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Define an infinite matrix D α = (d n , v α) by d n , v α = { v α σ (α) ⁢ (n) , v ∣ n , 0 , v ∤ n , where σ (α) ⁢ (n) is defined to be the sum of the 훼-th power of the positive divisors of n ∈ N , and construct the matrix domains ℓ p ⁢ (D α) ( 0 < p < ∞ ), c 0 ⁢ (D α) , c ⁢ (D α) and ℓ ∞ ⁢ (D α) defined by the matrix D α . We develop Schauder bases and determine 훼-, 훽- and 훾-duals of these new spaces. We characterize some matrix transformation from ℓ p ⁢ (D α) , c 0 ⁢ (D α) , c ⁢ (D α) and ℓ ∞ ⁢ (D α) to ℓ ∞ , 푐, c 0 and ℓ 1 . Furthermore, we determine some criteria for compactness of an operator (or matrix) from X ∈ { ℓ p ⁢ (D α) , c 0 ⁢ (D α) , c ⁢ (D α) , ℓ ∞ ⁢ (D α) } to ℓ ∞ , 푐, c 0 or ℓ 1 . [ABSTRACT FROM AUTHOR]

AbstractThis paper describes infinite sets of linear and polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a solution for the infinite set of equations. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the notions of the Köthe–Toeplitz duals of sequence spaces by introducing the concepts of bicomplex α-dual, bicomplex β-dual and bicomplex γ-dual, and also we compute them for some bicomplex sequence spaces l p ⁢ (픹 ⁢ ℂ) for 1 ≤ p ≤ ∞ , c 0 ⁢ (픹 ⁢ ℂ) and c ⁢ (픹 ⁢ ℂ) . Furthermore, we define a concept of bicomplex multiplier space as the bicomplex version of multiplier space of two sequence spaces and support this definition with examples. [ABSTRACT FROM AUTHOR]

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]

This research paper aims to introduce the concept of lacunary ideal Cauchy sequences of fuzzy variables in a credibility space. We establish the interrelationships between this notion with lacunary ideal convergent sequences in the same structure from several aspects of credibility. Furthermore, we explore the concepts of strongly lacunary Cauchy, strongly ℐ -lacunary Cauchy, and strongly ℐ ∗ -lacunary Cauchy sequences of fuzzy variables within the context of credibility. We also examine the interconnections between these concepts and analyze their relationships. [ABSTRACT FROM AUTHOR]

We present a generalization of the classical Orlicz–Pettis theorem about subseries convergence in topological vector spaces. In preparation we review some aspects of the theory of locally convex cones, a generalization of locally convex topological vector spaces. We introduce conical extensions of the classical sequence spaces and a version of Schur’s theorem about weak and norm convergence of sequences in the extension of ℓ 1. Our version of the Orlicz–Pettis theorem refers to series of bounded elements of a locally convex cone. [ABSTRACT FROM AUTHOR]

Recently, some new sequence spaces ℓ p (A α) (0 < p < ∞) , c 0 (A α) , c (A α) , and ℓ ∞ (A α) have been studied by Yaying et al. (Forum Math., 2024, https://doi.org/10.1515/forum-2023-0138) as matrix domains of A α = (a n , v α) , where a m , v α = { v α ρ (α) (m) , v ∣ m , 0 , v ∤ m , and ρ (α) (m) : = sum of the α th power of the positive divisors of m ∈ N . They obtained their duals, matrix transformations and associated compact matrix operators for these matrix classes. This article deals with some geometric properties of these sequence spaces. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of λ -limited sets and λ -L-sets in a Banach space X and its dual X ∗ respectively, using the vector valued sequence spaces λ w ∗ (X ∗) and λ w (X) . We find characterizations for these sets in terms of absolutely λ -summing operators and investigate the relationship between λ -compact sets and λ -limited sets, with a particular focus on the crucial role played by a norm iteration property. We also consider λ -limited operators and show that this class is an operator ideal containing the ideal of λ -compact operators for a suitably restricted λ . Furthermore, we define a generalized Gelfand-Philips property for Banach spaces corresponding to an abstract sequence space. [ABSTRACT FROM AUTHOR]

