Mbekhta's subspaces and a spectral theory of compact operators

Let $A$ be an operator on an infinite-dimensional complex Banach space. By means of Mbekhta's subspaces $H_{0}(A)$ and $K(A)$, we give a spectral theory of compact operators. The main results are: Let $A$ be compact. $1$. The following assertions are all equivalent: (1) 0 is an isolated point in the spectrum of $A;$(2) $K(A)$ is closed; (3) $K(A)$ is of finite dimension; (4) $K(A^{\ast })$ is closed; (5) $K(A^{\ast })$ is of finite dimension; $2$. sufficient conditions for $0$ to be an isolated point in $\sigma (A)$; $3$. sufficient and necessary conditions for $0$ to be a pole of the resolvent of $A$.