On unbounded operators and applications

Assume that $Au=f,\quad (1)$ is a solvable linear equation in a Hilbert space $H$, $A$ is a linear, closed, densely defined, unbounded operator in $H$, which is not boundedly invertible, so problem (1) is ill-posed. It is proved that the closure of the operator $(A^*A+\a I)^{-1}A^*$, with the domain $D(A^*)$, where $\a>0$ is a constant, is a linear bounded everywhere defined operator with norm $\leq 1$. This result is applied to the variational problem $F(u):= ||Au-f||^2+\a ||u||^2=min$, where $f$ is an arbitrary element of $H$, not necessarily belonging to the range of $A$. Variational regularization of problem (1) is constructed, and a discrepancy principle is proved.