Generalized Orlicz-Lorentz sequence spaces and corresponding operator ideals

2014 ◽  
Vol 64 (6) ◽  
Author(s):  
Manjul Gupta ◽  
Antara Bhar
Keyword(s):  
Banach Space ◽  
Structural Properties ◽  
Orlicz Function ◽  
Sequence Spaces ◽  
Operator Ideals ◽  
Orlicz Sequence Spaces ◽  
Vector Valued

AbstractIn this paper we introduce generalized or vector-valued Orlicz-Lorentz sequence spaces l p,q,M(X) on Banach space X with the help of an Orlicz function M and for different positive indices p and q. We study their structural properties and investigate cross and topological duals of these spaces. Moreover these spaces are generalizations of vector-valued Orlicz sequence spaces l M(X) for p = q and also Lorentz sequence spaces for M(x) = x q for q ≥ 1. Lastly we prove that the operator ideals defined with the help of scalar valued sequence spaces l p,q,M and additive s-numbers are quasi-Banach operator ideals for p < q and Banach operator ideals for p ≥ q. The results of this paper are more general than the work of earlier mathematicians, say A. Pietsch, M. Kato, L. R. Acharya, etc.

(KK)-Properties, Normal Structure and Fixed Points of Nonexpansive Mappings in Orlicz Sequence Spaces

1986 ◽  
Vol 38 (3) ◽  
pp. 728-750 ◽  
Author(s):  
D. van Dulst ◽  
V. de Valk
Keyword(s):  
Fixed Point ◽  
Fixed Point Theory ◽  
Nonexpansive Mappings ◽  
Orlicz Function ◽  
Normal Structure ◽  
Point Theory ◽  
Sequence Spaces ◽  
Orlicz Sequence Space ◽  
Orlicz Sequence Spaces ◽  
Vector Valued

In this paper we investigate Orlicz sequence spaces with regard to certain geometric properties that have proved to be important in fixed point theory. In particular, we shall consider various Kadec-Klee type properties, and weak and weak* normal structure. It turns out that many of these properties, though generally distinct, coincide in Orlicz sequence spaces and that all of them are intimately related to the so-called Δ2-condition. Some of our results extend to vector-valued Orlicz sequence spaces. For example, we prove a rather powerful theorem on the preservation of weak normal structure under the formation of substitution spaces. There is also a fixed point theorem: the Orlicz sequence space hM has the fixed point property if the complementary Orlicz function M* satisfies theΔ2-condition. Another one of our results implies that, under this assumption on M*, hM has weak normal structure if and only if M also satisfies the Δ2-condition.

scholarly journals On some new modular sequence spaces

2013 ◽  
Vol 31 (2) ◽  
pp. 55 ◽  
Author(s):  
Cigdem Asma Bektas ◽  
Gülcan Atıci
Keyword(s):  
Sequence Space ◽  
Orlicz Function ◽  
Sequence Spaces ◽  
Topological Properties ◽  
Orlicz Sequence Space ◽  
Orlicz Functions ◽  
Orlicz Sequence Spaces

Lindenstrauss and Tzafriri [7] used the idea of Orlicz function to define the sequence space ℓM which is called an Orlicz sequence space. Another generalization of Orlicz sequence spaces is due to Woo [9]. An important subspace of ℓ (M), which is an AK-space, is the space h (M) . We define the sequence spaces ℓM (m) and ℓ N(m), where M = (Mk) and N = (Nk) are sequences of Orlicz functions such that Mk and Nk be mutually  complementary for each k. We also examine some topological properties of these spaces. We give the α−, β− and γ− duals of the sequence space h (M) and α− duals of the squence spaces ℓ (M, λ) and ℓ (N, λ).

