{"title":"Curious ill-posedness phenomena in the composition of non-compact linear operators in Hilbert spaces","authors":"Stefan Kindermann, Bernd Hofmann","doi":"10.1515/jiip-2024-0007","DOIUrl":null,"url":null,"abstract":"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.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"20 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Inverse and Ill-Posed Problems","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/jiip-2024-0007","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
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.
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
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