The primary objective of this paper is to create a novel infinite Toeplitz matrix by leveraging Tetranacci numbers. This matrix serves as the foundation for defining new sequence spaces denoted as c 0 ⁢ ( G ) {c_{0}(G)} , c ⁢ ( G ) {c(G)} , ℓ ∞ ⁢ ( G ) {\ell_{\infty}(G)} , and ℓ p ⁢ ( G ) {\ell_{p}(G)} , where 1 ≤ p < ∞ {1\leq p<\infty} . By utilizing this newly constructed matrix, the paper also explores and examines various algebraic and topological properties inherent to the sequence spaces c 0 ⁢ ( G ) {c_{0}(G)} , c ⁢ ( G ) {c(G)} , ℓ ∞ ⁢ ( G ) {\ell_{\infty}(G)} , and ℓ p ⁢ ( G ) {\ell_{p}(G)} for values of p within the range of 1 ≤ p < ∞ {1\leq p<\infty} . At last, we also prove existence theorem with example for infinite systems of differential equations in ℓ p ⁢ ( G ) {\ell_{p}(G)} . [ABSTRACT FROM AUTHOR]

We study and characterize the inclusion relations of global classes in the general weight matrix framework in terms of growth relations for the defining weight matrices. We consider the Roumieu and Beurling cases, and as a particular case, we also treat the classical weight function and weight sequence cases. Moreover, we construct a weight sequence which is oscillating around any weight sequence which satisfies some minimal conditions and, in particular, around the critical weight sequence (p !) 1 / 2 , related with the non-triviality of the classes. Finally, we also obtain comparison results both on classes defined by weight functions that can be defined by weight sequences and conversely. [ABSTRACT FROM AUTHOR]

In this article, we use the notion of lacunary statistical convergence of order (α,β)\left(\alpha ,\beta ) to introduce new sequence spaces by lacunary sequence, invariant means defined by Musielak-Orlicz function ℳ=(ℵk){\mathcal{ {\mathcal M} }}=\left({\aleph }_{k}). We also examine some topological properties and prove inclusion relations between newly constructed sequence spaces.

In this paper, we introduce inductive limits of the Fréchet spaces ℓ (p +) , ces (p +) , and d (p +) ( 1 ≤ p < ∞ ) and projective limits of the (LB)-spaces ℓ (p -) , ces (p -) , and d (p -) ( 1 < p ≤ ∞ ). After having established some topological properties of such spaces as acyclicity and ultrabornologicity, we prove that the generalized Cesàro operators C t ( 0 ≤ t ≤ 1 ) act continuously in these sequence spaces, and we determine the spectra. Finally, we study the ergodic properties, that is, power boundedness, (uniform) mean ergodicity, and supercyclicity, of the operators C t acting in the (LF)-spaces and in the (PLB)-spaces mentioned above. [ABSTRACT FROM AUTHOR]

In this paper we investigate the concept of ℐ {\mathcal{I}} -monotonic convergence of sequences of bi-complex numbers. We also discuss some partial order relations ≼ i 2 {\preccurlyeq_{i_{2}}} and ≼ i id {\preccurlyeq_{i_{\rm id}}} between two bi-complex numbers and ℐ {\mathcal{I}} -convergence, ℐ {\mathcal{I}} -monotonic sequences with respect to different partial order relations, ℐ {\mathcal{I}} -Cauchy sequences, ℐ {\mathcal{I}} -bounded monotonic sequences of bi-complex numbers. The analysis indicates that ℐ {\mathcal{I}} -monotonicity with respect to ≼ i 2 {\preccurlyeq_{i_{2}}} is not comparable with ℐ {\mathcal{I}} -monotonicity with respect to ≼ i id {\preccurlyeq_{i_{\rm id}}} . [ABSTRACT FROM AUTHOR]

