Strict singularity of weighted composition operators on derivative Hardy spaces

We prove that the weighted composition operator $W_{\varphi ,\phi }$ fixes an isomorphic copy of ${\mathrm{\ell }}^p$ if the operator $W_{\varphi ,\phi }$ is not compact on the derivative Hardy space $S^p$. In particular, this implies that the strict singularity of the operator $W_{\varphi ,\phi }$ coincides with the compactness of it on $S^p$. Moreover, when $p\mathrm{\ne }2$, we characterize the conditions for those weighted composition operators $W_{\varphi ,\phi }$ on $S^p$ which fix an isomorphic copy of ${\mathrm{\ell }}^2$ .