Operator and Spectral Theory for AI: Operators, Spectra, Resolvents, Semigroups, and Infinite-Dimensional Learning

Modern AI systems increasingly operate through objects that are better understood as operators than as finite matrices: convolution and state-space layers, learned dynamics, kernel and covariance maps, neural operators, implicit models, and discretised approximations of continuum systems. Their behavior cannot always be read from eigenvalues alone.

Operator and Spectral Theory for AI develops the operator-theoretic language needed to reason about these systems rigorously. Beginning with bounded and unbounded operators, spectra, resolvents, compactness, Fredholm theory, functional calculus, and semigroups, the book builds toward perturbation theory, nonnormality and pseudospectra, spectral approximation and pollution, inverse problems, kernel and covariance operators, Koopman methods, operator learning, and modern sequence and implicit architectures.

Throughout, the emphasis is not only on what a theorem says, but on when a finite, learned, or discretised model genuinely represents an underlying operator. The text connects classical theory to practical questions in model development: stability, long-horizon amplification, low-rank adaptation, representation geometry, discretisation transfer, learned dynamics, and residual-based verification.

Designed for graduate students and researchers in mathematical machine learning, scientific machine learning, operator learning, inverse problems, dynamical systems, and AI theory, this volume provides a rigorous bridge between classical operator and spectral theory and the finite approximations used in modern AI.