This paper deals with Fredholm properties of the one-sided coupled operator matrix \({\mathcal {M}}=\left( \begin{array}{cc} A &{} B \\ 0 &{} D \end{array} \right) \left( \begin{array}{cc} I &{} 0 \\ L &{} I \end{array} \right)\) by means of generalized Schur factorization and the associated space decompositions. For \(\lambda \in {\mathbb {C}},\) some sufficient conditions are given for \(\lambda -{\mathcal {M}}\) to be Fredholm (resp. left or right Fredholm), and these conclusions are further used to determine the essential spectra of a delay equation and a wave equation with acoustic boundary conditions.
