Spectral Decomposition of Bounded Self-Adjoint Operators

This lecture covers the spectral decomposition of a self-adjoint operator defined on a separable Hilbert space. By applying functional calculus, the space is decomposed into invariant subspaces. The process involves defining successive subspaces and demonstrating the existence of measures that lead to an isomorphism between the Hilbert space and a space of square-integrable functions. The lecture concludes with the spectral decomposition theorem for bounded self-adjoint operators, which involves a family of measures, an isomorphism mapping, and the multiplication operator. The proof involves expressing elements of the Hilbert space as a sum of components, each associated with a function in the square-integrable space. The lecture also explores the convergence properties and the relationship between the spectral decomposition and diagonalization in finite dimensions.