Extension of $p$-compact operators in Banach spaces

We analyze various consequences in relation to the extension of operators $T:X\to Y$ that are $p$-compact, as well as the extension of operators $T:X\to Y$ whose adjoints $T^{\ast }:Y^{\ast }\to X^{\ast }$ are $p$-compact. In most cases, we discuss these extension properties when the underlying spaces, either domain or codomain, are $P_{\lambda }$ spaces. We also answer if these extensions are almost norm-preserving in such circumstances where the extension $\tilde{T}$ of a $T$ exists. It is observed that an operator can often be extended to a larger domain when the codomain is appropriately extended as well. Specific assumptions might enable us to obtain an extension of an operator that maintains the same range. Necessary and sufficient conditions are derived for a Banach space to be $L_1$-predual.