Continuous spectrum-shrinking maps and applications to preserver problems
Alexandru Chirvasitu, Ilja Gogić, Mateo Tomašević
TL;DR
The paper investigates continuous spectrum-shrinking maps between standard matrix spaces and their implications for preservation problems. It proves that such maps ${\\phi}: {\\mathcal{X}}_{n} \to M_m$ exist if and only if $n$ divides $m$, with the characteristic polynomials related by $k_{\\phi(X)}(x) = k_X(x)^{m/n}$, thereby enforcing spectrum preservation when the divisibility condition holds. For $n \ge 3$, any continuous commutativity-preserving and spectrum-shrinking map from ${\\mathcal{X}}_{n}$ to $M_n$ must be an inner automorphism up to transposition, i.e. $\\phi(X) = T X T^{-1}$ or $\\phi(X) = T X^{t} T^{-1}$ for some $T \in GL(n)$; analogous statements hold for semisimple elements in $GL(n)$ or $SL(n)$, while SU(n) fails. These results unify and strengthen existing preserver results, including a broadened characterization of Jordan automorphisms via preserved properties, and provide a framework for spectrum-shrinking to spectrum-preserving transitions in matrix algebras.
Abstract
For a positive integer $n$ let $\mathcal{X}_n$ be either the algebra $M_n$ of $n \times n$ complex matrices, the set $N_n$ of all $n \times n$ normal matrices, or any of the matrix Lie groups $\mathrm{GL}(n)$, $\mathrm{SL}(n)$ and $\mathrm{U}(n)$. We first give a short and elementary argument that for two positive integers $m$ and $n$ there exists a continuous spectrum-shrinking map $φ: \mathcal{X}_n \to M_m$ (i.e.\ $\mathrm{sp}(φ(X))\subseteq \mathrm{sp}(X)$ for all $X \in \mathcal{X}_n$) if and only if $n$ divides $m$. Moreover, in that case we have the equality of characteristic polynomials $k_{φ(X)}(\cdot) = k_{X}(\cdot)^\frac{m}{n}$ for all $X \in \mathcal{X}_n$, which in particular shows that $φ$ preserves spectra. Using this we show that whenever $n \geq 3$, any continuous commutativity preserving and spectrum-shrinking map $φ: \mathcal{X}_n \to M_n$ is of the form $φ(\cdot)=T(\cdot)T^{-1}$ or $φ(\cdot)=T(\cdot)^tT^{-1}$, for some $T\in \mathrm{GL}(n)$. The analogous results fail for the special unitary group $\mathrm{SU}(n)$ but hold for the spaces of semisimple elements in either $\mathrm{GL}(n)$ or $\mathrm{SL}(n)$. As a consequence, we also recover (a strengthened version of) Šemrl's influential characterization of Jordan automorphisms of $M_n$ via preserving properties.