Comparing the ill-posedness for linear operators in Hilbert spaces

IF 0.8 Q2 MATHEMATICS
Peter Mathé, Bernd Hofmann
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引用次数: 0

Abstract

The difficulty for solving ill-posed linear operator equations in Hilbert space is reflected by the strength of ill-posedness of the governing operator, and the inherent solution smoothness. In this study we focus on the ill-posedness of the operator, and we propose a partial ordering for the class of all bounded linear operators which lead to ill-posed operator equations. For compact linear operators, there is a simple characterization in terms of the decay rates of the singular values. In the context of the validity of the spectral theorem the partial ordering can also be understood. We highlight that range inclusions yield partial ordering, and we discuss cases when compositions of compact and non-compact operators occur. Several examples complement the theoretical results.

Abstract Image

Hilbert空间中线性算子的病态性比较
希尔伯特空间中求解病态线性算子方程的困难体现在控制算子的病态强度和固有的解的平滑性上。在本研究中,我们关注算子的病态性,并提出了一类导致病态算子方程的所有有界线性算子的偏序。对于紧线性算子,有一个关于奇异值衰减率的简单描述。在谱定理有效性的背景下,偏序也可以被理解。我们强调了范围包含产生偏序,并讨论了紧算子和非紧算子组合的情况。几个例子补充了理论结果。
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来源期刊
CiteScore
1.60
自引率
0.00%
发文量
55
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