Mbekhta’s subspaces and a spectral theory of compact operators

Abstract

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*) is closed; (5) K(A*) is of finite dimension; 2. sufficient conditions for 0 to be an isolated point in σ(A); 3. sufficient and necessary conditions for 0 to be a pole of the resolvent of A.