Matrix Mappings Research Articles

New sequence spaces derived from the Catalan–Motzkin matrix and related matrix transformations

UDC 512.6 We consider the domain of a conservative matrix involving Catalan and Motzkin numbers in the sequence spaces c and c 0 . Moreover, the α -, β -, and γ -duals are given and some matrix mappings are presented on the resulting spaces.

A study on Fibo-Pascal sequence spaces and associated matrix transformations and applications of Hausdorff measure of non-compactness

Abstract In this article, we introduce Fibo-Pascal sequence spaces P p F {P_{p}^{F}} , 0 < p < ∞ {0<p<\infty} , and P ∞ F {P_{\infty}^{F}} through the utilization of the Fibo-Pascal matrix P F {P^{F}} . We establish that both P p F {P_{p}^{F}} and P ∞ F {P_{\infty}^{F}} are BK-spaces, enjoying a linear isomorphism with the classical spaces ℓ p {\ell_{p}} and ℓ ∞ {\ell_{\infty}} , respectively. Further contributing to the depth of our investigation, we proceed to derive the Schauder basis of the space P p F {P_{p}^{F}} , alongside an exhaustive computation of the α-, β-, and γ-duals for both spaces P p F {P_{p}^{F}} and P ∞ F {P_{\infty}^{F}} . Additionally, we undertake the task of characterizing certain classes of matrix mappings pertaining to the spaces P p F {P_{p}^{F}} and P ∞ F {P_{\infty}^{F}} . The final section of this study is dedicated to the meticulous characterization of compact operators acting in the spaces P 1 F {P_{1}^{F}} , P p F {P_{p}^{F}} , and P ∞ F {P_{\infty}^{F}} by using Hausdorff measure of non-compactness.

A paranormed fractional ordered Euler–Riesz difference sequence space

In this study, we introduce a novel sequence space denoted as [Formula: see text] with a fractional order [Formula: see text]. This new space is defined by the matrix [Formula: see text], which is a composition of the Euler–Riesz matrix [Formula: see text] and the fractional ordered difference operator [Formula: see text]. We explore its topological properties along with its [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-duals. Furthermore, we provide characterizations for certain matrix mappings from [Formula: see text] to the sequence spaces of Maddox.

A study on q-analogue of Catalan sequence spaces

In this study, we construct q-analog C(q) of Catalan matrix and study the sequence spaces c0(C(q)) and c(C(q)) defined as the domain of q-Catalan matrix C(q) in the spaces c0 and c, respectively. We exhibit some topological properties, obtain Schauder bases and determine ?-, ?-, and ?-duals of the spaces c0(C(q)) and c(C(q)). Finally, we characterize certain class of matrix mappings from the spaces c0(C(q)) and c(C(q)) to the space ? = {??, c0, c, ?1} and give the necessary and sufficient conditions for a matrix operator to be compact.

Hausdorff measure of noncompactness of matrix mappings on certain difference sequence spaces

The well-known difference sequence spaces were introduced by Kızmaz [17] in 1981 and have been generalized by many authors uptill now. These spaces were fi rst time extended by Sarıgöl [31] to the sequence spaces$l_{\infty }(\Delta _{q}),$ $c(\Delta _{q})$ or $c_{0}(\Delta _{q})$. The aim of this paper is to establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the extended spaces and also to characterize some classes of compact operators by using the Hausdorff measure of noncompactness.

Four dimensional matrix mappings and applications

In this paper, we characterize the classes (L,Lk) , (Lk,L) and (L∞,Lk) , 1 ≤ k < ∞, of all four dimensional infinite matrices, where Lk and L∞ are the spaces of all absolutely k-summable and bounded double sequences, respectively. Using them, we establish some relations between N, pn, qn and N, p′ n, q′n k summability methods which extend some results of Bosanquet (1950), Sarıg¨ol (1993), Sarıg¨ol and Bor (1995), and Sunouchi (1949) to double summability methods, and give a relation between single and double summability methods.

Matrix mappings and compact operators for Schröder sequence spaces

In this paper, we discuss the domain of a recently defined conservative matrix, constructed by means of the Schröder numbers in the spaces of $p-$absolutely summable sequences and bounded sequences. We determine the $\beta-$duals of the Banach spaces, introduced here, and present characterization of some matrix operators. Moreover, we give the characterization of certain compact operators via the Hausdorff measure of noncompactness.

