Bilateral composition operators on vector-valued Hardy spaces

Let $T$ be a bounded operator on the Banach space $X$ and $\ph$ be an analytic self-map of the unit disk $\Bbb{D}$‎. ‎We investigate some operator theoretic properties of‎ ‎bilateral composition operator $C_{\ph‎, ‎T}‎: ‎f \ri T \circ f \circ \ph$ on the vector-valued Hardy space $H^p(X)$ for $1 \leq p \leq‎ ‎+\infty$.‎ ‎Compactness and weak compactness of $C_{\ph‎, ‎T}$ on $H^p(X)$‎ ‎are characterized and when $p=2$‎, ‎a concrete formula for its adjoint is given‎.