$L$-limited-like properties on Banach spaces

We study weakly precompact sets and operators. We show that an operator is weakly precompact if and only if its adjoint is pseudo weakly compact. We study Banach spaces with the $p$-$L$-limited$^*$ and the $p$-(SR$^*$) properties and characterize these classes of Banach spaces in terms of $p$-$L$-limited$^*$ and $p$-Right$^*$ subsets. The $p$-$L$-limited$^*$ property is studied in some spaces of operators.