Essential approximate point spectra and Weyl's theorem for operator matrices

When A E B(H) and B E B(K) are given, we denote by MC the operator acting on the infinite dimensional separable Hilbert space H circle plus K of the form M-C = ((AC)(0B)). In this paper, it is shown that a 2 x 2 operator matrix MC is upper semi-Fredholm and ind(M-C) <= 0 for some C is an element of B(K, H) if and only if A is upper semi-Fredholm and [GRAPHICS] We also give the necessary and sufficient conditions for which M-C is Weyl or M-C is lower semi-Fredholm with nonnegative index for some C is an element of B(K, H). In addition, we explore 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) 2004 Elsevier Inc. All rights reserved.