We give a universal formula describing derivation operators on a Hilbert space for a large class of interpolation methods. It is based on a simple new technique on "critical points" where all the derivations attain the maximum. We deduce from this a version of Kalton uniqueness theorem for such methods, in particular, for the real method. As an application of our ideas is the construction of a weak Hilbert space induced by the real J-method. Previously, such space was only known arising from the complex method. To complete the picture, we show, using a breakthrough of Johnson and Szankowski, nontrivial derivations whose values on the critical points grow to infinity as slowly as we wish. [ABSTRACT FROM AUTHOR]

In this research paper, our aim is to introduce the concepts of α , β -statistical convergence of order γ and α , β -statistical boundedness of order γ by using unbounded modulus functions in metric spaces. Furthermore, we obtain some relations between these concepts, and we provide some counterexamples to support our results. [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]

Hahn (Math Phys 32:3–88, 1922) defined the sequence space h. The main purpose of this study is to introduce the new Hahn sequence space h (C m) as the domain of Cesàro mean of order m and give some topological properties of the space h (C m) . Moreover, we determine the alpha-, beta- and gamma-duals of the space h (C m) and characterize the classes (ℓ 1 : h) , (h : ℓ p) , (h (C m) : V 1) and (V 2 : h (C m)) of matrix transformations, where 1 < p < ∞ , V 1 ∈ { ℓ ∞ , c , c 0 , ℓ p } and V 2 is any given sequence space. Finally, we compute the norm of the operators belonging to B (ℓ 1 , h (C m)) and determine the Hausdorff measure of noncompactness of the operators in B (ℓ 1 , h (C m)) . [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 this study, the concept of ρ-statistical convergence with respect to the neutrosophic norm in the neutrosophic normed spaces is introduced. Some properties and some inclusion theorems related to this concept are investigated. [ABSTRACT FROM AUTHOR]

For more than 50 years, the author has asked himself why Lorentz spaces are only defined for two parameters. Has this choice been made just for simplicity or is it a natural bound that cannot be exceeded? This question is principal and has nothing to do with usefulness. Now, I discovered a way to produce Lorentz sequence spaces for any finite number of parameters. Having found the right approach, everything turns out to be elementary; the presentation becomes an orgy of mathematical induction. Unfortunately, the new spaces are only of theoretical interest, since we do not know any handy description of their members. This dilemma is, most likely, the reason for the restriction to two, regretted above. However, by the axiom of choice, mathematicians are used to deals with objects that exist only formally; see Banach limits. Therefore, our situation is much more comfortable. It is recommended that, as a first step, readers should have a short glance at the last section, where historical aspects and the interplay between basic concepts are described. Apart from proved theorems, the paper contains many open problems. It is motivated by the same spirit as my very last bibitem in the references. Senior mathematicians should show the way into the future. [ABSTRACT FROM AUTHOR]

This article intends to develop q-Catalan sequence spaces l p (C (q)) and l ∞ (C (q)) due to q-Catalan matrix C (q) in l p and l ∞ , respectively. Apart from obtaining some basic topological properties and Schauder basis, we compute α-, β-, and γ-duals of the spaces l p (C (q)) and l ∞ (C (q)). We state and prove a theorem that characterize certain matrix classes (X, Y), where X is either of the space l p (C (q)) or l ∞ (C (q)) and Y ∈ { l ∞ , c 0 , c , l 1 }. The final section is devoted to determination of certain conditions by which a matrix operator becomes compact. [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]

Kwapień’s theorem asserts that every continuous linear operator from ℓ 1 to ℓ p is absolutely r , 1 -summing for 1 / r = 1 - 1 / p - 1 / 2. When p = 2 it recovers the famous Grothendieck’s theorem. In this paper we investigate multilinear variants of these theorems and related issues. Among other results we present a unified version of Kwapień’s and Grothendieck’s results that encompasses the cases of multiple summing and absolutely summing multilinear operators. [ABSTRACT FROM AUTHOR]

