Vector-valued Sequence Spaces Research Articles

Generalized Cesàro Summable Vector Valued Sequence Space of Bounded Type

This work aims to introduce and study new classes of generalized Cesàro summable vector-valued sequence spaces of bounded type. Besides exploring the completeness of the classes Ces(X, p) and Ces(X, p) ∞ when topologized with suitable natural p-normed, our primary interest is to study the β-dual of Ces(X, p).

Köthe Dual of Some Vector-valued Sequence Spaces

We study some properties of the spaces λ(E) of weakly λ-summable sequences and λ⟨E⟩ of strongly λ-summable sequences of a locally convex space E. For example, after proving results on bounded sets of these spaces, we express the elements of their K öthe duals in terms of sequences in the continuous dual E′ of E, then we prove that these spaces possess the AK property if and only if the K ̈othe dual coincides with the continuous dual.

Nuclearity of a Class of Vector-valued Sequence Spaces

In this note, we deal with a perfect sequence space $\lambda$ and a bornological convex space $E$ to introduce and study the space $\lambda(E)$ of totally $\lambda$-summable sequences from $E$. We prove that $\lambda(E)$ is complete if and only if $\lambda$ an $E$ are complete, nuclear if and only if $\lambda$ an $E$ are nuclear. and we make use of a result of Rolnald C. Rosier to give a similar characterization of the nuclearity of $\Lambda\{E\}$ of all absolutely $\lambda-$ summable sequences in a locally convex $E$.

An anisotropic summability and mixed sequences

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.

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

<abstract><p>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.</p></abstract>

Generalized vector-valued paranormed sequence spaces defined by a sequence of Orlicz functions

UDC 517.9 In this paper, we introduced a class of generalized vector-valued paranormed sequence space X [ E , A , Δ v m , M , p ] by using a sequence of Orlicz functions M = ( M k ) , a non-negative infinite matrix A = [ a n k ] , generalized difference operator Δ v m and bounded sequence of positive real numbers p k with inf k p k > 0 . Properties related to this space are studied under certain conditions. Some inclusion relations are obtained and a result related to subspace with Orlicz functions satisfying Δ 2 -condition has also been proved.

Uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces

This paper is devoted to providing a unifying approach to the study of the uniqueness of unconditional bases, up to equivalence and permutation, of infinite direct sums of quasi-Banach spaces. Our new approach to this type of problem permits to show that a wide class of vector-valued sequence spaces have a unique unconditional basis up to a permutation. In particular, solving a problem from Albiac and Leránoz (J Math Anal Appl 374(2):394–401, 2011. https://doi.org/10.1016/j.jmaa.2010.09.048) we show that if X is quasi-Banach space with a strongly absolute unconditional basis then the infinite direct sum ell _{1}(X) has a unique unconditional basis up to a permutation, even without knowing whether X has a unique unconditional basis or not. Applications to the uniqueness of unconditional structure of infinite direct sums of non-locally convex Orlicz and Lorentz sequence spaces, among other classical spaces, are also obtained as a by-product of our work.

Nuclear embeddings in general vector-valued sequence spaces with an application to Sobolev embeddings of function spaces on quasi-bounded domains

We study nuclear embeddings for spaces of Besov and Triebel-Lizorkin type defined on quasi-bounded domains Ω⊂Rd. The counterpart for such function spaces defined on bounded domains has been considered for a long time and the complete answer was obtained only recently. Compact embeddings for function spaces defined on quasi-bounded domains have already been studied in detail, also concerning their entropy and s-numbers. We now prove the first and complete nuclearity result in this context. The concept of nuclearity has already been introduced by Grothendieck in 1955. Our second main contribution is the generalisation of the famous Tong result (1969) which characterises nuclear diagonal operators acting between sequence spaces of ℓr type, 1≤r≤∞. We can now extend this to the setting of general vector-valued sequence spaces of type ℓq(βjℓpMj) with 1≤p,q≤∞, Mj∈N0 and a weight sequence with βj>0. In particular, we prove a criterion for the embedding idβ:ℓq1(βjℓp1Mj)↪ℓq2(ℓp2Mj) to be nuclear.

Strongly Summable Vector-Valued Sequence Spaces Defined by 2-modular

Summability is an important concept in sequence spaces. One summability concept is strongly Cesaro summable. In this paper, we study a subset of the set of all vector-valued sequence in 2-modular space. Some facts that we investigated in this paper include linearity, the existence of modular and completeness with respect to these modular.

