## A survey on variational characterizations for nonlinear eigenvalue problems

Jörg Lampe and Heinrich Voss

### Abstract

Variational principles are very powerful tools when studying self-adjoint linear operators on a Hilbert space $\mathcal H$. Bounds for eigenvalues, comparison theorems, interlacing results, and monotonicity of eigenvalues can be proved easily with these characterizations, to name just a few. In this paper we consider generalizations of these principles to families of linear, self-adjoint operators depending continuously on a scalar in a real interval.

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### Key words

nonlinear eigenvalue problem, variational characterization, iterative projection methods, AMLS, quantum dots, viscoelastic damping, total least-squares problems, fluid-solid interaction

### AMS subject classifications

35P30, 47A52, 47A75, 47J10, 65F15, 65F17

### Links to the cited ETNA articles

 [47] Vol. 28 (2007-2008), pp. 149-167 Julianne Chung, James G. Nagy, and Dianne P. O'Leary: A weighted-GCV method for Lanczos-hybrid regularization [118] Vol. 40 (2013), pp. 82-93 Aleksandra Kostić and Heinrich Voss: On Sylvester's law of inertia for nonlinear eigenvalue problems [131] Vol. 31 (2008), pp. 12-24 Jörg Lampe and Heinrich Voss: A fast algorithm for solving regularized total least squares problems [135] Vol. 42 (2014), pp. 13-40 Jörg Lampe and Heinrich Voss: Large-scale dual regularized total least squares [173] Vol. 13 (2002), pp. 106-118 Volker Mehrmann and David Watkins: Polynomial eigenvalue problems with Hamiltonian structure [214] Vol. 36 (2009-2010), pp. 113-125 Markus Stammberger and Heinrich Voss: On an unsymmetric eigenvalue problem governing free vibrations of fluid-solid structures

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