Convexity of sums of eigenvalues of a segment of unitaries
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916375414439936 |
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| author | Larotonda, Gabriel Miglioli, Martin |
| author_facet | Larotonda, Gabriel Miglioli, Martin |
| contents | For a $n\times n$ unitary matrix $u=e^z$ with $z$ skew-Hermitian, the angles of $u$ are the arguments of its spectrum, i.e. the spectrum of $-iz$. For $1\le m\le n$, we show that $s_m(t)$, the sum of the first $m$ angles of the path $t\mapsto e^{tx}e^y$ of unitary matrices, is a convex function of $t$ (provided the path stays in a vecinity of the identity matrix). This vecinity is described in terms of the opertor norm of matrices, and it is optimal. We show that the when all the maps $t\mapsto s_m(t)$ are linear, then $x$ commutes with $y$. Several application to unitarily invariant norms in the unitary group are given. Then we extend these applications to $Ad$-invariant Finsler norms in the special unitary group of matrices. This last result is obtained by proving that any $Ad$-invariant Finsler norm in a compact semi-simple Lie group $K$ is the supremum of a family of what we call orbit norms, induced by the Killing form of $K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16906 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convexity of sums of eigenvalues of a segment of unitaries Larotonda, Gabriel Miglioli, Martin Functional Analysis Differential Geometry Metric Geometry 15A42 (Primary) 26A51, 51F25 (Secondary) For a $n\times n$ unitary matrix $u=e^z$ with $z$ skew-Hermitian, the angles of $u$ are the arguments of its spectrum, i.e. the spectrum of $-iz$. For $1\le m\le n$, we show that $s_m(t)$, the sum of the first $m$ angles of the path $t\mapsto e^{tx}e^y$ of unitary matrices, is a convex function of $t$ (provided the path stays in a vecinity of the identity matrix). This vecinity is described in terms of the opertor norm of matrices, and it is optimal. We show that the when all the maps $t\mapsto s_m(t)$ are linear, then $x$ commutes with $y$. Several application to unitarily invariant norms in the unitary group are given. Then we extend these applications to $Ad$-invariant Finsler norms in the special unitary group of matrices. This last result is obtained by proving that any $Ad$-invariant Finsler norm in a compact semi-simple Lie group $K$ is the supremum of a family of what we call orbit norms, induced by the Killing form of $K$. |
| title | Convexity of sums of eigenvalues of a segment of unitaries |
| topic | Functional Analysis Differential Geometry Metric Geometry 15A42 (Primary) 26A51, 51F25 (Secondary) |
| url | https://arxiv.org/abs/2408.16906 |