$r$-pseudoinverse for a compact operator

The notion of a generalized $r$-solution for a system of linear algebraic equations is modified for approximation of the generalized inverse for a compact linear operator from a separable Hilbert space $X$ into a separable Hilbert space $Y$. Approximation of the $r$-pseudoinverse by projection method is proposed and justified. Stability of the results with respect to data perturbations and approximation error of the operator is analyzed.