The Schur multiplier norm and its dual norm

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 $$\mathrm{\parallel }X{\mathrm{\parallel }}_S=\min \{\mathrm{\parallel }\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(P){\mathrm{\parallel }}_{\mathrm{\infty }}:-P\le X\le P\}\mathrm{.}$$ The dual space of $(M_n(\text{bc}),\mathrm{\parallel }\mathrm{.}{\mathrm{\parallel }}_S)$ is $(M_n(\text{bc}),\mathrm{\parallel }\mathrm{.}{\mathrm{\parallel }}_{cbB})\mathrm{.}$ For $X=X^{\ast }:$ $$\mathrm{\parallel }X{\mathrm{\parallel }}_{cbB}=\min \{{\mathrm{T}\mathrm{r}}_n(\mathrm{\Delta }(\lambda )):\lambda \in {\text{br}}^n,-\mathrm{\Delta }(\lambda )\le X\le \mathrm{\Delta }(\lambda )\}\mathrm{.}$$