we establish a novel category of sequence spaces ℓ p q λ and ℓ ∞ q λ by utlizing q-analogue Λ q of Λ-matrix. Our investigation outlines several topological characteristics and inclusion results of these newly introduced sequence spaces, specifically identifying them as BK-spaces. Subsequently, we demonstrate that these novel sequence spaces are of nonabsolute type and establish their isometric isomorphism with ℓ p and ℓ ∞ . Moreover, we obtain the α-, β-, and γ-duals of these sequence spaces. We further characterize the class (ℓ p q λ , X) of matrices, where X is any of the spaces ℓ ∞ , c, or c 0 . Lastly, our study delves into the exploration of specific geometric properties exhibited by the space ℓ p q λ . [ABSTRACT FROM AUTHOR]

In this research paper, we undertake an investigation into Cesàro q -difference sequence spaces X (C 1 δ ; q) , where X ∈ { ℓ ∞ , c , c 0 }. These spaces are generated using the matrix C 1 δ , q , which is a product of the Cesàro matrix C 1 of the first-order and the second-order q -difference operator ∇ q 2 defined by (∇ q 2 f) k = f k - (1 + q) f k - 1 + q f k - 2 , (k ∈ N 0) , where q ∈ (0 , 1) and f k = 0 for k < 0. Our endeavor includes the establishment of significant inclusion relationships, the determination of bases for these spaces, the investigation of their α -, β -, and γ -duals, and the formulation of characterization results pertaining to matrix classes (X , Y) , with X chosen from the set { ℓ ∞ (C 1 δ ; q) , c (C 1 δ ; q) , c 0 (C 1 δ ; q) } and Y chosen from the set { ℓ ∞ , c , c 0 , ℓ 1 }. The final section of our study is dedicated to the meticulous spectral analysis of the weighted q -difference operator ∇ q 2 ; z over the space c 0 of null sequences. [ABSTRACT FROM AUTHOR]

In the current work, it is constructed the Motzkin matrix obtained by using Motzkin numbers M = (m rs) and is examined the sequence spaces c (M) and c 0 (M) described as the domain of Motzkin matrix M in the spaces c and c0, respectively. It is investigated topological properties, established Schauder basis and stated Köthe duals of the aforementioned spaces. Additionally, it is characterized the matrix classes from c (M) and c 0 (M) to the classical sequence spaces and vice versa. Finally, Motzkin core of any sequence is presented and it is elaborated some inclusion relation of just described new type of core. [ABSTRACT FROM AUTHOR]

In the present paper, we characterize certain compact and matrix operators on the Fibonacci spaces F θ q , studied by Gökçe and Sarıgöl (Kragujevac J. Math. 44(2), 273–286 (2020)), together with their norms and identities or estimates for the Hausdorff measures of noncompactness. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a new regular matrix B ~ and define the sequence space ℓ p ( B ~ ). We also examine some topological properties for this space and determine α - , β - , γ - duals. As well, we characterize some matrix classes and investigate some geometric properties such as uniform convexity, strict convexity and super-reflexivity. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new conservative matrix involving Schröder numbers and investigate its matrix domain in the sequence spaces c and c 0 . Furthermore, the Köthe–Toeplitz dual, generalized Köthe–Toeplitz dual and Garling dual are expressed and several matrix transformations are given based on the new Banach spaces. [ABSTRACT FROM AUTHOR]

The prime objective of this paper is to define a new double difference operator with arbitrary order via which new classes of difference double sequences are introduced. Results on topological structures, dual spaces and four-dimensional matrix mappings related to the proposed difference double sequence spaces are discussed. As an application of this work, the proposed operator is being used to approximate partial derivatives of fractional orders. Some numerical examples are also given in support of the validity or the clear visualization of the results obtained. [ABSTRACT FROM AUTHOR]

In this paper we generalize the space ℓ ˆ k of absolutely almost convergent series (J. Math. Anal. Appl. 161:50–56, 1991) via weighted mean transformations. We study some inclusion relations and their topological properties. Further we characterize certain matrix transformations. [ABSTRACT FROM AUTHOR]

Bor [1], Bor and Thorpe [2] and the author [8] established the set of necessary and sufficient condition for the equivalence of absolute weighted mean summability methods of infinite series. In the present paper we extend their result to doubly infinite series by two dimensional weighted mean, which also includes the result of Rhoades [7]. [ABSTRACT FROM AUTHOR]

In the present paper, we show that ϑ -convergent double series spaces CS ϑ and the series space BV of double sequences of bounded variation are BDK-spaces and investigate their A K (ϑ) -space property, where ϑ ∈ { p , b p , r } . [ABSTRACT FROM AUTHOR]

In this study, as the domain of four-dimensional backward difference matrix in the space L u (t) , we introduce the complete paranormed space BV (t) of bounded variation double sequences and examine some properties of that space. Also, we determine the γ -dual and β (ϑ) -dual of the space BV (t) . Finally, we characterize the classes (BV (t) : M u) , (BV (t) : C ϑ) and (L u (t) : μ) with μ ∈ { BS , CS ϑ , M u (Δ) , C ϑ (Δ) } , where M u (Δ) and C ϑ (Δ) denote the spaces of all double sequences whose Δ -transforms are in the spaces M u and C ϑ , respectively. [ABSTRACT FROM AUTHOR]

In this paper, the Nörlund–Orlicz difference sequence space N t (F , Δ n m , u , q) of nonabsolute type is introduced as a domain of Nörlund means which is isomorphic to the space ℓ (p) and the basis of the space is constructed. Additionally, the α−, β−, and γ-duals of the spaces are computed and their matrix transformations are given. Finally, the properties like rotundity, modularity of the newly formed spaces are established. [ABSTRACT FROM AUTHOR]

Let λ denote any one of the spaces ℓ∞ and ℓp and λ(Ť) be the domain of the band matrix Ť. We study ℓp(Ť) for 1 ≤ p ≤ ∞ and give some inclusions and its topological properties. Also, we define the alpha−, beta− and gamma− duals of the space ℓp(Ť). Finally, we give some matrix mappings.

