scholarly journals The weak cotype 2 and the Orlicz property of the Lorentz sequence space d(a, 1)

1992 ◽  
Vol 34 (3) ◽  
pp. 271-276
Author(s):  
J. Zhu
Keyword(s):  
Banach Space ◽  
Function Space ◽  
Sequence Space ◽  
Affirmative Answer ◽  
Sequence Spaces ◽  
Similar Question ◽  
Lorentz Sequence Space ◽  
Symmetric Basis ◽  
Orlicz Property

The question “Does a Banach space with a symmetric basis and weak cotype 2 (or Orlicz) property have cotype 2?” is being seriously considered but is still open though the similar question for the r.i. function space on [0, 1] has an affirmative answer. (If X is a r.i. function space on [0, 1] and has weak cotype 2 (or Orlicz) property then it must have cotype 2.) In this note we prove that for Lorentz sequence spaces d(a, 1) they both hold.

scholarly journals On Domain of Nörlund Sequences

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 268 ◽  
Author(s):  
Kuddusi Kayaduman ◽  
Fevzi Yaşar
Keyword(s):  
Banach Space ◽  
Bounded Variation ◽  
Sequence Space ◽  
Convergent Series ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Nörlund Matrix ◽  
Inclusion Relations ◽  
The Matrix ◽  
New Space

In 1978, the domain of the Nörlund matrix on the classical sequence spaces lp and l∞ was introduced by Wang, where 1 ≤ p < ∞. Tuğ and Başar studied the matrix domain of Nörlund mean on the sequence spaces f0 and f in 2016. Additionally, Tuğ defined and investigated a new sequence space as the domain of the Nörlund matrix on the space of bounded variation sequences in 2017. In this article, we defined new space and and examined the domain of the Nörlund mean on the bs and cs, which are bounded and convergent series, respectively. We also examined their inclusion relations. We defined the norms over them and investigated whether these new spaces provide conditions of Banach space. Finally, we determined their α­, β­, γ­duals, and characterized their matrix transformations on this space and into this space.

On Some Convexity Properties of Generalized Cesáro Sequence Spaces

2003 ◽  
Vol 10 (1) ◽  
pp. 193-200 ◽  
Author(s):  
Suthep Suantai
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Sequence Spaces ◽  
Luxemburg Norm ◽  
Convexity Properties ◽  
Generalized Cesáro Sequence Space

Abstract We define a generalized Cesáro sequence space and consider it equipped with the Luxemburg norm under which it is a Banach space, and we show that it is locally uniformly rotund.

scholarly journals Approximation numbers of matrix transformations and inclusion maps

2011 ◽  
Vol 42 (2) ◽  
pp. 193-203
Author(s):  
M. Gupta ◽  
L. R. Acharya
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Diagonal Operator ◽  
Approximation Numbers ◽  
Vector Valued

In this paper we establish relationships of the approximation numbers of matrix transformations acting between the vector-valued sequence spaces spaces of the type $\lambda(X)$ defined corresponding to a scalar-valued sequence space $\lambda$ and a Banach space $(X,\|.\|)$ as $$\lambda(X)=\{\overline x=\{x_i\}: x_i\in X, \forall~i\in \mathbb{N},~\{\|x_i\|_X\}\in \lambda\};$$ with those of their component operators. This study leads to a characterization of a diagonal operator to be approximable. Further, we compute the approximation numbers of inclusion maps acting between $\ell^p(X)$ spaces for $1\leq p\leq \infty$.

A Class of Reflexive Symmetric Bk-Spaces

1969 ◽  
Vol 21 ◽  
pp. 602-608 ◽  
Author(s):  
D. J. H. Garling
Keyword(s):  
Banach Space ◽  
Linear Space ◽  
Sequence Space ◽  
Linear Subspace ◽  
Sequence Spaces ◽  
Complex Numbers ◽  
Bk Space ◽  
Bk Spaces

We denote by ω the linear space of all sequences of real or complex numbers. A linear subspace of ω is called a sequence space. A sequence space E is a BK-space (9) if it is equipped with a norm under which: first, E is a Banach space and second, each of the coordinate maps x → xi is continuous. Let ∑ be the group of all permutations of Z+ = {1, 2, 3, …}. If x ∈ ω and σ ∈ ∑, the sequence xσ is defined by (xσ)i = xσ(i)). A sequence space E is symmetric if xσ ∈ E whenever x ∈ E and σ ∈ ∑. Accounts of symmetric sequence spaces occur in (3; 7; 8).

