Functional Calculus: Operator Definition and Properties

This lecture covers the definition of the functional calculus for a self-adjoint and bounded operator A, based on the properties of the spectrum. It explains the process of defining f(A) for f € C(σ(A), R) using a sequence of real coefficient polynomials that converge uniformly on σ(A) to f. The lecture also discusses the convergence of sequences of polynomials to f(A) in L(H) and the properties of the norm. The theorem of Stone-Weierstrass is presented, along with the concept of morphism unitary and isometric operators in L(H) for a self-adjoint operator A. Various inequalities and properties related to the functional calculus are explored.