scholarly journals Approximative Compactness and Radon-Nikodym Property inw∗Nearly Dentable Banach Spaces and Applications

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Shaoqiang Shang ◽  
Yunan Cui
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Generalized Inverse ◽  
Orlicz Function ◽  
Function Spaces ◽  
Inverse Theory ◽  
Nikodym Property ◽  
Projector Operator ◽  
Approximative Compactness

Authors definew∗nearly dentable Banach space. Authors study Radon-Nikodym property, approximative compactness and continuity metric projector operator inw∗nearly dentable space. Moreover, authors obtain some examples ofw∗nearly dentable space in Orlicz function spaces. Finally, by the method of geometry of Banach spaces, authors give important applications ofw∗nearly dentability in generalized inverse theory of Banach space.

scholarly journals Strongly exposed points in bases for the positive cone of ordered Banach spaces and characterizations of l1(Г)

1986 ◽  
Vol 29 (2) ◽  
pp. 271-282 ◽  
Author(s):  
Ioannis A. Polyrakis
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Sets ◽  
Extreme Points ◽  
Positive Cone ◽  
Infinite Dimensional ◽  
Strongly Exposed Points ◽  
Nikodym Property ◽  
Choquet Theory ◽  
Ordered Banach Spaces

The study of extreme, strongly exposed points of closed, convex and bounded sets in Banach spaces has been developed especially by the interconnection of the Radon–Nikodým property with the geometry of closed, convex and bounded subsets of Banach spaces [5],[2] . In the theory of ordered Banach spaces as well as in the Choquet theory, [4], we are interested in the study of a special type of convex sets, not necessarily bounded, namely the bases for the positive cone. In [7] the geometry (extreme points, dentability) of closed and convex subsets K of a Banach space X with the Radon-Nikodým property is studied and special emphasis has been given to the case where K is a base for acone P of X. In [6, Theorem 1], it is proved that an infinite-dimensional, separable, locally solid lattice Banach space is order-isomorphic to l1 if, and only if, X has the Krein–Milman property and its positive cone has a bounded base.

Weakly compact, unconditionally converging, and Dunford-Pettis operators on spaces of vector-valued continuous functions

1984 ◽  
Vol 95 (1) ◽  
pp. 101-108 ◽  
Author(s):  
Paulette Saab
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Representing Measure ◽  
Weakly Compact ◽  
Bounded Linear ◽  
Borel Set ◽  
Nikodym Property ◽  
Weakly Compact Operators

Given a compact Hausdorff space X, E and F two Banach spaces, let T: C(X, E) → F denote a bounded linear operator (here C(X, E) stands for the Banach space of all continuous E-valued functions defined on X under supremum norm). It is well known [4] that any such operator T has a finitely additive representing measure G that is defined on the σ–field of Borel subsets of X and that G takes its values in the space of all bounded linear operators from E into the second dual of F. The representing measure G enjoys a host of many important properties; we refer the reader to [4] and [5] for more on these properties. The question of whether properties of the operator T can be characterized in terms of properties of the representing measure has been considered by many authors, see for instance [1], [2], [3] and [6]. Most characterizations presented (see [3] concerning weakly compact operators or [3] and [6] concerning unconditionally converging operators) were given under additional assumptions on the Banach space E. The aim of this paper is to show that one cannot drop the assumptions on E, indeed as we shall soon show many of the operator characterizations characterize the Banach space E itself. More specifically, it is known [3] that if E* and E** have the Radon-Nikodym property then a bounded linear operator T: C(X, E) → F is weakly compact if and only if the measure G is continuous at Ø (also called strongly bounded), i.e. limn ||G|| (Bn) = 0 for every decreasing sequence Bn ↘ Ø of Borel subsets of X (here ||G|| (B) denotes the semivariation of G at B), and if for every Borel set B the operator G(B) is a weakly compact operator from E to F. In this paper we shall show that if one wants to characterize weakly compact operators as those operators with the above mentioned properties then E* and E** must both have the Radon-Nikodym property. This will constitute the first part of this paper and answers in the negative a question of [2]. In the second part we consider unconditionally converging operators on C(X, E). It is known [6] that if T: C(X, E) → F is an unconditionally converging operator, then its representing measure G is continuous at 0 and, for every Borel set B, G(B) is an unconditionally converging operator from E to F. The converse of the above result was shown to be untrue by a nice example (see [2]). Here again we show that if one wants to characterize unconditionally converging operators as above, then the Banach space E cannot contain a copy of c0. Finally, in the last section we characterize Banach spaces E with the Schur property in terms of properties of Dunford-Pettis operators on C(X, E) spaces.

scholarly journals On the weakly strongly exposed property and some smoothness properties of Orlicz spaces

1996 ◽  
Vol 54 (3) ◽  
pp. 431-440
Author(s):  
Yunan Cui ◽  
Henry K. Hudzik ◽  
Hongwei Zhu
Keyword(s):  
Banach Space ◽  
Orlicz Function ◽  
Orlicz Spaces ◽  
Function Spaces ◽  
Luxemburg Norm ◽  
Orlicz Norm ◽  
Dual Property ◽  
Order Continuous

The notion of a weakly strongly exposed Banach space is introduced and it is shown that this property is the dual property of very smoothness. Criteria for this property in Orlicz function spaces equipped with the Orlicz norm are presented. Criteria for strong smoothness and very smoothness of their subspaces of order continuous elements in the case of the Luxemburg norm are also given.

