A pair (u, α) in X × X ′ , where X is an infinite dimensional Banach space and X ′ its topological dual space, induces in a natural way two multiplication operators L α,u and R α,u on the Banach space L(X), defined by L α,u T (x) = α T(x) u, and R α,u T (x) = α(x)T(u), for all T in L(X) and x in X. In this paper, we present necessary and sufficient conditions for the compactness, demicompactness, stongly demicompactess, power compactness and Riesz property of this family of operators. We also establish sufficient conditions for the quasi-compactness and weak compactness of these operators. Finally, we show that the Dunford-Pettis property fails for the Banach space L(X) whenever either X or L(X) is reflexive.