On the Courant--Fischer theory for Krein spaces
If \(\mathbb C^n\) is equipped with a positive definite inner product \([\cdot,\cdot]\) and \(A\) is a Hermitian \(n\times n\)-matrix (i.e., \([Ax,x] = [x,Ax]\)), then, by the Courant--Fischer minimax principle, the values \[ \min_{\dim S=k}\max_{x\in S}R(x)\quad\text{with the Rayleigh ratio}\quad R(x):=\frac{[Ax,x]}{[x,x]}\tag{1} \] give the eigenvalues of \(A\). The present paper generalizes this result to a situation where \([\cdot,\cdot]\) is not positive definite anymore, but a Krein space inner product on \(\mathbb C^n\). In this case, according to a previous paper by the authors, additional conditions must be imposed assuring that the values of \(R(x)\) are semibounded at least for all \(x\) with \([x,x] > 0\) or for all \(x\) with \([x,x] < 0\). In particular, it is required that the spectrum is real and all eigenvalues of positive and of negative type (i.e., with a positive or negative eigenspace) are not ``interlacing'' (i.e., one group of eigenvalues is on the left of the other). Additionally, only one eigenvalue with a Jordan chain is allowed (increasing the difficulties). Here perhaps, the terminology of definitizable operators in Krein spaces and their critical points could have been helpful. The authors obtain characterizations of the eigenvalues by expressions like (1) with restrictions to \(x \in S\) where \([x,x] > 0\) (or \([x,x] < 0\), respectively). Finally, the results are applied to Hamiltonian dynamics. Note that there is a certain overlap with earlier results [cf.\ \textit{P.\,Binding, B.\,Najman} and \textit{Q.\,Ye}, Integral Equations Oper.\ Theory 35, No.\,4, 398--422 (1999; Zbl 0971.47008) and some of the references there].
scientific article; zbMATH DE number 5529492
scientific article