On the Regularization Method to Stable Approximate Solution of Equations of the First Kind with Unbounded Operators

Let $A:D\left(A\right)？X\to Y$ be a linear, closed, densely defined unbounded operator, where $X$ and $Y$ are Hilbert spaces. Assume that $A$ is not boundedly invertible. If equation (1) $Au=f$ is solvable, and $\Vert f_{\delta }？f\Vert \le \delta$ then the following results are provided: Problem $F_{\alpha ,\delta }\left(u\right):={\Vert Au？f_{\delta }\Vert }^2+\alpha {\Vert u\Vert }^2$ has a unique global minimizer $u_{\alpha ,\delta }$ for any $f_{\delta }\in Y$ , and $u_{\alpha ,\delta }=A^{*}{\left(A{A}^{*}+\alpha I_Y\right)}^{？1}{f}_{\delta }$