Spectra of operators with Bishop's property $(\beta)$

Let $X$ be a Banach space and let $\mathcal{A}(X)$ be the class that consists of all operators $T\in\mathcal{L}(X)$ such that for every $\lambda\in\mathbb{C}$, the range of $(T-\lambda I)$ has a finite-codimension when it is closed. For an integer $n\in\mathbb{N}$, we define the class $\mathcal{A}_{n}(X)$ as an extension of $\mathcal{A}(X)$. We then study spectral properties of such operators, and we extend some known results of multi-cyclic operators with $(\beta)$.