Norms of Hadamard Multipliers

Abstract

For B a fixed matrix, the authors consider the problem of finding the norm of the map $X \mapsto X \bullet B$, where $ \bullet $ is the Hadamard or entrywise product of matrices and the norm of a matrix is its spectral norm. Using techniques from the theory of Kren spaces, the problem is rewritten for Hermitian matrices as a minimization problem whose solution, for small matrices, can be obtained from standard optimization software. The Hadamard multiplier norm for an arbitrary matrix is given in terms of a Hermitian extension. The results are applied to refute a conjecture of R. V. McEachin concerning the value of a constant in an operator inequality.