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 $$\Vert X{\Vert }_S=\min \{\Vert \mathrm{diag}(P){\Vert }_{\infty }:-P\le X\le P\}.$$ The dual space of $( M_n(\bc), \|.\|_S)$ is $(M_n(\bc), \|.\|_{cbB}).$ For $X=X^{\ast }:$ $$ \|X\|_{cbB} = \min \{ \, \mathrm{Tr}_n\big(Δ(λ)\big)\, :\, λ\in \br^n, \, - Δ(λ) \leq X \leq Δ(λ)\,\}. $$

arXiv commentRewritten, such that much is based on R. Mathias' work on pairs of dual norms. An example is included