A note on meromorphic operators

Let $X$ be a complex Banach space and $T$ a bounded linear operator on $X$. $T$ is called meromorphic if the spectrum $\sigma(T)$ of $T$is a countable set, with $0$ the only possible point of accumulation, such that all the nonzero points of $\sigma(T)$ are poles of $(\lambda I-T)^{-1}$. By means of the analytical core $K(T)$ we give a spectral theory of meromorphic operators. Our results are a generalization of some results obtained by Gong and Wang (2003).