Fredholm consistency of upper-triangular operator matrices

In this paper, for given operators A\in\mathcal B(\mathcal H) and B\in\mathcal B(\mathcal K) , we characterize the set of all C\in\mathcal B(\mathcal K,\mathcal H) such that the operator matrix M_C= \left[ {\begin{array}{cc} A & C \\ 0 & B \\ \end{array} } \right] is Fredholm consistent. We completely describe the sets \bigcap_{C\in \mathcal B(\mathcal K,\mathcal H)}\sigma_{\mathrm{FC}}(M_C) and \bigcup_{C\in \mathcal B(\mathcal K,\mathcal H)}\sigma_{\mathrm{FC}}(M_C) . Also, we prove that \bigcap_{C\in \mathcal B(\mathcal K,\mathcal H)}\sigma_{\mathrm{FC}}(M_C)=\sigma_{\mathrm{FC}}(M_0) .