Classical Sequence Spaces Research Articles

Solvability of infinite system of third-order differential equations in the sequence space c using Meir–Keeler condensing operator

In this paper, we present an existence result for ω-periodic solutions for an infinite system of third-order differential equations in a classical Banach sequence space c. This result is obtained using techniques related to measures of noncompactness together with the concept of Meir–Keeler condensing operators, considering the respective Green’s function corresponding to the system. To illustrate the significance of our results, we provide some concrete examples.

The uniqueness of unconditional basis of the 2-convexified Tsirelson space, revisited

One of the hallmarks in the study of the classification of Banach spaces with a unique (normalized) unconditional basis was the unexpected result by Bourgain, Casazza, Lindenstrauss, and Tzafriri from their 1985 Memoir that the 2-convexified Tsirelson space T(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}^{(2)}$$\\end{document} had that property (up to equivalence and permutation). Indeed, on one hand, finding a “pathological” space (i.e., not built out as a direct sum of the only three classical sequence spaces with a unique unconditional basis) shattered the hopeful optimism of attaining a satisfactory description of all Banach spaces which enjoy that important structural feature. On the other hand it encouraged furthering a research topic that had received relatively little attention until then. After forty years, the advances on the subject have shed light onto the underlying patterns shared by those spaces with a unique unconditional bases belonging to the same class, which has led to reproving the original theorems with fewer technicalities. Our motivation in this note is to revisit the aforementioned result on the uniqueness of unconditional basis of T(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}^{(2)}$$\\end{document} from the current state-of-art of the subject and to fill in some details that we missed from the original proof.

Some properties of new sequence spaces based on Riordan numbers

In this paper, we define a new class of sequence spaces via Riordan numbers and prove their topological properties, and inclusion relations, obtain Schauder basis, and describe α,β and γ duals of them. We have given conditions under which there is matrix transformation between those new sequence spaces and the well-known classical sequence spaces. In the last part, we are given some results related to some special operator classes, such as approximable operators, nuclear operators, and ideal operators.

The fine spectra of a difference of weighted shift operators acting in some classical sequence spaces

In this paper we investigate the fine spectra, the ergodic properties and the linear dynamics of the operatorC(x)=(−nxn+n(n−1)xn−2)n∈N0,x=(xn)n∈N0∈CN0, and of its formal adjoint C′ acting in the classical sequence spaces s, s′, ω and φ. The operators C and C′ acting in s are conjugate to the one-dimensional Ornstein-Uhlenbeck differential operator Ou=u″−xu′ and to the differential operator O′u=u″+xu′+u acting in S(R), respectively.

Mersenne Matrix Operator and Its Application in $p-$Summable Sequence Space

In this study, it is introduced the regular Mersenne matrix operator which is obtained by using Mersenne numbers and examined sequence spaces described as the domain of this matrix in the space of $p$-summable sequences for $1\leq p \leq \infty$. After that, it investigated some properties and inclusion relations, established the Schauder basis, and stated $\alpha-$, $\beta-$, and $\gamma-$duals of the aforementioned spaces. Additionally, it is characterized by the matrix classes from newly described spaces to classical sequence spaces. Finally, we studied the compactness of matrix operators on related sequence spaces.

Asymptotically Stable Solutions of Infinite Systems of Quadratic Hammerstein Integral Equations

In this paper, we present a result on the existence of asymptotically stable solutions of infinite systems (IS) of quadratic Hammerstein integral equations (IEs). Our study will be conducted in the Banach space of functions, which are continuous and bounded on the half-real axis with values in the classical Banach sequence space consisting of real bounded sequences. The main tool used in our investigations is the technique associated with the measures of noncompactness (MNCs) and a fixed point theorem of Darbo type. The applicability of our result is illustrated by a suitable example at the end of the paper.

Lipschitz p-Approximate Schauder Frames

With the aim of representing subsets of Banach spaces as an infinite series using Lipschitz functions, we study a variant of metric frames which we call Lipschitz p-approximate Schauder frames (Lipschitz p-ASFs). We characterize Lipschitz p-ASFs and their duals completely using the canonical Schauder basis for classical sequence spaces. Similarity of Lipschitz p-ASF is introduced and characterized.