Kadec-Klee property in Musielak-Orlicz function spaces equipped with the Orlicz norm

2021 ◽  
Author(s):  
Yunan Cui ◽  
Li Zhao
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Orlicz Function ◽  
Function Spaces ◽  
Fixed Point Property ◽  
Point Property ◽  
Orlicz Norm

AbstractIt is well-known that the Kadec-Klee property is an important property in the geometry of Banach spaces. It is closely connected with the approximation compactness and fixed point property of non-expansive mappings. In this paper, a criterion for Musielak-Orlicz function spaces equipped with the Orlicz norm to have the Kadec-Klee property are given. As a corollary, we obtain that a class of non-reflexive Musielak-Orlicz function spaces have the Fixed Point property.

scholarly journals P-Convexity Property in Musielak-Orlicz Function Space of Bohner Type

2017 ◽  
Vol 3 (1) ◽  
pp. 221
Author(s):  
Yulia Romadiastri
Keyword(s):  
Banach Space ◽  
Function Space ◽  
Orlicz Function ◽  
Function Spaces ◽  
Convexity Property

<div style="text-align: justify;">In this paper, we described about Musielak-Orlicz function spaces of Bochner type. It has been obtained that Musielak-Orlicz function space <a href="https://www.codecogs.com/eqnedit.php?latex=L_\phi(\mu,X)" target="_blank"><img title="L_\phi(\mu,X)" src="https://latex.codecogs.com/gif.latex?L_\phi(\mu,X)" alt="" /></a> of Bochner type becomes a Banach space. It is described also about P-convexity of Musielak-Orlicz function space <a href="https://www.codecogs.com/eqnedit.php?latex=\small&amp;space;L_\phi(\mu,X)" target="_blank"><img title="\small L_\phi(\mu,X)" src="https://latex.codecogs.com/gif.latex?\small&amp;space;L_\phi(\mu,X)" alt="" /></a> of Bochner type. It is proved that the Musielak-Orlicz function space <a href="https://www.codecogs.com/eqnedit.php?latex=\small&amp;space;L_\phi(\mu,X)" target="_blank"><img title="\small L_\phi(\mu,X)" src="https://latex.codecogs.com/gif.latex?\small&amp;space;L_\phi(\mu,X)" alt="" /></a> of Bochner type is P-convex if and only if both spaces <a href="https://www.codecogs.com/eqnedit.php?latex=\small&amp;space;L_\phi" target="_blank"><img title="\small L_\phi" src="https://latex.codecogs.com/gif.latex?\small&amp;space;L_\phi" alt="" /></a> and X are P-convex.©2017 JNSMR UIN Walisongo. All rights reserved.</div>

Uniqueness of norm-preserving extensions of functionals on the space of compact operators

2019 ◽  
Vol 125 (1) ◽  
pp. 67-83
Author(s):  
Julia Martsinkevitš ◽  
Märt Põldvere
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Functional ◽  
Dual Space ◽  
Closed Subspace ◽  
Compact Operators ◽  
Bounded Linear ◽  
Nikodym Property ◽  
Space Of Compact Operators ◽  
Bounded Linear Functional

Godefroy, Kalton, and Saphar called a closed subspace $Y$ of a Banach space $Z$ an ideal if its annihilator $Y^\bot $ is the kernel of a norm-one projection $P$ on the dual space $Z^\ast $. If $Y$ is an ideal in $Z$ with respect to a projection on $Z^\ast $ whose range is norming for $Z$, then $Y$ is said to be a strict ideal. We study uniqueness of norm-preserving extensions of functionals on the space $\mathcal{K}(X,Y) $ of compact operators between Banach spaces $X$ and $Y$ to the larger space $\mathcal{K}(X,Z) $ under the assumption that $Y$ is a strict ideal in $Z$. Our main results are: (1) if $y^\ast $ is an extreme point of $B_{Y^{\ast} }$ having a unique norm-preserving extension to $Z$, and $x^{\ast\ast} \in B_{X^{\ast\ast} }$, then the only norm-preserving extension of the functional $x^{\ast\ast} \otimes y^\ast \in \mathcal {K}(X,Y)^\ast $ to $\mathcal {K}(X,Z)$ is $x^{\ast\ast} \otimes z^\ast $ where $z^\ast \in Z^\ast $ is the only norm-preserving extension of $y^\ast $ to $Z$; (2) if $\mathcal{K}(X,Y) $ is an ideal in $\mathcal{K}(X,Z) $ and $Y$ has Phelps' property $U$ in its bidual $Y^{\ast\ast} $ (i.e., every bounded linear functional on $Y$ admits a unique norm-preserving extension to $Y^{\ast\ast} $), then $\mathcal{K}(X,Y) $ has property $U$ in $\mathcal{K}(X,Z) $ whenever $X^{\ast\ast} $ has the Radon-Nikodým property.

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