NORMS OF HADAMARD MULTIPLIERS

For B a fixed matrix, the authors consider the problem of finding the norm of the map X --> X.B, where.is the Hadamard or entrywise product of matrices and the norm of a matrix is its spectral norm. Using techn iques from the theory of Krein spaces, the problem is rewritten for He rmitian matrices as a minimization problem whose solution, for small m atrices, can be obtained from standard optimization software. The Hada mard multiplier norm for an arbitrary matrix is given in terms of a He rmitian extension. The results are applied to refute a conjecture of R . V. McEachin concerning the value of a constant in an operator inequa lity.