希尔伯特空间中非紧凑线性算子组成中的奇异非问题现象

IF 0.9 4区数学 Q2 MATHEMATICS

Journal of Inverse and Ill-Posed Problems Pub Date : 2024-08-03 DOI:10.1515/jiip-2024-0007

Stefan Kindermann, Bernd Hofmann

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引用次数: 0

摘要

我们考虑了希尔伯特空间中具有非封闭范围的算子的组成，以及它们的组成如何影响问题的性质。具体而言，我们研究了豪斯多夫算子、塞萨罗算子、积分算子和它们的邻接算子，以及它们的一些组合。对于豪斯多夫算子和塞萨罗算子的组合，我们给出了相应奇异值衰减的估计值。有趣的是，这还提供了两个与实际相关的非紧凑算子的例子，对于这两个算子，它们的组合是紧凑的。此外，我们还描述了那些与非紧凑算子组成后会得到紧凑算子的算子的特征。

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Curious ill-posedness phenomena in the composition of non-compact linear operators in Hilbert spaces

We consider the composition of operators with non-closed range in Hilbert spaces and how the nature of ill-posedness is affected by their composition. Specifically, we study the Hausdorff-, Cesàro-, integration operator, and their adjoints, as well as some combinations of those. For the composition of the Hausdorff- and the Cesàro-operator, we give estimates of the decay of the corresponding singular values. As a curiosity, this provides also an example of two practically relevant non-compact operators, for which their composition is compact. Furthermore, we characterize those operators for which a composition with a non-compact operator gives a compact one.

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来源期刊

Journal of Inverse and Ill-Posed Problems MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

2.60

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： This journal aims to present original articles on the theory, numerics and applications of inverse and ill-posed problems. These inverse and ill-posed problems arise in mathematical physics and mathematical analysis, geophysics, acoustics, electrodynamics, tomography, medicine, ecology, financial mathematics etc. Articles on the construction and justification of new numerical algorithms of inverse problem solutions are also published. Issues of the Journal of Inverse and Ill-Posed Problems contain high quality papers which have an innovative approach and topical interest. The following topics are covered: Inverse problems existence and uniqueness theorems stability estimates optimization and identification problems numerical methods Ill-posed problems regularization theory operator equations integral geometry Applications inverse problems in geophysics, electrodynamics and acoustics inverse problems in ecology inverse and ill-posed problems in medicine mathematical problems of tomography