Normal generalized selfadjoint operators in Krein spaces
A linear and bounded operator \(A\) on the Krein space \((H,[.,.])\) is said to be \(n\)-selfadjoint if \[ \sum^n_{k=0} (-1)^k{n\choose k} A^{*k} A^{n-k}= 0, \] where \(A^*\) stands for the Krein adjoint of \(A\), i.e. \([Ax, y]= [x,A^* y]\). The main theorem gives a plain characterization of the normal \((A^*A= AA^*)\) and \(n\)-selfadjoint operators, namely \(A= T+ N\), with \(T= T^*\), \(N= -N^*\), \(TN= NT\), and \(N^m= 0\), where \(m= \min(2q+ 1,n)\), and \(q\) denotes the defect of \(H\). In particular, in Hilbert spaces \(q=0\), hence normal and \(n\)-selfadjoint operators are simply selfadjoint. The paper also contains properties of the classes of \(n\)-selfadjoint operators.
scientific article; zbMATH DE number 1370994