On unbounded operators and applications

2005/08/29 by Ramm, A. G.
#35R30 #45A50 #47A05 #FOS: Mathematics #Numerical Analysis (math.NA) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.math/0508587

Abstract

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

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