Schur null preserving maps

TL;DR

This paper characterizes Schur null preserving maps on matrix spaces, showing that surjective Schur multiplicative contractions are isometries, and extends these results to various matrix and multiplier contexts.

Contribution

It provides a comprehensive characterization of Schur null preserving maps and generalizes the result that surjective Schur multiplicative contractions are isometries.

Findings

Surjective Schur multiplicative contractions are isometries

Characterization of Schur null preserving maps on matrices

Extension of results to Schur multipliers

Abstract

We provide a characterisation of Schur multiplicative maps on both finite and infinite dimensional matrix spaces, and show that every surjective Schur multiplicative contraction is automatically an isometry. We also generalise this result and provide a characterisation of Schur null preserving maps on $m\times n$ matrices and on Schur multipliers.