On the domain of Riesz mean in the space Ls

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 925-940 ◽  
Author(s):  
Medine Yeşilkayagil ◽  
Feyzi Başar
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Double Sequence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Double Sequences ◽  
Riesz Mean ◽  
Double Sequence Space ◽  
Necessary And Sufficient ◽  
Barrelled Space

Let 0 < s < ?. In this study, we introduce the double sequence space Rqt(Ls) as the domain of four dimensional Riesz mean Rqt in the space Ls of absolutely s-summable double sequences. Furthermore, we show that Rqt(Ls) is a Banach space and a barrelled space for 1 ? s < 1 and is not a barrelled space for 0 < s < 1. We determine the ?- and ?(?)-duals of the space Ls for 0 < s ? 1 and ?(bp)-dual of the space Rqt(Ls) for 1 < s < 1, where ? ? {p, bp, r}. Finally, we characterize the classes (Ls:Mu), (Ls:Cbp), (Rqt(Ls) : Mu) and (Rqt(Ls):Cbp) of four dimensional matrices in the cases both 0 < s < 1 and 1 ? s < 1 together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.

A SPHERICAL VERSION OF THE KOWALSKI–SŁODKOWSKI THEOREM AND ITS APPLICATIONS

2020 ◽  
pp. 1-26
Author(s):  
SHIHO OI
Keyword(s):  
Banach Algebra ◽  
Function Space ◽  
Affirmative Answer ◽  
List Type ◽  
Private Communication ◽  
Type Number ◽  
Uniform Algebras ◽  
Surjective Isometry ◽  
Complex Linear ◽  
Local Map

Abstract Li et al. [‘Weak 2-local isometries on uniform algebras and Lipschitz algebras’, Publ. Mat.63 (2019), 241–264] generalized the Kowalski–Słodkowski theorem by establishing the following spherical variant: let A be a unital complex Banach algebra and let $\Delta : A \to \mathbb {C}$ be a mapping satisfying the following properties: (a) $\Delta $ is 1-homogeneous (that is, $\Delta (\lambda x)=\lambda \Delta (x)$ for all $x \in A$ , $\lambda \in \mathbb C$ ); (b) $\Delta (x)-\Delta (y) \in \mathbb {T}\sigma (x-y), \quad x,y \in A$ . Then $\Delta $ is linear and there exists $\lambda _{0} \in \mathbb {T}$ such that $\lambda _{0}\Delta $ is multiplicative. In this note we prove that if (a) is relaxed to $\Delta (0)=0$ , then $\Delta $ is complex-linear or conjugate-linear and $\overline {\Delta (\mathbf {1})}\Delta $ is multiplicative. We extend the Kowalski–Słodkowski theorem as a conclusion. As a corollary, we prove that every 2-local map in the set of all surjective isometries (without assuming linearity) on a certain function space is in fact a surjective isometry. This gives an affirmative answer to a problem on 2-local isometries posed by Molnár [‘On 2-local *-automorphisms and 2-local isometries of B(H)', J. Math. Anal. Appl.479(1) (2019), 569–580] and also in a private communication between Molnár and O. Hatori, 2018.

Entropy numbers of embedding maps between Besov spaces with an application to eigenvalue problems

1981 ◽  
Vol 90 (1-2) ◽  
pp. 63-70 ◽  
Author(s):  
Bernd Carl
Keyword(s):  
Banach Space ◽  
Asymptotic Behaviour ◽  
Besov Spaces ◽  
Eigenvalue Problems ◽  
Function Spaces ◽  
Sequence Spaces ◽  
Entropy Numbers ◽  
Diagonal Operators

