A unified concept of the degree of ill-posedness for compact and non-compact linear operators in Hilbert spaces under the auspices of the spectral theorem
<|reference_start|>A unified concept of the degree of ill-posedness for compact and non-compact linear operators in Hilbert spaces under the auspices of the spectral theorem: Covering ill-posed problems with compact and non-compact operators regarding the degree of ill-posedness is a never ending story written by many authors in the inverse problems literature. This paper tries to add a new narrative and some new facets with respect to this story under the auspices of the spectral theorem. The latter states that any self-adjoint and bounded operator is unitarily equivalent to a multiplication operator on some (semi-finite) measure space. We will exploit this fact and derive a distribution function from the corresponding multiplier, the growth behavior of which at zero allows us to characterize the degree of ill-posedness. We prove that this new concept coincides with the well-known one for compact operators (by means of their singular values), and illustrate the implications along examples including the Hausdorff moment operator and convolutions.<|reference_end|>
@article{werner2024a,
    title={A unified concept of the degree of ill-posedness for compact and
  non-compact linear operators in Hilbert spaces under the auspices of the
  spectral theorem},
    author={Frank Werner and Bernd Hofmann},
    journal={arXiv preprint arXiv:2408.01148},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.01148},
    primaryClass={math.NA cs.NA}
}