Schur multipliers of Schatten–von Neumann classes Sp

Abstract We study in this paper properties of Schur multipliers of Schatten von Neumann classes S p . We prove that for p ≤ 1 , Schur multipliers of S p are necessarily completely bounded. We also introduce for p ≤ 1 a scale W p of tensor products of l ∞ and prove that matrices in W p are Schur multipliers of S p . We compare this sufficient condition with the sufficient condition of membership in the p-tensor product of l ∞ spaces.