We also study the behaviour of the r-pseudoinverse when the original operator is perturbed inverse (orthogonal generalized inverse, pseudoinverse) operator for a compact operator that acts from the separable Hilbert space X to the separable Hilbert space Y using the concept of the generalized normal r-solution (solution rank r) which was introduced in [2] for systems of linear equations. In this paper we suggest and substantiate the approach to the approximation of the generalized We suggest and substantiate the approximation of the r-pseudoinverse operator by the projection method. We have obtained the estimates that characterize the deviation of the r-solution when there are errors in the right-hand side.