In this paper, we extent the classical spectral approximation theory for compact and bounded operators to general linear operators, and then apply it to polynomial eigenvalue problems (PEP). We also study the essential spectrum in PEPs, and prove that this spectrum is stable under relatively compact perturbations. Based on this analysis, we give some suggestions to make algorithms for solving PEPs more efficient.
Lu, Z., van der Vegt, J. J. W., & Xu, Y. (2018). Spectral approximation for polynomial eigenvalue problems. Computers and mathematics with applications, 76(5), 1184 - 1197. https://doi.org/10.1016/j.camwa.2018.06.007