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

We introduce the concepts of Pe{\l}czy'{n}ski’s property ($V$) of order $p$ and Pe{\l}czy'{n}ski’s property ($V^{\ast }$) of order $p$. It is proved that, for each $1<p<\mathrm{\infty }$, the James $p$-space $J_p$ enjoys Pe{\l}czy'{n}ski’s property ($V^{\ast }$) of order $p$ and the James $p^{\ast }$-space $J_{p^{\ast }}$ (where $p^{\ast }$ 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 ($V^{\ast }$).