scholarly journals Properties of modulus of monotonicity and Opial property in direct sums

2017 ◽  
Vol 71 (2) ◽  
pp. 69
Author(s):  
Joanna Markowicz ◽  
Stanisław Prus
Keyword(s):  
Banach Spaces ◽  
Banach Lattice ◽  
Direct Sums ◽  
Opial Property ◽  
Convex Modulus ◽  
Made In

We give an example of a Banach lattice with a non-convex modulus of monotonicity, which disproves a claim made in the literature. Results on preservation of the non-strict Opial property and Opial property under passing to general direct sums of Banach spaces are established.

scholarly journals Banach spaces with property (w)

1993 ◽  
Vol 35 (2) ◽  
pp. 207-217 ◽  
Author(s):  
Denny H. Leung
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Banach Lattice ◽  
Banach Lattices ◽  
Weakly Compact ◽  
Type Spaces

A Banach space E is said to have Property (w) if every operator from E into E' is weakly compact. This property was introduced by E. and P. Saab in [9]. They observe that for Banach lattices, Property (w) is equivalent to Property (V*), which in turn is equivalent to the Banach lattice having a weakly sequentially complete dual. Thus the following question was raised in [9].Does every Banach space with Property (w) have a weakly sequentially complete dual, or even Property (V*)?In this paper, we give two examples, both of which answer the question in the negative. Both examples are James type spaces considered in [1]. They both possess properties stronger than Property (w). The first example has the property that every operator from the space into the dual is compact. In the second example, both the space and its dual have Property (w). In the last section we establish some partial results concerning the problem (also raised in [9]) of whether (w) passes from a Banach space E to C(K, E).

scholarly journals On orthogonally decomposable ordered Banach spaces

1984 ◽  
Vol 30 (3) ◽  
pp. 357-380 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Banach Spaces ◽  
Banach Lattice ◽  
Order Structure ◽  
Duality Map ◽  
Ordered Banach Spaces

In a Banach lattice or the hermitian part of a C*-algebra, every element a admits a decomposition a = a+ − a− such that and N(−a) = ‖a−‖, where N is the canonical half-norm of the positive cones. In general ordered Banach spaces, this property is related to the order structure of the duality map and the metric projectability of the positive cones, and it turns out to be equivalent to an “orthogonal” decomposability.

Countable Direct Sums of Banach Spaces

1989 ◽  
Vol 141 (1) ◽  
pp. 73-79
Author(s):  
M. Valdivia
Keyword(s):  
Banach Spaces ◽  
Direct Sums

scholarly journals Strongly exposed points and a characterization of l1 (Γ) by the Schur property

1987 ◽  
Vol 30 (3) ◽  
pp. 397-400 ◽  
Author(s):  
Ioannis A. Polyrakis
Keyword(s):  
Banach Spaces ◽  
Banach Lattice ◽  
Positive Cone ◽  
Schur Property ◽  
Positive Elements ◽  
Strongly Exposed Points ◽  
Ordered Banach Spaces ◽  
Convex Subsets ◽  
Quasi Interior

In this paper we study the existence of strongly exposed points in unbounded closed and convex subsets of the positive cone of ordered Banach spaces and we prove the following characterization for the space l1(Γ): A Banach lattice X is order-isomorphic to l1(Γ) iff X has the Schur property and X* has quasi-interior positive elements.

scholarly journals Applications of Bregman-Opial Property to Bregman Nonspreading Mappings in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Eskandar Naraghirad ◽  
Ngai-Ching Wong ◽  
Jen-Chih Yao
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Fixed Point Theorems ◽  
Weak And Strong Convergence ◽  
Bregman Distances ◽  
Opial Property ◽  
Nonspreading Mappings

The Opial property of Hilbert spaces and some other special Banach spaces is a powerful tool in establishing fixed point theorems for nonexpansive and, more generally, nonspreading mappings. Unfortunately, not every Banach space shares the Opial property. However, every Banach space has a similar Bregman-Opial property for Bregman distances. In this paper, using Bregman distances, we introduce the classes of Bregman nonspreading mappings and investigate the Mann and Ishikawa iterations for these mappings. We establish weak and strong convergence theorems for Bregman nonspreading mappings.

scholarly journals Deviation from weak Banach–Saks property for countable direct sums

2014 ◽  
Vol 68 (2) ◽  
Author(s):  
Andrzej Kryczka
Keyword(s):  
Banach Spaces ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Direct Sums ◽  
Bounded Linear

AbstractWe introduce a seminorm for bounded linear operators between Banach spaces that shows the deviation from the weak Banach-Saks property. We prove that if (X

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