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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]

We study properties of the boundary trace operator on the Sobolev space W 1 1 (Ω) . Using the density result by Koskela and Zhang (Arch. Ration. Mech. Anal. 222(1), 1-14 2016), we define a surjective operator T r : W 1 1 (Ω K) → X (Ω K) , where Ω K is von Koch's snowflake and X (Ω K) is a trace space with the quotient norm. Since Ω K is a uniform domain whose boundary is Ahlfors-regular with an exponent strictly bigger than one, it was shown by L. Malý (2017) that there exists a right inverse to Tr, i.e. a linear operator S : X (Ω K) → W 1 1 (Ω K) such that T r ∘ S = I d X (Ω K) . In this paper we provide a different, purely combinatorial proof based on geometrical structure of von Koch's snowflake. Moreover we identify the isomorphism class of the trace space as ℓ 1 . As an additional consequence of our approach we obtain a simple proof of the Peetre's theorem (Special Issue 2, 277-282 1979) about non-existence of the right inverse for domain Ω with regular boundary, which explains Banach space geometry cause for this phenomenon. [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]

In this paper, we study the problem of extending surjective isometries between the positive unit spheres of two (∑ ℓ p) ℓ q -spaces for 1 ≤ p , q ≤ ∞ . Let S (∑ ℓ p) ℓ q + = x ∈ (∑ ℓ p) ℓ q : x ≥ 0 ; ‖ x ‖ = 1 be the positive unit sphere of (∑ ℓ p) ℓ q and f : S (∑ ℓ p) ℓ q + → S (∑ ℓ p) ℓ q + be an isometry, i.e., ‖ f (x) - f (y) ‖ = ‖ x - y ‖ , for all x , y ∈ S (∑ ℓ p) ℓ q +. We prove that every surjective isometry f : S (∑ ℓ p) ℓ q + → S (∑ ℓ p) ℓ q + can be extended to a linear surjective isometry from (∑ ℓ p) ℓ q onto itself. This conclusion also holds for the case of (∑ ℓ p) c 0 , where 1 ≤ p ≤ ∞ . [ABSTRACT FROM AUTHOR]

The aim of this work is to describe subsets of Banach limits in terms of a certain functional characteristic. We compute radii and cardinalities for some of these subsets. [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]

For 1 ≤ p < ∞ , we prove that the dense subspace Y p of ℓ p (Γ) comprising all elements y such that y ∈ ℓ q (Γ) for some q ∈ (0 , p) admits a C ∞ -smooth norm which locally depends on finitely many coordinates. Moreover, such a norm can be chosen as to approximate the · p -norm. This provides examples of dense subspaces of ℓ p (Γ) with a smooth norm which have the maximal possible linear dimension and are not obtained as the linear span of a biorthogonal system. Moreover, when p > 1 or Γ is countable, such subspaces additionally contain dense operator ranges; on the other hand, no non-separable operator range in ℓ 1 (Γ) admits a C 1 -smooth norm. [ABSTRACT FROM AUTHOR]

We characterize Saphar's approximation property of order p in terms of a density of finite rank operators in the space of absolutely p-summing operators. We also study the approximation property of the tensor norm associated with the ideal of p-dominated operators. [ABSTRACT FROM AUTHOR]

In this paper, we introduce generalized difference weak sequence space classes by utilizing the difference operator Δ ı ȷ \Delta^{\jmath}_{\imath} and the de la Vallée–Poussin mean, denoted as [ ( V , λ ) w , Δ ı ȷ ] m [(\mathscr{V},\lambda)_{w},\Delta^{\jmath}_{\imath}]_{m} for m = 0 m=0 , 1, and ∞. Further, we explore some algebraic and topological properties of these spaces, including their nature as linear, normed, Banach, and BK spaces. Additionally, we examine properties such as solidity, symmetry, and monotonicity. Finally, we define and establish some inclusion relations among generalized difference weak statistical convergence, generalized difference weak 휆-statistical convergence, and generalized difference weak [ V , λ ] [\mathscr{V},\lambda] -convergence. [ABSTRACT FROM AUTHOR]

