2015/09/22 by Hana Krulišová, Krulišová, Hana · 2 citations
Mathematics · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1509.06610
openalex publication_date 2015/09/22 · openalex created_date 2022/08/06 · openalex updated_date 2026/07/28
A Banach space X has Pełczy' nski's property (V) if for every Banach space Y every unconditionally converging operator T\colon X→ Y is weakly compact. In 1962, Aleksander Pełczy' nski showed that C(K) spaces for a compact Hausdorff space K enjoy the property (V), and some generalizations of this theorem have been proved since then. We introduce several possibilities of quantifying the property (V). We prove some characterizations of the introduced quantitative versions of this property, which allow us to prove a quantitative version of Pelczynski's result about C(K) spaces and generalize it. Finally, we study the relationship of several properties of operators including weak compactness and unconditional convergence, and using the results obtained we establish a relation between quantitative versions of the property (V) and quantitative versions of other well known properties of Banach spaces.