Polynomial versions of weak Dunford–Pettis properties in Banach lattices

Wang, Yu; Shi, Zhongrui; Bu, Qingying

doi:10.1007/s11117-021-00837-2

Cited by 4 publications

(5 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recent works studied polynomial versions of some classes of sets and properties in Banach lattices [19,20]. In this section, we are going to give a few contributions concerning polynomials and the properties studied in Section 2.…”

Section: Polynomial Considerationsmentioning

confidence: 99%

Grothendieck-type subsets of Banach lattices

Galindo¹,

Miranda²

2021

Preprint

View full text Add to dashboard Cite

show abstract

Section: Polynomial Considerationsmentioning

confidence: 99%

Grothendieck-type subsets of Banach lattices

Galindo¹,

Miranda²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Inspired by the results obtained for polynomial versions of the wDP P and of the wDP * P in [35], especially [35,Theorems 3.8 and 4.3], we investigate in this section the behavior of the disjoint DP P p and of the disjoint DP * P p when linear operators and linear functionals are replaced with homogeneous polynomials. In particular, polynomial versions of our Theorems 2.2, 3.3 and 3.4 shall be proved.…”

Section: Polynomial Propertiesmentioning

confidence: 99%

“…To prove the first theorem of the section we need two lemmas. The first one can be found within the proof of [35,Theorem 3.8].…”

Section: Polynomial Propertiesmentioning

confidence: 99%

“…Using once again that the correspondence (1) is an isometric isomorphism and a lattice homomorphism, (P n ) n is a positive weakly null sequence in P r ( k E). By [35,Lemma 3.2] there exists a disjoint weakly null sequence (…”

Section: Polynomial Propertiesmentioning

confidence: 99%

“…For instance, polynomial versions of the DP P and of the DP * P can be found in [13,20,30,31]. In the context of Banach lattices, the positive polynomial Schur property was investigated in [8] and polynomials versions of the wDP P and of the wDP * P appeared in [35]. In Section 4 we investigate the polynomial behavior of the disjoint DP P p and of the disjoint DP * P p .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Schur, positive Schur and weak Dunford–Pettis properties in Fréchet lattices

Botelho

Luiz

2021

Ann. Fenn. Math.

View full text Add to dashboard Cite

We prove some general results on sequential convergence in Fréchet lattices that yield, as particular instances, the following results regarding a closed ideal I of a Banach lattice E: (i) If two of the lattices E, I and E/I have the positive Schur property (the Schur property, respectively) then the third lattice has the positive Schur property (the Schur property, respectively) as well; (ii) If I and E/I have the dual positive Schur property, then E also has this property; (iii) If I has the weak Dunford-Pettis property and E/I has the positive Schur property, then E has the weak Dunford-Pettis property. Examples and applications are provided.Definition 1.1. A property P of Banach lattices is a 3-lattice property if the following holds: given a closed ideal I of the Banach lattice E, if two of the lattices E, I and E/I have P, then the third one has P as well.A Banach lattice has the positive Schur property if weakly null sequences formed by positive vectors are norm null. A lot of research has been done on this property, for some recent contributions see, e.g., [6,7,13,18,24,25,26,28]. Among other results, in this paper we prove that, contrary to the case of the Schur property for Banach spaces, the Schur and the positive Schur properties are 3-lattice properties. These results will appear as applications of general results on the Schur and the positive Schur properties in Fréchet lattices.

show abstract

Grothendieck-type subsets of Banach lattices

Galindo

Miranda

2022

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Polynomial versions of weak Dunford–Pettis properties in Banach lattices

Cited by 4 publications

References 21 publications

Grothendieck-type subsets of Banach lattices

Grothendieck-type subsets of Banach lattices

On the Schur, positive Schur and weak Dunford–Pettis properties in Fréchet lattices

Grothendieck-type subsets of Banach lattices

Contact Info

Product

Resources

About