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Rad započinjemo razmatranjem ograničenih operatora na Hilbertovim prostorima, te njihovih spektralnih svojstva. Posebno nas zanimaju sektralne reprezentacije hermitskih i unitranih operatora. Zatim uvodimo pojam neograničenih operatora u Hilbertovim prostorima, te promatramo neke njhove posebne klase. Nakon toga proučavamo spektralna svojstva i spektralnu reprezentaciju neograničenih operatora. Posebno promatramo svojstva operatora množenja i deriviranja na \(L^{2}\)(\({- \infty }, {+\infty }\)). Ove objekte i njihova svojstva proučavamo unutar teorije Hilbertovih operatora, te kada je potrebno koristimo poznate teoreme iz teorije mjere., We start the theses by considering bounded operators on Hilbert spaces, as well as their spectral properties. We examine in detail spectral representations of self-adjoint and unitary operators. We Then introduce unbounded operators in Hilbert spaces and analyze some special classes of these operators. We also study spectral properties and spectral representations of unbounded operators, as well as the properties of multiplication and differentiation operators on \(L^{2}\)(\({- \infty }, {+\infty }\)). We study these objects within the theory of Hilbert spaces, using standard theorems from measure theory when neccesary.