Weyl's theorem for upper triangular operator matrices

Let sigma(ab) (T) = {lambda is an element of C : T - lambda I is not an upper semi-Fredholm operator with finite ascent} be the Browder essential approximate point spectrum of T is an element of B(H) and let sigma(a) (T) = {lambda is an element of C : T - lambda I is not surjective} be the surjective spectrum of T. In this paper it is shown that if M-C = ((A)(0) (C)(B)) is a 2 x 2 upper triangular operator matrix acting on the Hilbert space H circle plus K, then the passage from sigma(ab)(A) boolean OR sigma(ab)(B) to sigma(ab)(M-C) is accomplished by removing certain open subsets of sigma(ab)(A) boolean AND sigma(ab)(B) from the former, that is, there is equality sigma(ab)(A) boolean OR sigma(ab)(B) = sigma(ab)(M-C) boolean OR G, where W is the union of certain of the holes in sigma(ab)(M-C) which happen to be subsets of sigma(ab)(A) boolean AND sigma(ab)(B). Weyl's theorem and Browder's theorem are liable to fail for 2 x 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 x 2 upper triangular operator matrices on the Hilbert space. (c) 2005 Elsevier Inc. All rights reserved.