Geometry and the norms of Hadamard multipliers (Q1805207)

As a matter of fact, the norm of a matrix \(B\) as a Hadamard multiplier is the norm of the map \(X \to X \cdot B\) where ``\(\cdot\)'' is the Hadamard or entrywise product of matrices. While \textit{U. Haagerup} [Decompositions of completely bounded maps on operator algebras (Preprint)] showed that if \(B\) is an \(n \times n\) matrix, then the norm of \(B\) as a Hadamard multiplier is \(K_ B = \min \{C(S) C(R) : S^* R = B,S,R \in {\mathcal M}_ n\}\), where \(C(P) = \max \{\| P_ 1 \|_ 2, \| P_ 2 \|_ 2, \dots, \| P_ n \|_ 2\}\), the authors give a geometric interpretation to a criterion of Haagerup: If \(B\) is an \(n \times n\) matrix, then \(K_ B\) is the radius of the smallest cube that contains an ellipsoid containing each column of the matrix \(B\), and then use this interpretation to find an explicit formula for the Hadamard multiplier norms of real \(2 \times 2\) matrices. For Hermitian matrices with one positive eigenvalue, necessary and sufficient conditions are given for the norm to be the magnitude of the largest entry.