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

TL;DR

This paper investigates how the non-compact Hausdorff moment operator affects the decay rates of singular values in composite linear operators, revealing faster decay than the compact integration operator alone.

Contribution

It provides the first example showing the impact of a non-compact factor on singular value decay in composite operators, highlighting a gap in existing bounds.

Findings

Singular values decay faster in the composition with the Hausdorff moment operator.

The composition's decay rate exceeds that of the integration operator alone.

Discussion on the gap between theoretical bounds and observed decay rates.

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 behaviour 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 composition decay faster than the ones 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.