The spectra of the 2 x 2 upper triangular operator matrix MC = (A C 0 B )
acting on a Hilbert space H1 ? H2 are investigated. We obtain a necessary
and sufficient condition of ?(MC) = ?(A)??(B) for every C ? B(H2,H1), in
terms of the spectral properties of two diagonal elements A and B of MC.
Also, the analogues for the point spectrum, residual spectrum and continuous
spectrum are further presented. Moveover, we construct some examples
illustrating our main results. In particular, it is shown that the inclusion
?r(MC) ? ?r(A) ? ?r(B) for every C ? B(H2,H1) is not correct in general. Note
that ?(T) (resp. ?r(T)) denotes the spectrum (resp. residual spectrum) of an
operator T, and B(H2,H1) is the set of all bounded linear operators from H2
to H1.