On unbounded operators and applications

Assume that Au = f is a solvable linear equation ill a Hilbert space H, A is a linear, closed, densely defined, unbounded operator ill H, which is not boundedly invertible, so problem (1) is ill-posed. It is proved that the Closure of the operator (A*A + alpha I)(-1) A*, with the domain D(A*), where alpha > 0 is a constant, is a linear bounded everywhere defined operator with norm <= 1/2 root alpha. This result is applied to the variational problem F(u) :=parallel to Au - f parallel to(2) +alpha parallel to u parallel to(2) = min, where f is all 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.