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.