On Generalizedp,q-Euler Matrix and Associated Sequence Spaces

In this study, we introduce newBK-spacesbsr,tp,qandb∞r,tp,qderived by the domain ofp,q-analogueBr,tp,qof the binomial matrix in the spacesℓsandℓ∞,respectively. We study certain topological properties and inclusion relations of these spaces. We obtain a basis for the spacebsr,tp,qand obtain Köthe-Toeplitz duals of the spacesbsr,tp,qandb∞r,tp,q.We characterize certain classes of matrix mappings from the spacesbsr,tp,qandb∞r,tp,qto spaceμ∈ℓ∞,c,c0,ℓ1,bs,cs,cs0.Finally, we investigate certain geometric properties of the spacebsr,tp,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.

Matrix mappings and Hausdorff measure of non-compactness on Riesz difference spaces of fractional order

Matrix mappings and Hausdorff measure of non-compactness on Riesz difference spaces of fractional order

Fark Seri Uzayları ve Matris Dönüşümleri

This paper deals with new series space |C_α |_p (∇) introduced by using Cesàro means and difference operator. It is shown that this newly defined space |C_α |_p (∇) is a BK- space and has Schauder basis. Furthermore, the α, β, and γ-duals of |C_α |_p (∇) are computed and the characterizations of classes of matrix mappings from |C_α |_p (∇) to X={l_∞,c,c_0} are also given.

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.

A new regular infinite matrix defined by Jordan totient function and its matrix domain in ℓp

In this paper, we first define a new regular matrix by using the arithmetic function called Jordan totient function and study the matrix domain of this newly introduced matrix in the Banach space ℓp. After computing the dual spaces of this new space, we characterize certain matrix mappings related to this space.

On matrix transformations and Hausdorff measure of noncompactness of Euler difference sequence spaces of fractional order

In the present paper, some results on matrix mappings and Hausdorff measure of noncompactness of certain generalized Euler difference sequence spaces of fractional order are discussed. Also, the Hausdorff measures of noncompactness of certain matrix operators that map an arbitrary BK-space into the classical sequence spaces are established. Furthermore, by using this measure, the characterization of some classes of Euler mean compact operators are determined in the BK-spaces.

Hypercomplex Widely Linear Estimation Through the Lens of Underpinning Geometry

We provide a rigorous account of the equivalence between the complex-valued widely linear estimation method and the quaternion involution widely linear estimation method with their vector-valued real linear estimation counterparts. This is achieved by an account of degrees of freedom and by providing matrix mappings between a complex variable and an isomorphic bivariate real vector, and a quaternion variable versus a quadri-variate real vector. Furthermore, we show that the parameters in the complex-valued linear estimation method, the complex-valued widely linear estimation method, the quaternion linear estimation method, the quaternion semi-widely linear estimation method, and the quaternion involution widely linear estimation method include distinct geometric structures imposed on complex numbers and quaternions, respectively, whereas the real-valued linear estimation methods do not exhibit any structure. This key difference explains, both in theoretical and practical terms, the advantage of estimation in division algebras (complex, quaternion) over their multivariate real vector counterparts. In addition, we discuss the computational complexities of the estimators of the hypercomplex widely linear estimation methods.

Matrix mappings and general bounded linear operators on the space bv

Abstract If X and Y are FK spaces, then every infinite matrix A ∈ (X, Y) defines a bounded linear operator LA ∈ B(X, Y) where LA (x) = Ax for each x ∈ X. But the converse is not always true. Indeed, if L is a general bounded linear operator from X to Y, that is, L ∈ B(X, Y), we are interested in the representation of such an operator using some infinite matrices. In this paper we establish the representations of the general bounded linear operators from the space bv into the spaces ℓ ∞ , c and c 0. We also prove some estimates for their Hausdorff measures of noncompactness. In this way we show the difference between general bounded linear operators between some sequence spaces and the matrix operators associated with matrix transformations.

The Application Domain of Infinite Matrices with Algorithms

The purpose of this paper is twofold. First, we define the new spaces and investigate some topological and structural properties. Also, we compute dual spaces of new spaces which are help us in the characterization of matrix mappings. Second, we give some examples related to new spaces. A flow chart of the stages of the newly constructed sequence spaces and the algorithms of the workings at each step are given.

Representations of functionals on absolute weighted spaces and adjoint operators

In the present paper, we establish general representations of continuous linear functionals, which play important roles in Functional Analysis, of the absolute weighted spaces which have recently been introduced in Sarıgöl (2016, 2011), and also determine their norms. Further making use of this we give adjoint operators of matrix mappings defined on these spaces.

On topological properties of spaces obtained by the double band matrix

Abstract 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.

Four-dimensional generalized difference matrix and some double sequence spaces

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 varthetain{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.