The authors prove that a separable Banach space \(X\) admits a representation as a sum \(X= X_1+ X_2\) of two infinite-codimensional closed subspaces \(X_1\) and \(X_2\) if and only if it is a sum \(X= A_1(Y_1)+ A_2(Y_2)\) of two infinite-codimensional operator ranges. In this case, \(X\) admits a representation of operator ranges \(X= T_1(Z_1)+ T_2(Z_2)\) such that neither range \(T_1(Z_1)\), \(T_2(Z_2)\) contains an infinite-dimensional closed subspace iff \(X\) does not contain \(\ell_1\) isomorphically.