The Schur multiplier norm and its dual norm

TL;DR

This paper derives explicit formulas for the Schur multiplier norm and its dual for complex self-adjoint matrices, providing new insights into their structure and duality properties.

Contribution

The paper introduces explicit formulas for the Schur multiplier norm and its dual for self-adjoint matrices, advancing understanding of their mathematical properties.

Findings

Schur multiplier norm is characterized by a minimization over diagonal matrices.

Dual norm is expressed via a trace minimization involving diagonal matrices.

Provides a duality relationship between two matrix norms.

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 $X=X^*:$ $$ \|X\|_{cbB} = \min \{ \, \mathrm{Tr}_n\big(\Delta(\lambda)\big)\, :\, \lambda \in \br^n, \, - \Delta(\lambda) \leq X \leq \Delta(\lambda)\,\}. $$