Some vector valued sequence spaces of Musielak-Orlicz functions and their operator ideals

2018 ◽  
Vol 68 (1) ◽  
pp. 115-134 ◽  
Author(s):  
Mohammad Mursaleen ◽  
Kuldip Raj
Keyword(s):  
Sequence Space ◽  
Sequence Spaces ◽  
Operator Ideals ◽  
Orlicz Sequence Space ◽  
Geometric Properties ◽  
Orlicz Functions ◽  
Opial Property ◽  
Uniform Opial Property ◽  
Vector Valued

AbstractIn the present paper we introduce generalized vector-valued Musielak-Orlicz sequence spacel(A,𝓜,u,p,Δr,∥·,… ,·∥)(X) and study some geometric properties like uniformly monotone, uniform Opial property for this space. Further, we discuss the operators ofs-type and operator ideals by using the sequence ofs-number (in the sense of Pietsch) under certain conditions on matrixA.

Rotundity In Köthe Spaces of Vector-Valued Functions

1989 ◽  
Vol 41 (4) ◽  
pp. 659-675 ◽  
Author(s):  
A. Kamińska ◽  
B. Turett
Keyword(s):  
Banach Space ◽  
Analogous Result ◽  
Function Spaces ◽  
Sequence Spaces ◽  
Uniform Rotundity ◽  
Köthe Spaces ◽  
Bochner Space ◽  
Köthe Space ◽  
Vector Valued Functions ◽  
Vector Valued

In this paper, Köthe spaces of vector-valued functions are considered. These spaces, which are generalizations of both the Lebesgue-Bochner and Orlicz-Bochner spaces, have been studied by several people (e.g., see [1], [8]). Perhaps the earliest paper concerning the rotundity of such Köthe space is due to I. Halperin [8]. In his paper, Halperin proved that the function spaces E(X) is uniformly rotund exactly when both the Köthe space E and the Banach space X are uniformly rotund; this generalized the analogous result, due to M. M. Day [4], concerning Lebesgue-Bochner spaces. In [20], M. Smith and B. Turett showed that many properties akin to uniform rotundity lift from X to the Lebesgue-Bochner space LP(X) when 1 < p < ∞. A survey of rotundity notions in Lebesgue-Bochner function and sequence spaces can be found in [19].

scholarly journals Some New Lacunary Strong Convergent Vector-Valued Sequence Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
M. Mursaleen ◽  
A. Alotaibi ◽  
Sunil K. Sharma
Keyword(s):  
Orlicz Function ◽  
Sequence Spaces ◽  
Complex Numbers ◽  
Infinite Matrix ◽  
Topological Properties ◽  
Inclusion Relations ◽  
Vector Valued

We introduce some vector-valued sequence spaces defined by a Musielak-Orlicz function and the concepts of lacunary convergence and strong (A)-convergence, whereA=(aik)is an infinite matrix of complex numbers. We also make an effort to study some topological properties and some inclusion relations between these spaces.

scholarly journals Vector valued multiple sequence spaces defined by Orlicz function

2014 ◽  
Vol 33 (1) ◽  
pp. 67 ◽  
Author(s):  
Binod Chandra Tripathy ◽  
Rupanjali Goswami
Keyword(s):  
Sequence Space ◽  
Orlicz Function ◽  
Sequence Spaces ◽  
Multiple Sequence ◽  
Vector Valued ◽  
Inclusion Results

In this article we define some vector valued multiple sequence space defined by Orlicz function. We study some of their properties like solidness, symmetry, completeness etc and prove some inclusion results.

scholarly journals Small Pre-Quasi Banach Operator Ideals of Type Orlicz-Cesáro Mean Sequence Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Awad A. Bakery ◽  
Mustafa M. Mohammed
Keyword(s):  
Orlicz Function ◽  
Sufficient Conditions ◽  
Operator Ideal ◽  
Linear Operators ◽  
Sequence Spaces ◽  
Operator Ideals ◽  
Bounded Linear ◽  
Cesàro Mean ◽  
Inclusion Relations ◽  
Cesaro Mean

In this paper, we give the sufficient conditions on Orlicz-Cesáro mean sequence spaces cesφ, where φ is an Orlicz function such that the class Scesφ of all bounded linear operators between arbitrary Banach spaces with its sequence of s-numbers which belong to cesφ forms an operator ideal. The completeness and denseness of its ideal components are specified and Scesφ constructs a pre-quasi Banach operator ideal. Some inclusion relations between the pre-quasi operator ideals and the inclusion relations for their duals are explained. Moreover, we have presented the sufficient conditions on cesφ such that the pre-quasi Banach operator ideal generated by approximation number is small. The above results coincide with that known for cesp  (1<p<∞).

