Preservers of Unitary Similarity Functions on Lie Products of Matrices

TL;DR

This paper characterizes preservers of certain unitary similarity functions on matrix Lie products, providing new insights into functions like the pseudo spectrum and spectral radius, and answering a question posed in 2014.

Contribution

It offers a novel characterization of maps preserving unitary similarity functions on Lie products of matrices, extending to special cases like pseudo spectrum and spectral radius.

Findings

Characterizations of surjective maps preserving $f(AB-BA)$

Results on pseudo spectrum preservation

Answers to Molnar's 2014 question

Abstract

Denote by $M_n$ the set of $n\times n$ complex matrices. Let $f: M_n \rightarrow [0,\infty)$ be a continuous map such that $f(\mu UAU^*)= f(A)$ for any complex unit $\mu$, $A \in M_n$ and unitary $U \in M_n$, $f(X)=0$ if and only if $X=0$ and the induced map $t \mapsto f(tX)$ is monotonic increasing on $[0,\infty)$ for any rank 1 nilpotent $X \in M_n$. Characterizations are given for surjective maps $\phi$ on $M_n$ satisfying $f(AB-BA) = f(\phi(A)\phi(B)-\phi(B)\phi(A))$. The general theorem are then used to deduce results on special cases when the function is the pseudo spectrum and the pseudo spectral radius, that answers a question of Molnar raised at the 2014 CMS summer meeting.