Spectra of quasisimilar operators

Abstract

We show that whenever densely similar operators on a Banach spaces, their approximate point spectra must have non-empty intersection. Also, we introduce the class $\mathcal A$ that consists of those operators for which the Goldberg spectrum coincides with the right essential spectrum. We study spectral properties of quasisimilar operators satisfying Bishop's property $(\beta)$ in the class $\mathcal A$. Finally, as an application to the class $\mathcal N$ that consists of those operators $T$ whose range $R(T)$ is contained in the linear span of finite number of orbits of $T$, we show that any two quasisimilar operators such that are in $\mathcal N$ and satisfying property $(\beta)$ must have the same approximate point spectrum.