Vector-valued Lorentz sequence spaces and their dual spaces

Vector-valued Lorentz sequence spaces and their dual spaces

\mathcal{I}- almost Lacunary vector valued sequence spaces in 2- normed spaces

One of the wide-ranging applications and research areas of Summability theory is the concept of statistical convergence. This concept was studied a related concept of convergence by using lacunary sequence by Fridy and Orhan. At the last quarter of the 20th century, lacunary statistical convergence has been discussed and captured significant aspect of creating the basis of several investigations conducted in many branches of mathematics. On the other hand, in 1961 Krasnoselskii and Rutisky presented the definition of Orlicz function. Also, in 1963 G\"{a}hler introduced the notion of 2-normed spaces. The main goal of this article is to introduce $\mathcal{I}-$ almost convergence of lacunary sequences with regard to an Orlicz function in 2-normed spaces and other sequence spaces by considering the concept of ideal that was presented by Kostyrko and others. Additionally, we examine the relationship between these sequence spaces and fundamental inclusion theorems are investigated.

Some Vector FK Sequence Spaces Generated by Modulus Function

In this paper, some vector valued sequence spaces and using modulus function are presented. Furthermore, we examined some topological properties of these sequence spaces equipped with a paranorm.

Sequential Evaluation Convergence of Operator Series and Its Application

For a class of classical vector-valued sequence spaces in Banach spaces, this article introduces a class of important subsets, which contain all the fully bounded sets and many non-bounded sets in the sequence spaces. Using this collection class, we obtain a convergence theorem for the assignment of series of operator series, and apply the conclusion to the characterization of infinite matrix transformations. In this article, the usual linear constraints on operators are completely removed, and the results obtained greatly increase the possibility of applications.

P-Summable Vector-Valued Sequence Spaces in Non-Archimedean 2-Normed Spaces

Let (X, ‖., .‖) be a non-Archimedean 2-normed spaces. In this paper, we construct p-summable vector-valued sequence spaces in non-Archimedean 2-normed spaces ℓ p (X) for 1 ≤ p ≤ ∞. Futhermore, we show that ℓ p (X) is a non-Archimedean 2-semi-normed spaces and we also give some inclusion relations on ℓ p (X).

On Some Modular Köthe Vector Valued Sequence Space using φ-Function

In this paper, we introduce some new modular vector valued sequence spaces defined by φ-function and give some basic result of this space.

Some Vector Valued Sequence Spaces Generated by Musielak- Phy Function Over 2-Normed Spaces

In this work, we introduced some new vector valued sequence spaces over 2-normed spaces using Musielak-Phy function Φ = (𝜑 𝑛 ) . We also studied some properties of these spaces.

Lacunary statistical convergence of order $$\alpha, \beta$$ for generalized vector-valued difference sequence spaces

In this paper, we defined space $$S_\theta ^{\alpha ,\beta } (\Delta _v^m,E,q)$$ of all vector-valued lacunary $$\Delta _v^m$$ -statistical convergent sequences of order $$(\alpha , \beta )$$ and space $$N_\theta ^{\alpha ,\beta } (\Delta _v^m,E,q,p)$$ of all vector-valued strongly $$\Delta _v^m$$ -lacunary summable sequences of order $$(\alpha ,\beta )$$ by taking sequence $$(E_k, q_k)$$ of normed spaces, where p is a positive real number and $$\alpha ,\beta$$ are real numbers with $$0<\alpha \leqslant \beta \leqslant 1$$ . Some inclusion relations between these spaces are obtained. We also studied space $$\omega _\theta ^{\alpha ,\beta } (\Delta _v^m,f,E,q,p)$$ of all vector-valued strongly $$\Delta _v^m$$ -lacunary summable sequences of order $$(\alpha , \beta )$$ with respect to modulus function f by taking a bounded sequence $$(p_k)$$ of strictly positive real numbers with $$\displaystyle \inf _k p_k>0$$ . The inclusion relations between spaces $$\omega _\theta ^{\alpha ,\beta } (\Delta _v^m,f,E,q,p)$$ and $$S_\theta ^{\alpha ,\beta } (\Delta _v^m,E,q)$$ are determined.

Structure and H-properties of Cesàro-φ vector valued sequence space.

In this work we introduce the generalization of Cesàro-Örlicz real sequence space i.e Cesàro-φ vector valued sequence space, where φ is a generalization of Örlicz function, i.e, cesφ(X) and prove that this space is belongs to the class of Köthe sequence spaces and has the Fatou property. Futhermore, we investigate that this space posses H-property.

Relationship with modular and norm of vector valued sequences spaces Cesàro-φ

In this paper, we introduce the generalization of Cesàro-Örlicz sequence space i.e Cesàro-φ vector valued sequence space, cesφ(X), and give the necessary and suffiecient conditions that the space is nontrivial. Furthermore, we will prove the relationship between modular and norm on this space.

Topological properties on generalization of vector valued sequence spaces defined by φ-function

Let M be a φ-function and E be a vector space. For N is the complementary of M, in this paper, we constuct the generalization of generalization of E-valued sequence space X (E, N) and investigate some topological properties of its space.