On unbounded operators and applications

Assume that equation(1)Au=fAu=f is a solvable linear equation in a Hilbert space HH, AA is a linear, closed, densely defined, unbounded operator in HH, which is not boundedly invertible, so problem (1) is ill-posed. It is proved that the closure of the operator (A∗A+αI)−1A∗(A∗A+αI)−1A∗, with the domain D(A∗)D(A∗), where α>0α>0 is a constant, is a linear bounded everywhere defined operator with norm ≤12α. This result is applied to the variational problem F(u)≔‖Au−f‖2+α‖u‖2=minF(u)≔‖Au−f‖2+α‖u‖2=min, where ff is an arbitrary element of HH, not necessarily belonging to the range of AA. Variational regularization of problem (1) is constructed, and a discrepancy principle is proved.