On the Approximation to Solutions of Operator Equations by the Least Squares Method

We consider the equation $Au=f$, where $A$ is a linear operator with compact inverse $A^{-1}$ in a separable Hilbert space $\mathfrak{H}$. For the approximate solution $u_n$ of this equation by the least squares method in a coordinate system $\{e_k\}_{k\in\mathbb{N}}$ that is an orthonormal basis of eigenvectors of a self-adjoint operator $B$ similar to $A$ ($\mathcal{D}(B)=\mathcal{D}(A)$), we give a priori estimates for the asymptotic behavior of the expressions $r_n=\|u_n-u\|$ and $R_n=\|Au_n-f\|$ as $n\to\infty$. A relationship between the order of smallness of these expressions and the degree of smoothness of $u$ with respect to the operator $B$ is established.