Invariant subspaces of two Hermitian structures on a Euclidean space
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. \(J_1(v)=-J_2(v))\).
scientific article; zbMATH DE number 1420049