Invariant subspaces of two Hermitian structures on a Euclidean space
type: Item
instance of: scholarly article
title: Invariant subspaces of two Hermitian structures on a Euclidean space
zbMATH Open document ID: 0946.15004
DOI: 10.1016/S0024-3795(99)00206-2
author: Hillel Gauchman
author: Patrick Coulton
published in: Linear Algebra and its Applications
publication date: 2000-10-22T00:00:00Z
review text: Let \(V\) be a real vector space \((\dim(V)=2n)\), \(J\) be a complex structure on \(V\) (i.e. a linear endomorphism such that \(J^2=-I)\). The orientation of \(J\) is the orientation of any basis \(\{e_1,\dots,e_{2n}\}\) adapted to \(J\) (i.e. \(J(e_{2i-1})= e_{2i}\); \(i=1, \dots,n)\). A Hermitian structure on a Euclidean space \(V\) is a pair \((\langle , \rangle,J)\), where \(\langle , \rangle\) is the inner product and \(\langle J(x),J(y) \rangle= \langle x,y \rangle\) \((x,y\in V)\). The authors establish that if \((\langle , \rangle,J_1)\) and \((\langle , \rangle,J_2)\) are Hermitian structures on \(V\) then: (1) if \(n\) is even and \(J_1,J_2\) have opposite orientations then there exist \(v,w\in V\setminus\{0\}\) such that \(J_1(v)= J_2(v)\), \(J_1(w)=-J_2(w)\); (2) if \(n\) is odd and the orientation of \(J_1,J_2\) is the same (resp. opposite) then there exists \(v\in V\setminus\{0\}\) such that \(J_1(v)= J_2(v)\) (resp. \(J_1(v)=-J_2(v))\).
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 1420049
zbMATH Keywords: invariant subspaces
zbMATH Keywords: complex structure
zbMATH Keywords: orientation
zbMATH Keywords: Hermitian structure
zbMATH Keywords: Euclidean space
reviewed by: V. G. Skobelev
MaRDI profile type: Publication
Wikidata QID: Q127633020
description: scientific article; zbMATH DE number 1420049