Some approximation methods for the equation \(Ax=y\) with a linear bounded operator \(A\) on a Hilbert space \(H\) are considered. An approximate method for \(A\) is treated as a sequence \(\{A_n\}\), \(A_n\in L(H)\). Two classes of sequences are investigated: Moore-Penrose and Fredholm sequences which correspond to normally solvable and Fredholm operators, respectively. Approximation methods are investigated in the frame of \(C^*\)-algebra theory of bounded sequences of linear bounded operators on \(H\).
The authors give a characterization of Moore-Penrose invertibility in a \(C^*\)-algebra which yields to the one of the main results of the paper on the behaviour of singular values of elements of Moore-Penrose sequences \(\{A_n\}\). For a Fredholm sequence the index and \(\alpha\)-number are introduced. The authors prove relations between the index and the \(\alpha\)-number of a Fredholm sequence and some other properties of this sequence, in particular, the asymptotic distribution of the singular values of \(A_n\).