On unbounded operators and applications

Assume that $Au=f,(1)$ 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 $(A^{\ast }A+\text{a}I{)}^{-1}{A}^{\ast }$, with the domain $D(A^{\ast })$, where $\text{a}>0$ is a constant, is a linear bounded everywhere defined operator with norm $\le 1$. This result is applied to the variational problem $F(u):=\mathrm{\mid }\mathrm{\mid }Au-f\mathrm{\mid }{\mathrm{\mid }}^2+\text{a}\mathrm{\mid }\mathrm{\mid }u\mathrm{\mid }{\mathrm{\mid }}^2=min$, 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.