Schur multiplier operator and matrix inequalities

In this note, we obtain a reverse version of the Haagerup Theorem. In particular, if A ∈ Mn has a 2 × 2− principal submatrix as [ 1 α β 1] with β = ¯α, then SA 1 where the operator SA : Mn −→ Mn is defined by SA(B) := A ◦ B where ” ◦ ” stands for Schur product.