Stability rates for linear ill-posed problems with compact and non-compact operators
Two types of ill-posedness of linear operator equations \(Ax= y\) in Hilbert spaces are considered: type I, if \(A\) is not compact, but for the range \(R(A)\) we have \(R(A)\neq \text{cl }R(A)\), and type II for compact operators \(A\). Ill-posedness of type I means that range \(R(A)\) contains a closed infinite-dimensional subspace. Conditions are presented that can be ensured for ill-posed equations of type II by appropriate choice of an orthonormal system, but for ill-posed equations of type I are never satisfied. Since noncompact operators \(A\) have no singular values, the authors introduce stability rates in order to have a common measure for both, compact and noncompact ones. Properties of these rates are analyzed for compact convolution operators and for noncompact multiplication operators \(A\).
scientific article; zbMATH DE number 1331698