On stabilized point spectra of multivalued systems (Q2494140)

From the author's abstract: Let \(H\) be an infinite-dimensional Hilbert space. Denote by \(\Lambda(E,F)\) the set of all \(\lambda \mathcal{R}\) for which the multivalued system \(0\in (F-\lambda E)(x)\) admits a nonzero solution \(x\in H\). Here, \(E:H\to H\) is a linear bounded operator while \(F:H\multimap H\) is such that \(gr F\) is a nonempty closed cone. One says that \(\Lambda(E,F)\) is the point spectrum of the pair \((E,F)\). It is well-known that \(\Lambda(E,F)\) does not behave in a stable manner with respect to perturbations in the argument \((E,F)\). The purpose of this note is to study the outer-semicontinuous hull (or graph closure) of the mapping \(\Lambda\). So, the stabilized point spectrum \(\Sigma(E,F):=\limsup\limits_{(D,G)\to (E,F)} \Lambda(D,G)\) is examined. At first, for every \((E,F)\), the equality \[ \Sigma_E(F)=\limsup\limits_{G\to F} \Lambda(E,F) = \Sigma(E,F) \] is checked. Moreover, \(\Lambda(E,F)\) is characterized in terms of \(\varepsilon\)-eigenvalues or `discriminant functions' \(\Phi(\lambda,E,F)\) and \(\Psi(\lambda,E,F)\). It is proved that they are continuous and Lipschitz continuous with respect to each couple of variables. The results obtained for the point spectrum are extended to the case of the Polynomial Eigenvalue Problem \[ \exists \lambda\in \mathcal{R}\;\exists x\in H\setminus \{0\} \;: \;[\lambda E_1+\ldots +\lambda^m E_m]x\in F(x). \]