Compact Operators: Properties and Theorems

This lecture covers the properties and theorems related to compact and relatively compact operators, including the RAGE theorem and the Kato-Rellich theorem. The slides discuss Wiener's theory, the concept of relatively bounded operators, and the generalization of results to compact operators. The lecture also delves into the definition of finite-rank operators and the compactness of operators. Perturbation theory for self-adjoint operators is introduced, emphasizing the robustness under appropriate perturbations. The characterization of spectral subspaces and the concept of relative boundedness are explored, along with the application of perturbation theory to self-adjoint operators.