Continuous spectrum, point spectrum and residual spectrum of operator matrices

Let MC denote a 2 × 2 upper triangular operator matrix of the form , which is acting on the sum of Banach spaces X⊕Y or Hilbert spaces H⊕K. In this paper, the sets and ⋂C∈B(K,H)σr(MC) are, respectively, characterized completely, where σc(·) denotes the continuous spectrum, σp(·) denotes the point spectrum and σr(·) denotes the residual spectrum. Moreover, some corresponding counterexamples are given.