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.)



Schur norm for self-adjoint operators
R. Bhatia has answered this question:


The answer is yes, and it's equation 3.7 of 


<http://www.ams.org/journals/proc/1993-117-04/S0002-9939-1993-1116267-7/S0002-9939-1993-1116267-7.pdf>


