Spectral Theory and Applications
Alexandre Girouard · Contemporary mathematics - American Mathematical Society · 2018
This thesis presents some results in spectral theory for bounded and unbounded linear operators, as well as results that are helpful in spectral analysis. The key concepts are essential and discrete spectrum along with spectral measures, functions of operators and spectral projections. The most important statements contained in this thesis are the spectral theorem about representing a self-adjoint operator as a simple L2 multiplication operator, and Weyl's essential spectrum theorem on the invariance of essential spectrum under suitable perturbations. There are also a few results that can be used to immediately decide the finiteness or infiniteness of the discrete spectrum for quite general types of Hamiltonian operators, and similar qualitative results for the absolutely continuous and essential spectrum types. The reader is assumed to have knowledge in functional analysis, real analysis, and elementary complex analysis.