Pe{\l}czy\'{n}ski's property ($V^{*}$) of order $p$ and its quantification

TL;DR

This paper introduces and studies Pe{2}czy
44ski's property ($V$) of order $p$ and its dual, demonstrating their presence in classical Banach spaces like James $p$-spaces and $L_1$, with a focus on quantitative aspects.

Contribution

It defines Pe{2}czy
44ski's property ($V$) of order $p$, proves its presence in James $p$-spaces and their duals, and establishes a quantitative version for $L_1$ and $ ext{l}_1$.

Findings

James $p$-spaces have Pe{2}czy
44ski's property ($V^{*}$) of order $p$.

James $p^{*}$-spaces have Pe{2}czy
44ski's property ($V$) of order $p$.

Finite measure $L_1$ and $ ext{l}_1$ spaces enjoy the quantitative version of Pe{2}czy
44ski's property ($V^{*}$).

Abstract

We introduce the concepts of Pe{\l}czy\'{n}ski's property ($V$) of order $p$ and Pe{\l}czy\'{n}ski's property ($V^{*}$) of order $p$. It is proved that, for each $1<p<\infty$, the James $p$-space $J_{p}$ enjoys Pe{\l}czy\'{n}ski's property ($V^{*}$) of order $p$ and the James $p^{*}$-space $J_{p^{*}}$ (where $p^{*}$ denotes the conjugate number of $p$) enjoys Pe{\l}czy\'{n}ski's property ($V$) of order $p$. We prove that both $L_{1}(\mu)$ ($\mu$ a finite positive measure) and $l_{1}$ enjoy the quantitative version of Pe{\l}czy\'{n}ski's property ($V^{*}$).