On Banach lattices with Levi norms

Ercan

2008

Positivity

A Riesz space E is said to have b-property if each subset which is order bounded in E ∼∼ is order bounded in E. The relationship between b-property and completeness, being a retract and the absolute weak topology |σ| (E ∼ , E) is studied. Perfect Riesz spaces are characterized in terms of b-property. It is shown that b-property coincides with the Levi property in Dedekind complete Frechet lattices. Mathematics Subject Classification (2000). Primary 46A40.

“…Combining the above theorem with Proposition 13, we obtain the following result of [6]. We thank the referee for valuable remarks.…”

Section: ş Alpay and Z Ercan Positivitymentioning

confidence: 58%

“…In [6] it is proved that the converse of this is also true for Banach lattices E and F , when F has the Levi property. We can generalize this as follows.…”

Section: Relation Between B-property and Order Bounded Operatorsmentioning

confidence: 92%

“…In [6] it is proved that a Riesz space E has b-property if and only if the Dedekind completion E δ of E has b-property. However, the proof given there depends on the Hahn-Banach Theorem.…”

Section: Proposition 9 Let E Be a Riesz Space Then E Is Perfect If mentioning

confidence: 99%

Section: B-property and Locally Solid Riesz Spacesmentioning

confidence: 99%

See 2 more Smart Citations

Characterizations of Riesz spaces with b-property

Ercan

2008

Positivity

“…b-weakly compact operators were defined in [3] and studied in [4][5][6][7][8][9][10][11]. The space of bweakly compact operators is denoted by W b (E, F ) whereas W (E, F ) denotes the space of weakly compact operators.…”

Section: B-property and Related Classes Of Operatorsmentioning

confidence: 99%

On Riesz spaces with b-property and strongly order bounded operators

Rend. Circ. Mat. Palermo

Altin

2011

A Riesz space E is said to have the b-property if each subset that is order bounded in the bidual remains to be order bounded in E. Properties of a Riesz space with the bproperty, the relationship between the b-property and various classes of operators are studied.Keywords b-property · b-weakly compact operator · Strongly order bounded operator · Preregular operator Mathematics Subject Classification (2000) 46A40 · 46B40 · 47B60 · 47B65

A Note on b-Weakly Compact Operators

Altin

2007

Positivity

We consider a continuous operator T : E → X where E is a Banach lattice and X is a Banach space. We characterize the b-weak compactness of T in terms of its mapping properties. (2000). 47B60, 47B10, 47B07. Mathematics Subject ClassificationLet E be a Riesz space. E ∼ and E ∼∼ will denote the order dual and order bidual of E respectively. The canonical embedding Q E : E → E ∼∼ is defined byx is an order bounded and order continuous linear functional on E ∼ . The canonical embedding Q E is a lattice homomorphism. If E ∼ separates the points of E, Q E is also one-to-one and hence, E can be considered as a Riesz subspace of E ∼∼ . Since all Banach lattices have separating order duals, we will not distinguish a Banach lattice E and its image in E , the bidual of E. We will assume all Riesz spaces considered in this note have separating order duals. Definition. Let A be a subset of the Riesz space E. If Q E (A) is order bounded inClearly, every order bounded subset of E is a b-order bounded subset of E. There are b-order bounded subsets of E which are far away from being order bounded. The canonical basis vectors {e n } of c 0 is b-order bounded but clearly not an order bounded subset of c 0 .The notion of b-order bounded subsets of a Riesz space is instrumental in distinguishing a family of Riesz spaces. Definition. A Riesz space E is said to have the b-property if every b-order bounded subset of E is order bounded in E.b-order boundedness was introduced in [3]. b-order boundedness and Riesz spaces with the b-property were also investigated in [4,5,6].