Recently, Devaiya and Srivastava [3] studied the T r -strong convergence of numerical sequences and Fourier series using a lower triangular matrix T = (b m , n) , and generalized the results of Kórus [8]. The main objective of this paper is to introduce [ T r , G , u , q ] -strongly convergent sequence spaces for r ∈ N , and defined by a sequence of modulus functions. We also provide a relationship between [ T , G , u , q ] and [ T r , G , u , q ] -strongly convergent sequence spaces. Further, we investigate some geometrical and topological characteristics and establish some inclusion relationships between these sequence spaces. In the last, we derive some results on characterizations for T r -strong convergent sequences, statistical convergence and Fourier series using the idea of [ T r , G , u , q ] -strongly convergent sequence spaces. [ABSTRACT FROM AUTHOR]

This work computes the spectrum of the backward fractional difference operator $ \Delta ^{(\alpha)} $ Δ (α) $ (0 (0 < α < 1) over various Banach spaces $ \ell _{1},c $ ℓ 1 , c and $ c_0 $ c 0 . Note that recently, the spectrum of the fractional difference operator $ \Delta _\nu ^{(\alpha)} $ Δ ν (α) , ( $ \nu = (\nu _k) $ ν = (ν k) being either a constant or strictly decreasing sequence of positive real numbers satisfying some conditions) on the sequence space $ \ell _{1} $ ℓ 1 has been computed by Baliarsingh P. [On the spectrum of fractional difference operator. Linear Multilinear Algebra. 2021;1–13] and it has been noticed that the results obtained are unified and more general, but lack of accuracy for the case $ 0 0 < α < 1. In this work, we improve the results by providing the sharper estimations and some illustrative examples. Also, individual sets describing the point, the residual and the continuous spectrum of the operator $ \Delta ^{(\alpha)} $ Δ (α) on the Banach space c and $ c_0 $ c 0 are computed. [ABSTRACT FROM AUTHOR]

In this article, we introduce and study sequence spaces of Cesàro–Nörlund operators of order n associated with a sequence of Orlicz functions. We obtain some topological properties and Schauder basis of these sequence spaces. Moreover, we compute the α-, β- and γ-duals and the matrix transformations of these newly formed sequence spaces. Finally, we prove that these sequence spaces are of Banach–Saks type p and have a weak fixed-point property. [ABSTRACT FROM AUTHOR]

Recently, we have obtained two general theorems dealing with | N ¯ , p n | k summability factors of infinite series and trigonometric Fourier series (Bor in Bull Sci Math 169:102990, 2021) by using almost increasing sequences. In this paper, we have generalized these theorems to | N ¯ , p n ; θ n | k summability methods. Some new and known results are also obtained. [ABSTRACT FROM AUTHOR]

We present two families of L-matrices and investigate their properties using intricate factorization techniques. These factorization schemes involve well-known matrices such as the classical Cesàro, Gamma, Hausdorff, and Hilbert matrices, which are utilized to assess the norm of L-matrices. Additionally, these factorizations give rise to novel and intriguing inequalities in sequence spaces. [ABSTRACT FROM AUTHOR]

For fixed sequences u = (u i) i ∈ N , φ = (φ i) i ∈ N , we consider the weighted composition operator W u , φ with symbols u , φ defined by x = (x i) i ∈ N ↦ u (x ∘ φ) = (u i x φ i) i ∈ N . We characterize the continuity and the compactness of the operator W u , φ when it acts on the weighted Banach spaces l p (v) , 1 ≤ p ≤ ∞ , and c 0 (v) , with v = (v i) i ∈ N a weight sequence on N . We extend these results to the case in which the operator W u , φ acts on sequence (LF)-spaces of type l p (V) and on sequence (PLB)-spaces of type a p (V) , with p ∈ [ 1 , ∞ ] ∪ { 0 } and V a system of weights on N . We also characterize other topological properties of W u , φ acting on l p (V) and on a p (V) , such as boundedness, reflexivity and to being Montel. [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 this paper, we introduce the Fibonomial sequence spaces b 0 r , s , F and b c r , s , F and show that these are linearly isomorphic to the spaces c 0 and c, respectively. In addition, we present α - dual, β - dual and γ - dual for those spaces and characterize certain matrix classes. In the final section, we obtain some criteria for the compactness of certain matrix operators via Hausdorff measure of noncompactness on the space b 0 r , s , F. [ABSTRACT FROM AUTHOR]