We present a complete characterization of the right-symmetric points in the one sum of two Banach spaces. We also obtain some basic properties of the left-symmetric (right-symmetric) points in the p sum, $ 1\leq p\not =2 \lt \infty $ 1 ≤ p ≠ 2 < ∞ ( $ 1 \lt p\not =2 \lt \infty $ 1 < p ≠ 2 < ∞), of two Banach spaces. Using these properties we (a) give examples of Banach spaces which do not have any non-zero left-symmetric points and (b) prove a complete characterization of those left-symmetric and right-symmetric points in the p sum, $ 1 \lt p\not =2 \lt \infty $ 1 < p ≠ 2 < ∞ , of two Banach spaces, whose components satisfy an additional norm assumption. We give examples of Banach spaces where all non-zero left-symmetric or right-symmetric points satisfy this additional norm assumption. We also present an alternative proof of the recently obtained characterization of the left-symmetric and the right-symmetric points in $ \ell _p^n $ ℓ p n , $ n\geq 3 $ n ≥ 3 , and $ \ell _p $ ℓ p , $ 1\leq p\not =2 \lt \infty $ 1 ≤ p ≠ 2 < ∞. [ABSTRACT FROM AUTHOR]

Given an infinite matrix M = (m nk) , we study a family of sequence spaces ℓ M p associated with it. When equipped with a suitable norm ‖ · ‖ M , p , we prove some basic properties of the Banach spaces of sequences (ℓ M p , ‖ · ‖ M , p) . In particular, we show that such spaces are separable and strictly/uniformly convex for a considerably large class of infinite matrices M for all p > 1 . A special attention is given to the identification of the dual space (ℓ M p) ∗ . Building on the earlier works of Bennett and Jägers, we extend and apply some classical factorization results to the sequence spaces ℓ M p . [ABSTRACT FROM AUTHOR]

Let 0 < α < 1 . We obtain necessary and sufficient conditions for the boundedness of the discrete fractional Hardy–Littlewood maximal operators M α on discrete weighted Lebesgue spaces. From this and a discrete variant of the Whitney decomposition theorem, necessary and sufficient conditions for the boundedness of the discrete Riesz potentials I α on discrete weighted Lebesgue spaces are discussed. As an application, the boundedness of I α on discrete weighted Morrey spaces is further obtained. [ABSTRACT FROM AUTHOR]

In the continuous setting, Morrey spaces have been studied extensively, especially since the late 1960s. Meanwhile, Morrey sequence spaces, which are also known as discrete Morrey spaces, have only been developed by Gunawan et al. since 2018. In this article, we extend some known results on their inclusion properties and their (lack of) uniform nonsquareness to mixed Morrey double-sequence spaces, i.e. Morrey double-sequence spaces equipped with a mixed norm. As in the calculation of three geometric constants of Morrey spaces by Gunawan et al. in 2019, we also compute three geometric constants, namely Von Neumann-Jordan constant, James constant, and Dunkl-Williams constant for mixed Morrey double-sequence spaces. These constants measure uniformly nonsquareness of any Banach space. Through the values of the three constants, we reveal that mixed Morrey double-sequence spaces are not uniformly nonsquare. A relation between mixed Morrey double-sequence spaces and mixed Morrey spaces is also discussed. [ABSTRACT FROM AUTHOR]

First, we introduce the concept of triple sequence space c 3 (▵) and we define a Hausdorff measure of noncompactness (MNC) on this space. Furthermore, by using this MNC we study the existence of solutions of infinite systems of Caputo–Hadamard fractional differential equations with three point integral boundary conditions in the triple sequence space c 3 (▵) . Finally, we give an example to show the effectiveness of our main result. [ABSTRACT FROM AUTHOR]

We introduce the concept of weighted sequence space b v p ω and we construct a Hausdorff measure of noncompactness (MNC) in this space. Then, by applying this MNC we study the existence of solutions of infinite systems of third-order three-point nonhomogeneous boundary value problem in b v p ω . Finally, we present two examples to show the usefulness of our results. [ABSTRACT FROM AUTHOR]

This paper presents two new geometric constants μ (X , a) and μ ′ (X , a) , which extend the rectangular constants μ (X) and μ ′ (X) . We firstly provide their bounds. Then the relationships between these geometric constants and the geometric properties of Banach spaces are discussed, including uniform nonsquareness, uniform convexity and uniform smoothness. Meanwhile, we provide several estimates of μ (l p , a) and obtain some new upper bound estimates on μ ′ (l p , a) . [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 paper we analyze, from a mathematical point of view, a simple processes with input in l p ⁢ (퐫) and output in l p ⁢ (퐬) . We characterize the functions u : ℕ → ℂ and φ : ℕ → ℕ which define weighted composition operators W φ , u having closed range when acting between two different weighted l p spaces. We also analyze when this operator is upper or lower semi-Fredholm. [ABSTRACT FROM AUTHOR]

We prove thet a Banach space X has the λ -bounded compact approximation property (respectively, weak λ -bounded approximation property) if and only if X has the λ -Lipschitz bounded compact approximation property (respectively, weak λ -Lipschitz bounded approximation property). Also, it is shown that the dual space of X has the approximation property if and only if for every separable reflexive Banach space Y, the space of finite rank Lipschitz maps from X to Y is dense in the Lipschitz norm in the space of compact operators from X to Y. [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]

