Extremal Schur Multipliers

Abstract: The Schur product of two complex m x n matrices is their entry wise product. We show that an extremal element X in the convex set of m x n complex matrices of Schur multiplier norm at most 1 satisfies the inequality rank(X) =< (m +n)^1/2. For positive n x n matrices with unit diagonal, we give a characterization of the extremal elements, and show that such a matrix satisfies rank(X) =< n^1/2.