Duality of uncomplemented null-space and non-closed range, and its impact on the regularization of Mazur-type operators

Jens Flemming; Bernd Hofmann; Jens Flemming; Bernd Hofmann

doi:10.3934/era.2026378

Electronic Research Archive

2026, Volume 34, Issue 12: 8987-8996. doi: 10.3934/era.2026378

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Duality of uncomplemented null-space and non-closed range, and its impact on the regularization of Mazur-type operators

Jens Flemming ¹,
Bernd Hofmann ^{2
,
,}

1.
Faculty of Informatics/Mathematics, HTWD–University of Applied Sciences, 01069 Dresden, Germany
2.
Faculty of Mathematics, Chemnitz University of Technology, 09107 Chemnitz, Germany

Received: 05 August 2026 Revised: 14 October 2026 Accepted: 15 October 2026 Published: 26 October 2026

Up to now, linear operators with uncomplemented null-space have been playing only a minor role in regularization theory for ill-posed inverse problems in Banach spaces. We show that ill-posedness due to an uncomplemented null-space and ill-posedness due to a non-closed range can be regarded as dual concepts, and we may switch between both views at will by restricting or extending the ill-posed operator under consideration in a suitable well-defined way. We apply the duality result to Mazur-type operators, which represent a class of bounded linear operators with uncomplemented null-space. Mazur-type operators in their natural form are not accessible to Tikhonov-type regularization, but after transferring ill-posedness to the range, Tikhonov regularization can be applied. On the other hand, the null-space view of ill-posedness may open up an alternative path for developing new regularization methods for operators ill-posed in the classical sense, that is, for operators ill-posed due to a non-closed range.
- linear ill-posed problems,
- non-closed range,
- uncomplemented null-space,
- regularization,
- Mazur-type operators
Citation: Jens Flemming, Bernd Hofmann. Duality of uncomplemented null-space and non-closed range, and its impact on the regularization of Mazur-type operators[J]. Electronic Research Archive, 2026, 34(12): 8987-8996. doi: 10.3934/era.2026378

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Abstract

Up to now, linear operators with uncomplemented null-space have been playing only a minor role in regularization theory for ill-posed inverse problems in Banach spaces. We show that ill-posedness due to an uncomplemented null-space and ill-posedness due to a non-closed range can be regarded as dual concepts, and we may switch between both views at will by restricting or extending the ill-posed operator under consideration in a suitable well-defined way. We apply the duality result to Mazur-type operators, which represent a class of bounded linear operators with uncomplemented null-space. Mazur-type operators in their natural form are not accessible to Tikhonov-type regularization, but after transferring ill-posedness to the range, Tikhonov regularization can be applied. On the other hand, the null-space view of ill-posedness may open up an alternative path for developing new regularization methods for operators ill-posed in the classical sense, that is, for operators ill-posed due to a non-closed range.

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