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

Bernd Hofmann and Peter Mathé

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.

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Key words

Hausdorff moment problem, linear inverse problem, degree of ill-posedness, composite operator, conditional stability

AMS subject classifications

47A52, 47B06, 65J20, 44A60