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

When A ∈ B(H) and B ∈ B(K) are given, we denote by MC the operator acting on the infinite dimensional separable Hilbert space H ⊕K of the form MC = A C 0 B . In this paper, it is shown that a 2×2 operator matrix MC is upper semi-Fredholm and ind(MC) 0 for some C ∈ B(K,H) if and only if A is upper semi-Fredholm and n(B) <∞and n(A)+ n(B) d(A) +d(B) or n(B) = d(A)=∞, if R(B) is closed, d(A)=∞, if R(B) is not closed. We also give the necessary and sufficient conditions for which MC is Weyl or MC is lower semi-Fredholm with nonnegative index for some C ∈ 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 × 2 upper triangular operator matrices on the Hilbert space.  2004 Elsevier Inc. All rights reserved.