Characterizations of perturbations of spectra of 2×22×2 upper triangular operator matrices

When A∈B(H)A∈B(H) and B∈B(K)B∈B(K) are given, we denote by MCMC the operator acting on the infinite dimensional separable Hilbert space H⊕KH⊕K of the form MC=(AC0B). In this paper, we first give some necessary and sufficient conditions for MCMC to be a left invertible operator (an upper semi-Weyl, upper semi-Fredholm) operator for some C∈B(K,H)C∈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.