The space was extended to the space by Maddox, where is a bounded sequence of positive numbers. The series space has more recently been studied by Hazar and Sarıgöl. In this article, following Maddox we derived a more general series space and show that it is a paranormed space and linearly isomorphic to the space Also, we construct its duals and Schauder basis. Further, we characterize the classes of certain matrix transformations on that space and obtain some well-known results as a particular case. [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 derive a series space |Cλ,μ|k using the well known absolute Cesàro summability |Cλ,μ|k of Das (Proc. Camb. Philol. Soc. 67:321-326, 1970), compute its β-dual, give some algebraic and topological properties, and characterize some matrix operators defined on that space. So we generalize some results of Bosanquet (J. Lond. Math. Soc. 20:39-48, 1945), Flett (Proc. Lond. Math. Soc. 7:113-141, 1957), Mehdi (Proc. Lond. Math. Soc. (3)10:180-199, 1960), Mazhar (Tohoku Math. J. 23:433-451, 1971), Orhan and Sarıgöl (Rocky Mt. J. Math. 23(3):1091-1097, 1993) and Sarıgöl (Commun. Math. Appl. 7(1):11-22, 2016; Math. Comput. Model. 55:1763-1769, 2012). [ABSTRACT FROM AUTHOR]

In this study, I introduce some new double sequence spaces $B(\mathcal {M}_{u})$ , $B(\mathcal{C}_{p})$ , $B(\mathcal{C}_{bp})$ , $B(\mathcal {C}_{r})$ and $B(\mathcal{L}_{q})$ as the domain of four-dimensional generalized difference matrix $B(r,s,t,u)$ in the spaces $\mathcal {M}_{u}$ , $\mathcal{C}_{p}$ , $\mathcal{C}_{bp}$ , $\mathcal{C}_{r}$ and $\mathcal{L}_{q}$ , respectively. I show that the double sequence spaces $B(\mathcal{M}_{u})$ , $B(\mathcal{C}_{bp})$ and $B(\mathcal{C}_{r})$ are the Banach spaces under some certain conditions. I give some inclusion relations with some topological properties. Moreover, I determine the α-dual of the spaces $B(M_{u})$ and $B(\mathcal{C}_{bp})$ , the $\beta(\vartheta)$ -duals of the spaces $B(M_{u})$ , $B(\mathcal{C}_{p})$ , $B(\mathcal{C}_{bp})$ , $B(\mathcal{C}_{r})$ and $B(\mathcal{L}_{q})$ , where $\vartheta\in\{p,bp,r\}$ , and the γ-dual of the spaces $B(\mathcal{M}_{u})$ , $B(\mathcal{C}_{bp})$ and $B(\mathcal{L}_{q})$ . Finally, I characterize the classes of four-dimensional matrix mappings defined on the spaces $B(\mathcal{M}_{u})$ , $B(\mathcal{C}_{p})$ , $B(\mathcal{C}_{bp})$ , $B(\mathcal{C}_{r})$ and $B(\mathcal{L}_{q})$ of double sequences. [ABSTRACT FROM AUTHOR]

In the present paper, the difference sequence spaces c s 0 λ ( Δ ) , c s λ ( Δ ) and bs λ ( Δ ) of nonabsolute type are generalized by introducing a generalized Λ difference operator Λ ( Δ ( α ˜ ) ) . Also, their Schauder basis are calculated and α -, β - and γ -duals of these spaces are investigated. Finally, some matrix transformations between these spaces and the basic sequence spaces ℓ p , c and c 0 are characterized, where 1 ≤ p ≤ ∞. [ABSTRACT FROM AUTHOR]

In this paper, we define new spaces as a generalization of the Fibonacci difference sequence spaces. Also, we establish some inclusion theorems related to these spaces and find the-,-,-duals. Lastly, we characterize some matrix classes on these spaces. [ABSTRACT FROM PUBLISHER]

In the present paper, new classes of difference sequence spaces X ( Γ , Δ α , u , p ) for X ∈ {ℓ ∞ , c , c 0 } are introduced by using the fractional difference operator Δ α , defined by Δ α ( x k ) = ∑ i = 0 ∞ ( - 1 ) i Γ ( α + 1 ) i ! Γ ( α - i + 1 ) x k + i , where ( u n ) is a sequence satisfying certain conditions and α is a proper fraction. In fact, the operator Δ α generalizes the difference operators used in several difference sequence spaces such as X ( Δ ), X ( Δ 2 ), Δ m ( X ), ΔX ( p ), Δ m X ( p ), X ( u ; Δ 2 ), X ( u , Δ , p ), X ( u , Δ 2 , p ) (see [3–12,20–22] ). Also, we investigate the topological structures and establish α −, β − and γ − duals of the spaces X ( Γ , Δ α , u , p ). Furthermore, the matrix transformations between these spaces and the basic sequence spaces ℓ ∞ ( q ), c 0 ( q ) and c ( q ) are characterized. [ABSTRACT FROM AUTHOR]