Properties of the Spectrum

This lecture covers the properties of the spectrum of an operator in L(H). It discusses the compactness of the spectrum within the disk D(0,||A||) and explores the inclusion of the boundary or parts of the boundary in the spectrum. The concept of the spectral radius is introduced, showing its relationship with the norm of the operator. The lecture also delves into the case of self-adjoint operators, demonstrating that -||A|| or ||A|| belongs to the spectrum. Various mathematical proofs are presented to support these properties, including the Cauchy-Schwarz inequality. The lecture concludes with theorems related to unitary and self-adjoint operators in L(H).