Variational principles for real eigenvalues of self-adjoint operator pencils
Let \(L(\lambda)\) be an operator-function defined on some real interval \([\alpha,\beta]\), whose values are bounded selfadjoint operators on a Hilbert space, and which is continuously differentiable in the operator norm. It is known [see \textit{A. S. Markus}, ``Introduction to the Spectral Theory of Polynomial Operator Pencils'', AMS, Transl. of Math. Monographs, Vol. 71 (1980; Zbl 0678.47005)] that variational principle given by the formulae \[ \lambda_j=\min_{S\in S_j} \max_{0\not=x\in S}p(x), \qquad \lambda_j'=\max_{S\in S_j} \min_{0\not=x\in S} p(x), \] where \(p(x)\) is the generalized Rayleigh functional, \(\lambda_j\) and \(\lambda_j'\) are the eigenvalues of \(L\) below or above, respectively, the essential spectrum of \(L\) in \((\alpha,\beta)\), is valid if a) \(L(\alpha)\gg 0\), i.e. \(L(\alpha)\geq\varepsilon I\) (\(\varepsilon>0\)), b) \(L(\beta)\ll 0\), c) each of the scalar functions \(\varphi_x:=(L(\lambda)x,x)\), \(x\not=0\), has exactly one simple zero \(p(x)\) on \((\alpha,\beta)\). The authors show that the variational principles for a pencil \(L\) remain valid under much less restrictive conditions then assumptions a)--c).
scientific article; zbMATH DE number 1542409
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