A Linear-Algebra-First Spectral Framework for Compact Operator Equations of the Form (I − λT)u = f Solvability, Resonance, and Constructive Representations via EigenCoordinates

Abstract

This paper develops a proof-driven, linear-algebra-first approach to the operator equation (I − λT)u = f,where T is a compact operator acting on a Hilbert space of functions. Our guiding viewpoint is that many analytic problems become structurally transparent once written as operator analogues of finite-dimensional linear systems. Working primarily in L 2 ([a, b]), we adopt a kernel-based model in which T is induced by a Hilbert–Schmidt integral operator and, under symmetry of the kernel, is compact self-adjoint. In this regime the spectral theorem yields a diagonalization-like expansion T u = P n µn⟨u, en⟩en, reducing (I − λT)u = f to an infinite diagonal coefficient system. We establish: (i) a non-resonance invertibility criterion and a constructive eigen-expansion formula for the solution; (ii) a Fredholm-type alternative stated in linear-algebraic language with explicit compatibility conditions at resonance; (iii) a Neumann-series criterion |λ| ∥T∥ < 1 providing an independent constructive inversion route together with explicit tail/error estimates. Beyond solvability, we derive sharp norm bounds in terms of spectral separation, give perturbation inequalities for stability of solutions under
operator/kernel perturbations, and present fully computed examples including rank-one kernels and a Green-kernel reformulation of a Dirichlet boundary-value problem. The results unify matrix diagonalization philosophy with compact operator spectral analysis and provide a practical blueprint for both rigorous proofs and computable approximations via spectral truncation and Galerkin projection.