Geometric and isometric properties of two classes of sequence spaces

We study geometric and isometric properties of two classes of sequence spaces. First, we consider the Banach sequence space X n X_n , the completion of c 00 c_{00} relatively to the norm \[ ‖ x ‖ n = sup { ∑ i ∈ F | x i | , with , F of cardinality at most n } . \|x\|_n = \sup \left \{\sum _{i \in F}|x_i|, \text { with }, F \text { of cardinality at most } n\right \}. \] For this space, we show that the isometry group of X n X_n consists of standard isometries. Second, we consider a Tsirelson-type space, T [ θ , A n ] T[\theta , A_n] ( 0 > θ > 1 0>\theta >1 ) associated with the regular family A n A_n , comprised of all subsets of N \mathbb {N} with at most n n elements. For this space, we characterize the extreme points and show that for θ ≤ 1 n \theta \leq \frac {1}{n} , T [ θ , A n ] T[\theta , A_n] supports standard isometries. We also derive the form for the surjective isometries of T [ θ , A 3 ] T[\theta , A_3] .

Spectral properties of generalized Cesàro operators in sequence spaces

The generalized Cesàro operators C_t, for tin [0,1], were first investigated in the 1980s. They act continuously in many classical Banach sequence spaces contained in {{mathbb {C}}}^{{{mathbb {N}}}_0}, such as ell ^p, c_0, c, bv_0, bv and, as recently shown in Curbera et al. (J Math Anal Appl 507:31, 2022) [26], also in the discrete Cesàro spaces ces(p) and their (isomorphic) dual spaces d_p. In most cases C_t (tnot =1) is compact and its spectra and point spectrum, together with the corresponding eigenspaces, are known. We study these properties of C_t, as well as their linear dynamics and mean ergodicity, when they act in certain non-normable sequence spaces contained in {{mathbb {C}}}^{{{mathbb {N}}}_0}. Besides {{mathbb {C}}}^{{{mathbb {N}}}_0} itself, the Fréchet spaces considered are ell (p+), ces(p+) and d(p+), for 1le p<infty , as well as the (LB)-spaces ell (p-), ces(p-) and d(p-), for 1<ple infty .

On some new vector valued sequence spaces $ E(X, \lambda, p) $

&lt;abstract&gt;&lt;p&gt;To define a new sequence space and determine the Köthe-Toeplitz duals of this sequence space, characterizing the matrix transformation classes between the defined sequence spaces and classical sequence spaces has been an important area of work for researchers. Defining and examining a new vector-valued sequence space is also a considerable field of study since it generalizes classical sequence spaces. In this study, new vector-valued sequence spaces $ E(X, \lambda, p) $ are introduced. The Köthe-Toeplitz duals of $ E(X, \lambda, p) $ spaces are identified. Also, necessary and sufficient conditions are determined for $ A = (A_{nk}) $ to belong to the matrix classes $ (E(X, \lambda, p), c(q)) $; where $ A_{ nk}\in B(X, Y) $, $ X\in \{c, \ell_\infty\} $ and $ Y $ is any Banach spaces.&lt;/p&gt;&lt;/abstract&gt;

On Certain Linear Structure of Orlicz Space c0 (M, (X, ||.||), ā, ᾱ) of Vector Valued Difference Sequence

In this article, we introduce and study a new class c0 (M, (X, ||.||), ā, ᾱ) of normed space (X, ||.||) valued difference sequences with the help of Orlicz function M . This is a generalization of the classical sequence space c0. Our primary interest is to explore some linear structures and investigate the conditions relating to the containment relation of the class c0 (M, (X, ||.||), ā, ᾱ) in terms of different ā and ᾱ.

On new difference sequence spaces via Cesàro mean

In the present article, we define certain a set of new classes of sequence spaces by using Ces\`{a}ro mean and difference operator $\Delta^r\;\;r\in\mathbb{N}_0=\{0,1,2,3,\dots\}$. Also, we study the topological structures of the defined classes and determine their $\alpha-,\beta-$ and $\gamma-$ duals. Matrix transformations of given classes with their basic sequence spaces are characterized.

Nonlinear classification of Banach spaces based on geometric structure spaces

We introduce a nonlinear equivalence of Banach spaces based on geometric structure spaces which is weaker than that based on the structure of the Birkhoff-James orthogonality. It turns out that finite-dimensional spaces are classified isomorphically under this equivalence, and that finite-dimensional spaces with the supremum norm are identified by their geometric structure spaces. Moreover, we introduce and study quasi-smooth Banach spaces that have well-behaved geometric structure spaces. As applications, we show that the usual isomorphic equivalence is (almost) equivalent to that based on geometric structure spaces, and that the classical sequence spaces have mutually different geometric structure spaces.

