Weyl's theorem for upper triangular operator matrices

Let σab(T)={λ∈C:T-λIisnotanuppersemi-Fredholmoperatorwithfiniteascent} be the Browder essential approximate point spectrum of T ∈ B(H) and let σd(T)={λ∈C:T-λIisnotsurjective} be the surjective spectrum of T. In this paper it is shown that if MC=AC0B is a 2 × 2 upper triangular operator matrix acting on the Hilbert space H ⊕ K, then the passage from σab(A) ∪ σab(B) to σab(MC) is accomplished by removing certain open subsets of σd(A) ∩ σab(B) from the former, that is, there is equalityσab(A)∪σab(B)=σab(MC)∪G,where G is the union of certain of the holes in σab(MC) which happen to be subsets of σd(A) ∩ σab(B). Weyl's theorem and Browder's theorem are liable to fail for 2 × 2 operator matrices. In this paper, it also explores how Weyl's theorem, Browder's theorem, a-Weyl's theorem and a-Browder's theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.