A Variant of the Kaplansky Problem for Maps on Positive Matrices

TL;DR

This paper characterizes injective order- and spectrum-shrinking maps on positive matrices as being implemented by unitary or antiunitary conjugations, with counterexamples showing all assumptions are necessary.

Contribution

It provides a complete characterization of spectrum-shrinking order-preserving maps on positive matrices, extending to Hermitian matrices, with necessary assumptions demonstrated by counterexamples.

Findings

Injective order- and spectrum-shrinking maps are unitary or antiunitary conjugations.

Counterexamples show all assumptions are essential.

Results generalize to Hermitian matrices.

Abstract

We prove that all injective maps on positive complex matrices which preserve order and shrink spectrum are implemented by unitary or antiunitary conjugations. We show by counterexamples that all assumptions are indispensable. The result easily generalizes to maps on hermitian matrices.