In this work we deal with the degree of ill-posedness
of linear operators in Hilbert spaces, where the
operator may be decomposed into a compact linear
integral operator with a well-known decay rate of
singular values and a multiplication operator.
This case occurs, for example, for nonlinear operator
equations. Then the local degree of ill-posedness is
investigated via the Fréchet derivative, providing
the situation described above.
If the multiplier function has got zeroes, the
determination of the local degree of ill-posedness is
not trivial. We are going to investigate this
situation, provide analytical tools as well as their
limitations. By using several numerical approaches
for computing the singular values we find that the
degree of ill-posedness does not change through those
multiplication operators. We provide a conjecture,
verified by several numerical studies, how those
operators influence the singular values.
Finally, we analyze the influence of these
multiplication operators on Tikhonov regularization
and corresponding convergence rates. In this context
we also provide a short summary on the relationship
between nonlinear problems and their linearizations.