The combinatorics of permuting and preserving curve-bound spectra
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arXiv
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| Format: | Preprint |
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2026
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| author | Chirvasitu, Alexandru |
| author_facet | Chirvasitu, Alexandru |
| contents | We prove that continuous spectrum- and commutativity-preserving maps to $\mathcal{M}_n(\mathbb{C})$ from the space of normal (real or complex) $n\times n$, $n\ge 3$ matrices with spectra contained in a given continuous-injection interval image $Λ\subseteq \mathbb{C}$ or $\mathbb{R}$ are (a) conjugations; (b) transpose conjugations, or (c) orderings of spectra according to an orientation of $Λ$, with fixed eigenspaces. This generalizes results of Petek's (self-maps of real or complex Hermitian matrices) and the author's (complex Hermitian matrices as the domain, $\mathcal{M}_n(\mathbb{C})$ as the codomain). An application rules out possibility (c) for normal matrices with spectra constrained to a simple closed curve, extending a result by the author, Gogić and Tomašević to the effect that continuous commutativity and spectrum preservers on unitary groups are (transpose) conjugations.
The involution preserving eigenspaces and complex-conjugating eigenvalues is a novel possibility beyond (a), (b) and (c) if the domain consists of all semisimple operators with $Λ$-bound spectra instead; its continuity (or lack thereof) and whether or not that map furthermore extends continuously to arbitrary $Λ$-constrained-spectrum matrices hinge on the geometry and regularity of $Λ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_01208 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The combinatorics of permuting and preserving curve-bound spectra Chirvasitu, Alexandru Spectral Theory Functional Analysis General Topology Group Theory Operator Algebras 47A10, 15B57, 54D05, 15A27, 46C05, 20B30, 54H15, 54F50 We prove that continuous spectrum- and commutativity-preserving maps to $\mathcal{M}_n(\mathbb{C})$ from the space of normal (real or complex) $n\times n$, $n\ge 3$ matrices with spectra contained in a given continuous-injection interval image $Λ\subseteq \mathbb{C}$ or $\mathbb{R}$ are (a) conjugations; (b) transpose conjugations, or (c) orderings of spectra according to an orientation of $Λ$, with fixed eigenspaces. This generalizes results of Petek's (self-maps of real or complex Hermitian matrices) and the author's (complex Hermitian matrices as the domain, $\mathcal{M}_n(\mathbb{C})$ as the codomain). An application rules out possibility (c) for normal matrices with spectra constrained to a simple closed curve, extending a result by the author, Gogić and Tomašević to the effect that continuous commutativity and spectrum preservers on unitary groups are (transpose) conjugations. The involution preserving eigenspaces and complex-conjugating eigenvalues is a novel possibility beyond (a), (b) and (c) if the domain consists of all semisimple operators with $Λ$-bound spectra instead; its continuity (or lack thereof) and whether or not that map furthermore extends continuously to arbitrary $Λ$-constrained-spectrum matrices hinge on the geometry and regularity of $Λ$. |
| title | The combinatorics of permuting and preserving curve-bound spectra |
| topic | Spectral Theory Functional Analysis General Topology Group Theory Operator Algebras 47A10, 15B57, 54D05, 15A27, 46C05, 20B30, 54H15, 54F50 |
| url | https://arxiv.org/abs/2601.01208 |