On the domain of difference double sequence spaces of arbitrary orders

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.

On generalized Fibonacci difference sequence spaces and compact operators

In this paper, we introduce Fibonacci backward difference operator [Formula: see text] of fractional order [Formula: see text] by the composition of Fibonacci band matrix [Formula: see text] and difference operator [Formula: see text] of fractional order [Formula: see text] defined by [Formula: see text] and introduce sequence spaces [Formula: see text] and [Formula: see text] We present some topological properties, obtain Schauder basis and determine [Formula: see text]-, [Formula: see text]- and [Formula: see text]-duals of the spaces [Formula: see text] and [Formula: see text] We characterize certain classes of matrix mappings from the spaces [Formula: see text] and [Formula: see text] to any of the space [Formula: see text] [Formula: see text] [Formula: see text] or [Formula: see text] Finally we compute necessary and sufficient conditions for a matrix operator to be compact on the spaces [Formula: see text] and [Formula: see text]

On Generalized p , q -Euler Matrix and Associated Sequence Spaces

In this study, we introduce new BK -spaces b s r , t p , q and b ∞ r , t p , q derived by the domain of p , q -analogue B r , t p , q of the binomial matrix in the spaces ℓ s and ℓ ∞ , respectively. We study certain topological properties and inclusion relations of these spaces. We obtain a basis for the space b s r , t p , q and obtain Köthe-Toeplitz duals of the spaces b s r , t p , q and b ∞ r , t p , q . We characterize certain classes of matrix mappings from the spaces b s r , t p , q and b ∞ r , t p , q to space μ ∈ ℓ ∞ , c , c 0 , ℓ 1 , b s , c s , c s 0 . Finally, we investigate certain geometric properties of the space b s r , t p , q .

On some new paranormed sequence spaces defined by the matrix (D )(r,0,0,s)

In this paper, we introduce some new paranormed sequence spaces and study some topological properties. Further, we determine α, β and γ-duals of the new sequence spaces and finally, we establish the necessary and sufficient conditions for characterization of matrix mappings.

Hausdorff measure of noncompactness of matrix mappings on Cesàro spaces

In this study we establish some identities or estimates for operator norms and the Hausdorff measure of noncompactness of certain operators on spaces |C_{α}|_{k}, which have more recently been introduced in [14]. Further, by applying the Hausdorff measure of noncompactness, we establish the necessary and sufficient conditions for such operators to be compact and so the some well known results are generalized.

Sequence spaces derived by the triple band generalized Fibonacci difference operator

In this article we introduce the generalized Fibonacci difference operator $\mathsf{F}(\mathsf{B})$ F ( B ) by the composition of a Fibonacci band matrix and a triple band matrix $\mathsf{B}(x,y,z)$ B ( x , y , z ) and study the spaces $\ell _{k}( \mathsf{F}(\mathsf{B}))$ ℓ k ( F ( B ) ) and $\ell _{\infty }(\mathsf{F}(\mathsf{B}))$ ℓ ∞ ( F ( B ) ) . We exhibit certain topological properties, construct a Schauder basis and determine the Köthe–Toeplitz duals of the new spaces. Furthermore, we characterize certain classes of matrix mappings from the spaces $\ell _{k}(\mathsf{F}(\mathsf{B}))$ ℓ k ( F ( B ) ) and $\ell _{\infty }(\mathsf{F}(\mathsf{B}))$ ℓ ∞ ( F ( B ) ) to space $\mathsf{Y}\in \{\ell _{\infty },c_{0},c,\ell _{1},cs_{0},cs,bs\}$ Y ∈ { ℓ ∞ , c 0 , c , ℓ 1 , c s 0 , c s , b s } and obtain the necessary and sufficient condition for a matrix operator to be compact from the spaces $\ell _{k}(\mathsf{F}(\mathsf{B}))$ ℓ k ( F ( B ) ) and $\ell _{\infty }(\mathsf{F}(\mathsf{B}))$ ℓ ∞ ( F ( B ) ) to $\mathsf{Y}\in \{ \ell _{\infty }, c, c_{0}, \ell _{1},cs_{0},cs,bs\} $ Y ∈ { ℓ ∞ , c , c 0 , ℓ 1 , c s 0 , c s , b s } using the Hausdorff measure of non-compactness.

On the Domain of the Four-Dimensional Sequential Band Matrix in Some Double Sequence Spaces

Let E represent any of the spaces M u , C ϑ ( ϑ = { b , b p , r } ) , and L q ( 0 < q < ∞ ) of bounded, ϑ -convergent, and q-absolutely summable double sequences, respectively, and E ˜ be the domain of the four-dimensional (4D) infinite sequential band matrix B ( r ˜ , s ˜ , t ˜ , u ˜ ) in the double sequence space E, where r ˜ = ( r m ) m = 0 ∞ , s ˜ = ( s m ) m = 0 ∞ , t ˜ = ( t n ) n = 0 ∞ , and u ˜ = ( u n ) n = 0 ∞ are given sequences of real numbers in the set c ∖ c 0 . In this paper, we investigate the double sequence spaces E ˜ . First, we determine some topological properties and prove several inclusion relations under some strict conditions. Then, we examine α -, β ( ϑ ) -, and γ -duals of E ˜ . Finally, we characterize some new classes of 4D matrix mappings related to our new double sequence spaces and conclude the paper with some significant consequences.

The application domain of difference type matrix D(r,0,s,0,t) on some sequence spaces

In this paper we introduce new sequence spaces with the help of domain of matrix D(r,0,s,0,t), and study some of their topological properties. Further, we determine ? and ? duals of the new sequence spaces and finally, we establish the necessary and sufficient conditions for characterization of the matrix mappings.

On topological properties of spaces obtained by the double band matrix

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.