Weyl’s theorem for upper triangular operator matrices Original Research Article

Let image be the Browder essential approximate point spectrum of T set membership, variant B(H) and let image be the surjective spectrum of T. In this paper it is shown that if image is a 2 × 2 upper triangular operator matrix acting on the Hilbert space H circled plus K, then the passage from σab(A) union or logical sum σab(B) to σab(MC) is accomplished by removing certain open subsets of σd(A) ∩ σab(B) from the former, that is, there is equalityimagewhere image 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.