Pe{\l}czy\'{n}ski's property ($V^{*}$) 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 ($V^{*}$) 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 ($V^{*}$).