The intersection of essential approximate point spectra of operator matrices

When A ∈ B(H) and B ∈ B(K) are given, we denote by MC the operator acting on the infinitedimensional separable Hilbert space H ⊕ K of the form MC = A C 0 B . In this paper, it is shown that there exists some operator C ∈ B(K,H) such that MC is upper semi-Fredholm and ind(MC) 0 if and only if there exists some left invertible operator C ∈ B(K,H) such that MC is upper semi-Fredholm and ind(MC) 0. A necessary and sufficient condition forMC to be upper semi-Fredholm and ind(MC) 0 for some C ∈ Inv(K,H) is given, where Inv(K,H) denotes the set of all the invertible operators of B(K,H). In addition, we give a necessary and sufficient condition forMC to be upper semi-Fredholm and ind(MC) 0 for all C ∈ Inv(K,H). © 2005 Elsevier Inc. All rights reserved