Characterizations of perturbations of spectra of 2 x 2 upper triangular operator matrices

When A epsilon B(H) and B epsilon B(K) are given, we denote by M-C the operator acting on the infinite dimensional separable Hilbert space H circle plus K of the form M-C = ((A)(0) (C)(B)). In this paper, we first give some necessary and sufficient conditions for M-C to be a left invertible operator (an upper semi-Weyl, upper semi-Fredholm) operator for some C epsilon B(K, H), which extend the corresponding results in Cao et al. (2006) [4], Cao and Meng (2005) [5], Hwang and Lee (2001) [12] and Li and Du (2006) [15]. Then we present some counter-examples. (C )2012 Elsevier Inc. All rights reserved.