The Spectrum: Operators and Observables in Quantum Physics

This lecture introduces the concept of the spectrum in quantum physics, where observables are represented by self-adjoint operators. Unlike classical physics, not all observables have simultaneous definite values due to non-commutativity. The lecture covers the invertibility of operators, the openness of the set of invertible operators, and the definition of the spectrum of an operator. Examples are provided to illustrate the spectrum in finite and infinite-dimensional Hilbert spaces. The lecture also discusses cases where an operator is not a bijection from H to H, distinguishing between injective and surjective properties. The concept of point, continuous, and residual spectra is explained, along with the relationship between eigenvalues and eigenvectors.