The problem of a spectral completion \(T=\left(\begin{smallmatrix} A & B\\ X & Y \end{smallmatrix}\right)\) of the partially specified operator matrix \(\left(\begin{smallmatrix} A & B\\ ? & ? \end{smallmatrix}\right)\), where \(X\) and \(Y\) are compact operators, is analyzed. If \( \Omega\) is a given open set containing \(0\) and every component of \(\Omega\) is simply connected then a sufficient and necessary condition for the existence of compact operators \(X\) and \(Y\) is found so that \(\sigma(T)\) is contained in \(\Omega\). An application of the main results concerning the power stabilizability to the theory of discrete time linear systems where \(X\) and \(Y\) are infinite dimensional is given.