Let L(X) be the space of bounded linear operators on the Banach space X. We study the strict singularity and cosingularity of the two-sided multiplication operators 5 → ASB on L(X), where A, B ∈ L(X) are fixed bounded operators and X is a classical Banach space. Let 1 < p < ∞ and p ≠ 2. Our main result establishes that the multiplication S → ASB is strictly singular on L(Lp(0, 1)) if and only if the non-zero operators A, B ∈ L(Lp(0,1)) are strictly singular. We also discuss the case where X is a ℒ1 - or a ℒ ∞-space, as well as several other relevant examples. ©Canadian Mathematical Society 2005.