{"title":"Convexity of sums of eigenvalues of a segment of unitaries","authors":"Gabriel Larotonda, Martin Miglioli","doi":"arxiv-2408.16906","DOIUrl":null,"url":null,"abstract":"For a $n\\times n$ unitary matrix $u=e^z$ with $z$ skew-Hermitian, the angles\nof $u$ are the arguments of its spectrum, i.e. the spectrum of $-iz$. For $1\\le\nm\\le n$, we show that $s_m(t)$, the sum of the first $m$ angles of the path\n$t\\mapsto e^{tx}e^y$ of unitary matrices, is a convex function of $t$ (provided\nthe path stays in a vecinity of the identity matrix). This vecinity is\ndescribed in terms of the opertor norm of matrices, and it is optimal. We show\nthat the when all the maps $t\\mapsto s_m(t)$ are linear, then $x$ commutes with\n$y$. Several application to unitarily invariant norms in the unitary group are\ngiven. Then we extend these applications to $Ad$-invariant Finsler norms in the\nspecial unitary group of matrices. This last result is obtained by proving that\nany $Ad$-invariant Finsler norm in a compact semi-simple Lie group $K$ is the\nsupremum of a family of what we call orbit norms, induced by the Killing form\nof $K$.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.16906","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
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 $K$.