On unbounded operators and applications

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 (AA + α I )−1A∗, with the domain D(A), where α > 0 is a constant, is a linear bounded everywhere defined operator with norm ≤ 1 2 √ α . This result is applied to the variational problem F(u) := ‖Au − f ‖2 + α‖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. c © 2007 Elsevier Ltd. All rights reserved.