SynopsisIn this paper we determine the asymptotic behaviour of entropy numbers of embedding maps between Besov sequence spaces and Besov function spaces. The results extend those of M. Š. Birman, M. Z. Solomjak and H. Triebel originally formulated in the language of ε-entropy. It turns out that the characterization of embedding maps between Besov spaces by entropy numbers can be reduced to the characterization of certain diagonal operators by their entropy numbers.Finally, the entropy numbers are applied to the study of eigenvalues of operators acting on a Banach space which admit a factorization through embedding maps between Besov spaces.The statements of this paper are obtained by results recently proved elsewhere by the author.

scholarly journals A study on Copson operator and its associated sequence space II

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Hadi Roopaei
Keyword(s):  
Lower Bound ◽  
Sequence Space ◽  
Sequence Spaces ◽  
Matrix Domains

AbstractIn this paper, we investigate some properties of the domains $c(C^{n})$ c ( C n ) , $c_{0}(C^{n})$ c 0 ( C n ) , and $\ell _{p}(C^{n})$ ℓ p ( C n ) $(0< p<1)$ ( 0 < p < 1 ) of the Copson matrix of order n, where c, $c_{0}$ c 0 , and $\ell _{p}$ ℓ p are the spaces of all convergent, convergent to zero, and p-summable real sequences, respectively. Moreover, we compute the Köthe duals of these spaces and the lower bound of well-known operators on these sequence spaces. The domain $\ell _{p}(C^{n})$ ℓ p ( C n ) of Copson matrix $C^{n}$ C n of order n in the sequence space $\ell _{p}$ ℓ p , the norm of operators on this space, and the norm of Copson operator on several matrix domains have been investigated recently in (Roopaei in J. Inequal. Appl. 2020:120, 2020), and the present study is a complement of our previous research.

scholarly journals Dual pairs of sequence spaces

2001 ◽  
Vol 28 (1) ◽  
pp. 9-23 ◽  
Author(s):  
Johann Boos ◽  
Toivo Leiger
Keyword(s):  
Sequence Space ◽  
Dual Pair ◽  
General Concept ◽  
Sequence Spaces ◽  
Dual Pairs ◽  
The Third ◽  
Convex Sequence ◽  
Starting Point ◽  
Alpha Beta ◽  
Locally Convex

The paper aims to develop for sequence spacesEa general concept for reconciling certain results, for example inclusion theorems, concerning generalizations of the Köthe-Toeplitz dualsE×(×∈{α,β})combined with dualities(E,G),G⊂E×, and theSAK-property (weak sectional convergence). TakingEβ:={(yk)∈ω:=𝕜ℕ|(ykxk)∈cs}=:Ecs, wherecsdenotes the set of all summable sequences, as a starting point, then we get a general substitute ofEcsby replacingcsby any locally convex sequence spaceSwith sums∈S′(in particular, a sum space) as defined by Ruckle (1970). This idea provides a dual pair(E,ES)of sequence spaces and gives rise for a generalization of the solid topology and for the investigation of the continuity of quasi-matrix maps relative to topologies of the duality(E,Eβ). That research is the basis for general versions of three types of inclusion theorems: two of them are originally due to Bennett and Kalton (1973) and generalized by the authors (see Boos and Leiger (1993 and 1997)), and the third was done by Große-Erdmann (1992). Finally, the generalizations, carried out in this paper, are justified by four applications with results around different kinds of Köthe-Toeplitz duals and related section properties.

Some criteria for nuclearity

1986 ◽  
Vol 100 (1) ◽  
pp. 151-159 ◽  
Author(s):  
M. A. Sofi
Keyword(s):  
Sequence Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Sequence Spaces ◽  
Bilinear Forms ◽  
Simple Formulation ◽  
Normal Topology ◽  
Nuclear Spaces ◽  
Locally Convex

For a given locally convex space, it is always of interest to find conditions for its nuclearity. Well known results of this kind – by now already familiar – involve the use of tensor products, diametral dimension, bilinear forms, generalized sequence spaces and a host of other devices for the characterization of nuclear spaces (see [9]). However, it turns out, these nuclearity criteria are amenable to a particularly simple formulation in the setting of certain sequence spaces; an elegant example is provided by the so-called Grothendieck–Pietsch (GP, for short) criterion for nuclearity of a sequence space (in its normal topology) in terms of the summability of certain numerical sequences.

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