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]

We introduce a discrete version of weighted Morrey spaces, and discuss the inclusion relations of these spaces. In addition, we obtain the boundedness of discrete weighted Hardy-Littlewood maximal operators on discrete weighted Lebesgue spaces by establishing a discrete Calderón-Zygmund decomposition for weighted l 1 -sequences. Furthermore, the necessary and sufficient conditions for the boundedness of the discrete Hardy-Littlewood maximal operators on discrete weighted Morrey spaces are discussed. Particularly, the necessary and sufficient conditions are also discussed for the discrete power weights. [ABSTRACT FROM AUTHOR]

We calculate the essential norm of bounded diagonal infinite matrices acting on Köthe sequence spaces. As a consequence of our result, we obtain a recent criteria for the compactness of multiplication operator acting on Köthe sequence spaces.

Define an infinite matrix D α = ( d n , v α ) \mathfrak{D}^{\alpha}=(d^{\alpha}_{n,v}) by d n , v α = { v α σ ( α ) ⁢ ( n ) , v ∣ n , 0 , v ∤ n , d^{\alpha}_{n,v}=\begin{cases}\dfrac{v^{\alpha}}{\sigma^{(\alpha)}(n)},&v\mid n,\\ 0,&v\nmid n,\end{cases} where σ ( α ) ⁢ ( n ) \sigma^{(\alpha)}(n) is defined to be the sum of the 훼-th power of the positive divisors of n ∈ N n\in\mathbb{N} , and construct the matrix domains ℓ p ⁢ ( D α ) \ell_{p}(\mathfrak{D}^{\alpha}) ( 0 < p < ∞ 0

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]

We prove when a Banach ideal of linear operators defined, or characterized, by the transformation of vector-valued sequences is maximal. Known results are recovered as particular cases and new information is obtained. To accomplish this task, we study a tensor quasi-norm determined by the underlying sequence classes. The duality theory for these tensor quasi-norms is also developed. [ABSTRACT FROM AUTHOR]

In this paper we define and study a vector-valued sequence space, called the space of anisotropic (s, q, r)-summable sequences, that generalizes the classical space of (s; q)-mixed sequences (or mixed (s; q)-summable sequences). Furthermore, we define two classes of linear operators involving this new space and one of them generalizes the class of (s; q) -mixed linear operators due A. Pietsch. Some characterizations, inclusion results and a Pietsch domination-type theorem are presented for these classes. It is worth mentioning that some of these results are new even in the particular cases of mixed summable sequences and mixed summing operators. [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]

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 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 this article, all sequences u{\boldsymbol{u}}, v{\boldsymbol{v}}, and w{\boldsymbol{w}} that define continuous and compact tridiagonal operators Tu,v,w{T}_{u,v,w} acting on the weighted sequence space lβ2{l}_{\beta }^{2} were characterized. Additionally, the essential norm of this operator, and as an important consequence of our results, the essential norm of multiplication operator Mu{M}_{u} acting on lβ2{l}_{\beta }^{2} spaces was calculated.

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 the present paper, we introduce and study a class N of norms α on the sequence space C 00 , called permutation symmetric gauge norms, which properly contains the classical class of { ‖ ⋅ ‖ p : 1 ≤ p ≤ ∞ }. For each α ∈ N , we define the generalized sequence space ℓ α , which is proved to be a Banach space, and then we obtain a characterization of ℓ α in terms of the projection onto C 00 . For the duality, the expected results in the classical sequence spaces ℓ p ( 1 ≤ p ≤ ∞) are still valid in these new settings, including the characterization of dual space and the existence of Hölder's inequality. It is worthy pointing out that the generalized sequence space ℓ α is not a reflexive space, which is different from the usual sequence spaces. [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 give the necessary and sufficient conditions for a bounded bilinear operator on X × c 0 Y to be nuclear. As application, we find the necessary and sufficient conditions for bilinear multiplication operators M V : E X × c 0 Y → F Z defined by M V x , y = V n x n , y n n ∈ N to be nuclear. For various other similar type of operators, we give the necessary and sufficient conditions to be nuclear. [ABSTRACT FROM AUTHOR]

We study Orlicz sequence algebras and their properties. In particular, we fully characterize biflat and biprojective Orlicz sequence algebras as well as weakly amenable and approximately (semi-)amenable Orlicz sequence algebras. As a consequence, we show the existence of a wide class of sequence algebras that behave differently—in terms of the amenability properties—from any of the algebras ℓ p , 1 ⩽ p ⩽ ∞ . [ABSTRACT FROM AUTHOR]