Spectral Theory of Normal Operators

U ovom radu proučavamo normalne operatore i njihova spektralna svojstva. U prva tri poglavlja navest ćemo osnovne pojmove, definirati projektore i spomenuti njihova svojstva te objasniti vezu normalnog operatora s dekompozicijom jedinice. Nakon toga, Cayleyevim transformacijama povezat ćemo hermitske i unitarne operatore te proširiti pojam adjungiranog operatora. U posljednja dva poglavlja objasnit ćemo konstrukciju projektora na područje vrijednosti operatora te upoznati se s pojmom kvazi inverznog operatora i njegovom primjenom kod rješavanja problema najmanjih kvadrata.In this bachelor’s thesis we will consider normal operators and their spectral properties. In the first three chapters we will investigate basic concepts, define projections and mention their properties. After that, we will connect Hermitian and unitary operators by Cayley transformations, and expand the concept of an adjoint operator. In the last two chapters the construction of a projection onto the image of an operator will be explained, and the concept of a generalized inverse and its usage in solving the least squares problem will be introduced