Abstract:
This paper presents a nonlinear spectral framework for analyzing monotone and nonexpansive operators in
Banach and Hilbert spaces. We construct a nonlinear spectral resolution for maximal monotone operators using
Yosida approximations and Fitzpatrick functions, leading to a family of nonlinear projections and an associated
spectral measure. For nonexpansive mappings, we establish an iterative spectral approximation based on
Krasnoselskii iterations, with proven convergence and recovery of nonlinear eigenvectors. We further extend
this framework to ReLU-based neural networks, analyzing spectral bounds, depth-dependent scaling, and
gradient alignment. These results bridge nonlinear operator theory and neural architectures, offering new tools
for theoretical analysis and applications in optimization, physics, and machine learning.
