How to characterize an operator $T$ that factors through a special space?

Question

Let $T\in \mathcal{L}(X,Y)$ and $1<p<\infty$. My question is: Is there a convienent and useful characterization of the operator $T$ factoring through a space $Z$ satisfying $\mathcal{L}(Z,l_{p})=\mathcal{K}(Z,l_{p})$? where $\mathcal{L}$ denotes linear bounded operator space, $\mathcal{K}$ denotes the compact operator space. This question seems open-ended. But it is useful. Thank you!