Spectral Analysis of Nonlinear Operators: Theoretical Foundations and Applications
Keywords:
Nonlinear spectral theory, maximal monotone operators, Yosida approximation, Fitzpatrick function, nonexpansive mappings, Krasnoselskii iteration, neural networks, spectral graph theoryAbstract
We review the theory and applications of spectral analysis of nonlinear operators. Classical linear spectral theory does not extend directly to nonlinear maps. Over the past decades, researchers have developed nonlinear analogues of eigenvalues, spectra and spectral measures. Key approaches include the Yosida approximation for maximal monotone operators, Fitzpatrick function techniques, and iterative schemes for nonexpansive maps. Other frameworks define nonlinear spectrum via fixed-point and bifurcation concepts (Neuberger, Rhodius, Kachurovskij, and more recent “spectrum at a point” definitions). We summarize major definitions and theorems: for example, every continuous map f has a spectrum σ(f,p) at a point p that is closed and shares many properties with linear spectrum. Maximal monotone operators admit a spectral resolution by Yosida approximations A_λ =〖(I+λA)〗^(-1). For nonexpansive maps T, the Krasnoselskii iteration x_(n+1) =(x_n +〖Tx〗_n )/2 converges to a fixed point in the spectrum. We discuss modern applications, including bounds on Lipschitz networks (ReLU nets), spectral clustering with graph Laplacians, and integral operators. We include examples and experiments using publicly available tools (e.g. computing eigenvalues for p-Laplacian or graph Laplacians in Python). This survey is intended to be comprehensive and accessible, explaining concepts and results with minimal jargon.
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