AI 中文总结
该研究揭示了巴拿赫空间中无补零空间与非闭值域导致的不适定性的对偶关系，将其应用于Mazur型算子，提出可通过转移不适定性实现Tikhonov正则化的应用，为新正则化方法开发提供替代路径。
AI 中文摘要
迄今为止，具有无补零空间的算子在巴拿赫空间不适定逆问题的正则化理论中仅发挥次要作用。我们证明，由无补零空间导致的不适定性与由非闭值域导致的不适定性可视为对偶概念，通过以合适且明确定义的方式限制或扩展所考虑的不适定算子，可在两种视角间自由切换。我们将该对偶性结果应用于Mazur型算子，这类算子是具有无补零空间的有界线性算子。自然形式下的Mazur型算子无法采用Tikhonov型正则化，但将不适定性转移至值域后，即可应用Tikhonov正则化。另一方面，不适定性的零空间视角或可为经典意义下不适定的算子（即因非闭值域而不适定的算子）开发新正则化方法开辟替代路径。
英文摘要
Up to now 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 are 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 transfering 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.
