ξ-completely continuous operators and ξ-Schur Banach spaces

Causey, Ryan M.; Navoyan, Khazhak

doi:10.1016/j.jfa.2018.09.002

Cited by 12 publications

(18 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…. < r s such that ∪ s i=1 supp(P M,ri ) is an initial segment of N , then P N,i = P M,ri for all 1 i s. This was proved in [10].…”

Section: Proofmentioning

confidence: 75%

“…The concepts behind these classes are that weakly null sequences are mapped by the operator to sequences which are "very" weakly null (completely continuous operators send weakly null sequences to 0-weakly null sequences, and weak Banach-Saks operators send weakly null sequences to 1-weakly null sequences). In [10], the notions of ξ-completely continuous operators and ξ-Schur Banach spaces were introduced. These notions are weakenings of the notions of completely continuous operators and Schur Banach spaces, respectively.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The $ξ,ζ$-Dunford Pettis property

Causey

2018

Preprint

View full text Add to dashboard Cite

Using the hierarchy of weakly null sequences introduced in [2], we introduce two new families of operator classes. The first family simultaneously generalizes the completely continuous operators and the weak Banach-Saks operators. The second family generalizes the class DP. We study the distinctness of these classes, and prove that each class is an operator ideal. We also investigate the properties possessed by each class, such as injectivity, surjectivity, and identification of the dual class. We produce a number of examples, including the higher ordinal Schreier and Baernstein spaces. We prove ordinal analogues of several known results for Banach spaces with the Dunford-Pettis, hereditary Dunford-Pettis property, and hereditary by quotients Dunford-Pettis property. For example, we prove that for any 0 ξ, ζ < ω 1 , a Banach space X has the hereditary ω ξ , ω ζ -Dunford Pettis property if and only if every seminormalized, weakly null sequence either has a subsequence which is an ℓ ω ξ 1 -spreading model or a c ω ζ 0 -spreading model.

show abstract

“…. < r s such that ∪ s i=1 supp(P M,ri ) is an initial segment of N , then P N,i = P M,ri for all 1 i s. This was proved in [10].…”

Section: Proofmentioning

confidence: 75%

Section: Introductionmentioning

confidence: 99%

The $ξ,ζ$-Dunford Pettis property

Causey

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…For 0 < ξ < ω 1 , fix a sequence ξ n ↑ ω ξ and a sequence ϑ It was shown in [6] that Z ξ is ζ -Schur for each ζ < ω ξ , but Z ξ is not ω ξ -Schur. Thus for every ξ < ω 1 , we have found a Banach space which is ζ -Schur for every ζ < ω ξ and which fails to be ω ξ -Schur.…”

Section: Examples Of ξ -Schur Banach Spacesmentioning

confidence: 99%

A ξ-weak Grothendieck compactness principle

Beanland¹,

Causey²

2021

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

For 0 ≤ ξ ≤ ω1, we define the notion of ξ-weakly precompact and ξ-weakly compact sets in Banach spaces and prove that a set is ξ-weakly precompact if and only if its weak closure is ξ-weakly compact. We prove a quantified version of Grothendieck’s compactness principle and the characterisation of Schur spaces obtained in [7] and [9]. For 0 ≤ ξ ≤ ω1, we prove that a Banach space X has the ξ-Schur property if and only if every ξ-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence. The ξ = 0 and ξ= ω1 cases of this theorem are the theorems of Grothendieck and [7], [9], respectively.

show abstract

“…In the terminology of [11], the ξ th level of the repeated averages hierarchy considered as a family of probability measures on S ξ are a ξ-sufficient, and therefore ξ-regulatory probability block (see [11,Corollary 4.8]). For 0 < ε 1 < ε, there exists N ∈ [N] such that for each F ∈ S ξ , there exists E) ε 1 (see the definition of ξ-full and the definitions of the sets E(f, P, P, ε) and G(f, P, M, ε) preceding [11,Lemma 4.4]). Since f (N(i), E) ε 1 > 0, then N(i) ∈ E.…”

Section: Properties Of Interestmentioning

confidence: 99%

Subprojectivity of Projective Tensor Products of Banach Spaces of Continuous Functions

Causey¹

2021

Glasgow Math. J.

Self Cite

View full text Add to dashboard Cite

Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-saturated if and only if it is subprojective if and only if K and L are both scattered. We remove the hypothesis of metrizability from their result and extend it from the case of the twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, K n , $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each K i is scattered.

show abstract

ξ-completely continuous operators and ξ-Schur Banach spaces

Cited by 12 publications

References 29 publications

The $ξ,ζ$-Dunford Pettis property

The $ξ,ζ$-Dunford Pettis property

A ξ-weak Grothendieck compactness principle

Subprojectivity of Projective Tensor Products of Banach Spaces of Continuous Functions

Contact Info

Product

Resources

About