Local variational min-sup characterizations are presented for the real spectrum of a selfadjoint operator pencil on a separable Hilbert space \(H\). Instead of minimizing over all subspaces of fixed codimensions as in classical results the new characterizations minimize over subspaces that are close to extremal subspaces. Particularly the following theorem is proved.
Theorem. Let \(\lambda_ 0\) be an \(A\)-positive eigenvalue of the operator pencil \(P: \lambda\to\lambda A-B\), \[ Bx=\lambda_ 0 Ax,\qquad x\in C_ +=\{x\in H\mid\;(Ax,x)>0\}, \] let \(E\) be the resolution of the identity for \(\lambda_ 0 A-B\) and \[ S_ 0={\mathfrak R}(E_ 0), \quad T_ 0={\mathfrak R}(I-E_ 0), \quad E_ 0=E(]-\infty,0[). \] Then there is \(\varepsilon>0\) such that \(\lambda_ 0=\min_{S\in N_ \varepsilon(S_ 0,T_ 0)} \sup_{x\in S\cap C_ +} (Bx,x)/(Ax,x)\).
Here \(S_ 0\), \(T_ 0\) are closed subspaces with orthonormal bases \(\{e_ n\}_{n=0}^{+\infty}\), \(\{e_ n\}_{-n=1}^{+\infty}\), \(S_ 0+T_ 0=H\), \[ N_ \varepsilon (S_ 0,T_ 0)=\{\text{span}\{e_ n+e_ n'\}_{n=0}^{+\infty}:\;e_ n'=\sum_{m=- \infty}^{+\infty}\varepsilon_{mn}e_ m,\;\sum_{m=- \infty}^{+\infty} |\varepsilon_{mn}|^ 2<\varepsilon\}. \]