Let \({\mathcal{H}}\) and \({\mathcal{K}}\) be complex infinite-dimensional separable Hilbert spaces and \({\mathcal{B}}({\mathcal{K}},{\mathcal{H}})\) be the algebra of all bounded linear operators from \({\mathcal{K}}\) into \({\mathcal{H}}\) . Given \(A\in {\mathcal{B}}({\mathcal{H}})\) , \(B\in {\mathcal{B}}({\mathcal{K}})\) and \(C\in {\mathcal{B}}({\mathcal{K}},{\mathcal{H}})\) , we denote by \(M_{C}=\left( \begin{array}{cc} A & C \\ 0 & B \\ \end{array} \right)\) the upper triangular operator matrix acting on \({\mathcal{H}}\oplus {\mathcal{K}}\) . In this paper, we give the characterization on the existence of \(C\in {\mathcal{B}}({\mathcal{K}},{\mathcal{H}})\) such that \(M_C\) to be upper semi-Fredholm with fixed nullity and to be Fredholm with fixed index, respectively. Besides, we also show that the existence of invertible \(C_0\in {\mathcal{B}}({\mathcal{K}},{\mathcal{H}})\) such that \(M_{C_0}\) is a CI operator(resp. CW operator) is equivalent with \(M_0\) is a CI operator (resp. CW operator).
