Continuous spectrum-shrinking maps and applications to preserver problems
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| Format: | Preprint |
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2025
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| _version_ | 1866908284736241664 |
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| author | Chirvasitu, Alexandru Gogić, Ilja Tomašević, Mateo |
| author_facet | Chirvasitu, Alexandru Gogić, Ilja Tomašević, Mateo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_06840 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Continuous spectrum-shrinking maps and applications to preserver problems Chirvasitu, Alexandru Gogić, Ilja Tomašević, Mateo Spectral Theory Group Theory Operator Algebras Rings and Algebras 47A10, 47B15, 47B49, 15A27, 54D05 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. |
| title | Continuous spectrum-shrinking maps and applications to preserver problems |
| topic | Spectral Theory Group Theory Operator Algebras Rings and Algebras 47A10, 47B15, 47B49, 15A27, 54D05 |
| url | https://arxiv.org/abs/2501.06840 |