GEOMETRY AND THE NORMS OF HADAMARD MULTIPLIERS

The norm of a matrix B as a Hadamard multiplier is the norm of the map X --> X . B where . is the Hadamard or entrywise product of matrices. We give a geometric interpretation to a criterion of Haagerup and use this interpretation to find an explicit formula for the Hadamard mult iplier norms of real 2 x 2 matrices. For Hermitian matrices with one p ositive eigenvalue, we give necessary and sufficient conditions for th e norm to be the magnitude of the largest entry.