Mutlak Cesa ́ro Seri Uzayı Ve Bazı Matris Dönüşümleri

On recent paper, the paranormed space |C_(λ,μ) |(p) which is defined as the domain of Cesa ́ro matrix in the Maddox’s space l(p) has been introduced and studied in (Gökçe and Sarıgöl, 2019). In this study, some characterizations of matrix operators from the Absolute Cesa ́ro series space |C_(λ,μ) |(p) to the classical sequence spaces c,c_0,l_∞ are given. Also, it is shown that the matrix operators between the absolute Cesa ́ro series space and the spaces c,c_0,l_∞ are bounded operators. Finally, certain results are obtained as a special case.

On some classical properties of normed spaces via generalized vector valued almost convergence

AbstractRecently, the authors interested some new problems on multiplier spaces of Lorentz’ almost convergence andfλ-convergence as a generalization of almost convergence.fλ-convergence is firstly introduced by Karakuş and Başar, and used for some new characterizations of completeness and barrelledness of the spaces through weakly unconditionally Cauchy series in a normed spaceXand its continuous dualX*. In the present paper, we deal withfλ-convergence to have some inclusion relations between the vector valued spaces obtained from this type convergence and corresponding classical sequence spaces, and to give new characterizations of some classical properties like completeness, reflexivity, Schur property and Grothendieck property of normed spaces. By the way, we give a characterization of finite-dimensional normed spaces.

Existence of solutions for an infinite system of Hilfer fractional boundary value problems in tempered sequence spaces

This article aims to study the existence of the solutions to the infinite system of Hilfer fractional differential equations in tempered sequence spaces. We look for a solution in Banach tempered sequence spaces that is more abstract than classical sequence space. Our approach to studying solvability is using the Meir-Keeler condensing operator. Finally, we illustrate our main results through some numerical examples.

The generalized Banach sequence spaces and their dual spaces based on permutation symmetric gauge norms

In the present paper, we introduce and study a class of norms α on the sequence space , called permutation symmetric gauge norms, which properly contains the classical class of . For each , we define the generalized sequence space , which is proved to be a Banach space, and then we obtain a characterization of in terms of the projection onto . For the duality, the expected results in the classical sequence spaces () 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 is not a reflexive space, which is different from the usual sequence spaces.

Matrix Operators on the Absolute Euler space $\left\vert E_{\phi }^{r}\right\vert (\mu)$

In recent paper, the space $ \left\vert E_{\phi}^{r}\right\vert (\mu)$ which is the generalization of the absolute Euler Space on the space $l(\mu)$, has been introduced and studied by Gökçe and Sarıgöl [3]. In this study, we give certain characterizations of matrix transformations from the paranormed space $ \left\vert E_{\phi}^{r}\right\vert (\mu)$ to one of the classical sequence spaces $c_{0},c,l_{\infty }.$ Also, we show that such matrix operators are bounded linear operators.

Nonlinear equivalence of Banach spaces based on Birkhoff-James orthogonality, II

We introduce the notion of geometric structure spaces of Banach spaces equipped with closure space structure. It is shown that the nonlinear equivalence of Banach spaces based on Birkhoff-James orthogonality induces homeomorphisms between geometric structure spaces. As applications, we prove that spaces of continuous functions are isomorphic with respect to the structure of Birkhoff-James orthogonality if and only if they are isometrically isomorphic, and that classical sequence spaces c0 and ℓp are completely classified from the viewpoint of Birkhoff-James orthogonality.

A Study on Strongly Almost Convergent and Strongly Almost Null Binomial Double Sequence Spaces

The 4 dimensional (4d) binomial matrix and its domains on the classical double sequence spaces $\mathcal{L}_{p}$, $\mathcal{M}_{u}$, $\mathcal{C}_{P}$, $\mathcal{C}_{bP}$, $\mathcal{C}_{r}$, $\mathcal{C}_{f}$ and $\mathcal{C}_{f_0}$ have been described and examined by Demiriz and Erdem in the papers \cite{e.d.2}-\cite{e.d.3}. In this article, we describe two double sequence spaces with the aid of the aforementioned matrix and study some properties of these. After giving inclusion relations, we compute $\alpha-$, $\beta(bp)-$ and $\gamma-$duals and give some new matrix classes related them.