Vector valued double sequence spaces defined by Orlicz function

2009 ◽  
Vol 59 (6) ◽  
Author(s):  
Binod Tripathy ◽  
Bipul Sarma
Keyword(s):  
Orlicz Function ◽  
Double Sequence ◽  
Sequence Spaces ◽  
Double Sequence Spaces ◽  
Vector Valued ◽  
Inclusion Results

AbstractIn this article we introduce some vector valued double sequence spaces defined by Orlicz function. We study some of their properties like solidness, symmetricity, completeness etc. and prove some inclusion results.

scholarly journals On (V*) sets and Pelczynski's property (V*)

1990 ◽  
Vol 32 (1) ◽  
pp. 109-120 ◽  
Author(s):  
Fernando Bombal
Keyword(s):  
Banach Space ◽  
Broad Class ◽  
Orlicz Function ◽  
Compact Sets ◽  
Integrable Functions ◽  
Sequential Completeness ◽  
Bounded Sets ◽  
Vector Valued ◽  
Order Continuous ◽  
Space Property

The concept of (V*) set was introduced, as a dual companion of that of (V)-set, by Pelczynski in his important paper [14]. In the same paper, the so called properties (V) and (V*) are defined by the coincidence of the (V) or (V*) sets with the weakly relatively compact sets. Many important Banach space properties are (or can be) defined in the same way; that is, by the coincidence of two classes of bounded sets. In this paper, we are concerned with the study of the class of (V*) sets in a Banach space, and its relationship with other related classes. To this general study is devoted Section I. A (as far as we know) new Banach space property (we called it property weak (V*)) is defined, by imposing the coincidence of (V*) sets and weakly conditionally compact sets. In this way, property (V*) is decomposed into the conjunction of the weak (V*) property and the weak sequential completeness. In Section II, we specialize to the study of (V*) sets in Banach lattices. The main result in the section is that every order continuous Banach lattice has property weak (V*), which extends previous results of E. and P. Saab ([16]). Finally, Section III is devoted to the study of (V*) sets in spaces of Bochner integrable functions. We characterize a broad class of (V*) sets in L1(μ, E), obtaining similar results to those of Andrews [1], Bourgain [6] and Diestel [7] for other classes of subsets. Applications to the study of properties (V*) and weak (V*) are obtained. Extension of these results to vector valued Orlicz function spaces are also given.

scholarly journals Contractive projections in Orlicz sequence spaces

2004 ◽  
Vol 2004 (2) ◽  
pp. 133-145 ◽  
Author(s):  
Beata Randrianantoanina
Keyword(s):  
Orlicz Function ◽  
Sequence Spaces ◽  
Complex Case ◽  
Square Function ◽  
Second Derivative ◽  
Orlicz Norm ◽  
The Real ◽  
Complemented Subspaces ◽  
Orlicz Sequence Spaces ◽  
Contractive Projections

We characterize norm-one complemented subspaces of Orlicz sequence spacesℓMequipped with either Luxemburg or Orlicz norm, provided that the Orlicz functionMis sufficiently smooth and sufficiently different from the square function. We measure smoothness ofMusingAC1andAC2classes introduced by Maleev and Troyanski in 1991, and the condition forMto be different from a square function is essentially a requirement that the second derivativeM″ofMcannot have a finite nonzero limit at zero. This paper treats the real case; the complex case follows from previously known results.

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