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 A + I)-1 A, with the domain D(A), where > 0 is a constant, is a linear bounded everywhere defined operator with norm 1 2 . This result is applied to the variational problem F(u) := Au - f 2 + 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.

• Type: Journal
• Year: 2008
• Where: APPML
• Authors: Alexander G. Ramm
• Keywords: APPML 2008, Linear, Operator, Solvable Linear Equation