On condition of quasisimilar, quasinormal and pure dominant operators in existence of essential spectra in an infinite dimensional Hilbert spaces

The problem of finding conditions of quasisimilar, pure dominant operators in connection to related essential spectrum has been considered by several authors. In this paper, we show that quasisimilar pure dominant operators have their essential spectra equal to their spectra provided one of the interfering quasiaffinities is compact. We will consider T as a pure dominant operator, as a compact operator having dense range and let so that we can investigate the conditions of the spectrum of and essential spectrum of In this study an effort will be made to give relevant examples to illustrate conditions of pure parts and hence deduce results of equality of essential spectra.