Quantification of Pe{\l}czy\'nski's property (V)

TL;DR

This paper introduces and characterizes quantitative versions of Pe{2}czy
44ski's property (V) in Banach spaces, extending classical results and exploring relationships with other operator properties.

Contribution

It develops new quantitative measures for property (V), provides characterizations, and generalizes classical theorems about $C(K)$ spaces.

Findings

Quantitative versions of property (V) are characterized.

A quantitative version of Pe{2}czy
44ski's theorem for $C(K)$ spaces is established.

Relationships between quantitative property (V) and other Banach space properties are analyzed.

Abstract

A Banach space $X$ has Pe{\l}czy\' nski's property (V) if for every Banach space $Y$ every unconditionally converging operator $T\colon X\to Y$ is weakly compact. In 1962, Aleksander Pe{\l}czy\' 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.