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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.)

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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

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