Abstract

 In this paper we introduce some class of bounded linear oper- ators and we show that if T is a normal operator on the separable Hilbert space and belongs to this class, then the spectrum of T coincides to the eigenvalues of T. We generalized spectral theorem for compact normal op- erators to normal operators that have countable spectra with converges to zero. We give conditions that for which two normal operators T and S that TS and ST are normal operators.