The Schur multiplier norm and its dual norm

Abstract: We present a formula for the Schur multiplier norm of a complex self-adjoint matrix, and a formula for the norm, which is dual to the Schur multiplier norm, of a self-adjoint matrix. For a complex self-adjoint $n \times n $ matrix $X$ we show that its Schur multiplier norm is determined by $$ |X|S = \min {\, |\mathrm{diag}(P)|\infty \, :\, - P \leq X \leq P \, }.$$ The dual space of $( M_n(\bc), |.|S)$ is $(M_n(\bc), |.|{cbB}).$ For $X=X^*:$ $$ |X|_{cbB} = \min { \, \mathrm{Tr}_n\big(\Delta(\lambda)\big)\, :\, \lambda \in \br^n, \, - \Delta(\lambda) \leq X \leq \Delta(\lambda)\,}. $$