If $A$ is a $n \times n$ complex matrix then the **Schur norm** of $A$ is given by $$ || A||_S := \max_{||B||=1} ||A*B||,$$ where $||. ||$ is the operator norm and $*$ is the Hadamard (entry-wise) product:

$$(A*B)_{ij}= A_{ij} B_{ij}.$$

In the case that $A$ is self-adjoint may we restrict the minimization over $B$ to minimization over self-adjoint $B$?

(I'm working on a problem that naturally involves self-adjoint B, but it is driving me crazy that the theorems I want to cite minimize over arbitrary, perhaps non-self-adjoint B.)