In this article, we define the notions of strongly I-deferred Cesàro summable and μ -deferred I-statistical convergence for real sequences and also introduce their corresponding sequence spaces D C (p , q) I and μ D S (p , q) I , respectively. Additionally, some properties, based results and relations among numerous spaces have been established under a few restrictions. [ABSTRACT FROM AUTHOR]

In this article we introduce the notion of relative uniform convergence of difference double sequence of positive linear functions. We define the difference double sequence spaces 2 ℓ ∞ (Δ , r u) , 2 c (Δ , r u) , 2 c 0 (Δ , r u) , 2 c 0 B (Δ , r u) , 2 c B (Δ , r u) , 2 c R (Δ , r u) , 2 c 0 R (Δ , r u) and study their topological properties. [ABSTRACT FROM AUTHOR]

Abstract We study the weak convergence of infinite products of convex combinations of operators in complete CAT(0) spaces. We provide a new approach to this problem by considering a constructive selection of convex combinations in CAT(0) spaces that does not depend on the order of the involved elements and retain continuity properties with respect to them.

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 a way different from traditional methods, recently, new series spaces have been established by using absolute summability methods. Among these absolute summability methods, the absolute generalized Nörlund summability method |N, pn, qn, θn|k is extremely important as it includes several summability methods such as absolute Nörlund, weighted, Cesàro and Euler summability methods. Now, it is natural to ask behavior of the space derived by the absolute summability of generalized Nörlund means. The purpose of this study is to derive a more general series space including some well known spaces such as the spaces | N ˜ q θ | k , |Cα|k, |Cα, β|k, | N p θ | k and | E r θ | k , and to investigate its topological structure, α-, β- and γ-duals. Beside this, some matrix transformations related to these spaces have been characterized. [ABSTRACT FROM AUTHOR]

We study the weak convergence of infinite products of convex combinations of operators in complete CAT(0) spaces. We provide a new approach to this problem by considering a constructive selection of convex combinations in CAT(0) spaces that does not depend on the order of the involved elements and retain continuity properties with respect to them. [ABSTRACT FROM AUTHOR]

We establish Arazy–Cwikel type properties for the family of couples (ℓ p , ℓ q) , 0 ≤ p < q ≤ ∞ , and show that (ℓ p , ℓ q) is a Calderón–Mityagin couple if and only if q ≥ 1 . Moreover, we identify interpolation orbits of elements with respect to this couple for all p and q such that 0 ≤ p < q ≤ ∞ and obtain a simple positive solution of a Levitina–Sukochev–Zanin problem, clarifying its connections with whether (ℓ p , ℓ q) has the Calderón–Mityagin property or not. [ABSTRACT FROM AUTHOR]

The aim of this article is to investigate the neutrosophic Nörlund ℐ-statistically convergent sequence space. We present some neutrosophic normed spaces (NNSs) in Nörlund convergent spaces. In addition, we also examine various topological and algebraic properties of these convergent sequence spaces. Theorems are proved in light of the NNS theory approach. Results are obtained via different perspectives and new examples are produced to justify the counterparts and show the existence of the introduced notions. The results established in this research work supply an exhaustive foundation in NNS and make a significant contribution to the theoretical development of NNS in the literature. The original aspect of this study is the first wholly up-to-date and thorough examination of the features and implementation of neutrosophic Nörlund ℐ-statistically convergent sequences in NNS, based upon the standard definition.