Classes of operators determined by ordinal indices

2015/07/22 by Kevin Beanland, Beanland, Kevin, Ryan M. Causey +5
Mathematics · #46B28 #Advanced Banach Space Theory #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1507.06285

openalex publication_date 2015/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce and study the Bourgain index of an operator between two Banach spaces. In particular, we study the Bourgain ℓ_p and c₀ indices of an operator. Several estimates for finite and infinite direct sums are established. We define classes determined by these indices and show that some of these classes form operator ideals. We characterize the ordinals which occur as the index of an operator and establish exactly when the defined classes are closed. We study associated indices for non-preservation of ℓ_p^ξ and c₀^ξ spreading models and indices characterizing weak compactness of operators between separable Banach spaces. We also show that some of these classes are operator ideals and discuss closedness and distinctness of these classes.

Citations

• Complexity of weakly null sequences

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