Abstract
We first show that if $\psi : M_n(B(H)) \to M_n (B(H))$ is a $D_n \otimes F(H)$-bimodule map, then there is a matrix $A \in M_n$ such that $\psi = S_A$. Secondly, we show that for an operator space $\varepsilon, A \in M_n$, the Schur product map $S_A : M_n(\varepsilon) \to M_n(\varepsilon)$ and $\phi_A : M_n(\varepsilon) \to \varepsilon$, defined by $\phi_A([x_{ij}]) = \sum^{n}_{i,j=1}{a_{ij}x_{ij}}$, we have $\Vert S_A \Vert = \Vert S_A \Vert_{cb} = \Vert A \Vert_S, \Vert \phi_A \Vert = \Vert \phi_A \Vert_{cb} = \Vert A \Vert_1$ and obtain some characterizations of A for which $S_A$ is contractive.