The degree of ill-posedness of composite linear ill-posed problems with focus on the impact of the non-compact Hausdorff moment operator

Abstract

We consider compact composite linear operators in Hilbert space, where the composition is given by some compact operator followed by some non-compact one possessing a non-closed range. Focus is on the impact of the non-compact factor on the overall behavior of the decay rates of the singular values of the composition. Specifically, the composition of the compact integration operator with the non-compact Hausdorff moment operator is considered. We show that the singular values of the composite operator decay faster than those of the integration operator, providing a first example of this kind. However, there is a gap between available lower bounds for the decay rate and the obtained result. Therefore we conclude with a discussion.