Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices
Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices
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极限算子、集合紧性和无限矩阵的谱理论
DOI：
10.1090/s0065-9266-2010-00626-4
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发表时间：
2010
影响因子：
2.4
通讯作者：
M. Lindner
中科院分区：
文献类型：
--
作者：
S. Chandler;M. Lindner
在这本回忆录的前半部分，我们探讨了极限算子的抽象理论（参见Rabinovich， Roch and Silbermann（2004）和Lindner（2006）的近期专著）与Chandler-Wilde和Zhang（2002）引入的广义集体紧算子理论的概念和结果之间的相互关系。我们将这个广义集体紧算子理论应用于算子的极限算子集（算子谱）得到了一些结果。在本回忆录的后半部分，我们研究了广义序列空间上的有界线性算子，其中和是复巴拿赫空间。我们对极限算子集的Fredholmness、可逆性、无穷处可逆性和可逆性或注入性之间的关系作了迄今为止似乎更完整的研究，并着重讨论了当算子是单位元的局部紧摄动时的情况。特别是，对于和的微妙极限情况，我们得到了比以前已知的更有力的结果。我们在本研究中的工具是回忆录前半部分的结果，以及对和之间的部分对偶及其对有界线性算子的含义的利用，这些算子对于回忆录前半部分引入的较弱拓扑（严格拓扑）也是连续的。回忆录后半部分的结果包括一个新的证明，证明所有极限算子的内射性（经典的Favard条件）意味着一般概周期算子的可逆性，以及在所谓的维纳代数中算子的无限可逆性和Fredholmness的特征。在最后两章中，我们的结果用具体的例子加以说明和应用。首先，我们研究了离散薛定谔算子（包括自伴随算子和非自伴随算子）的谱和本质谱，包括具有几乎周期和随机势的算子。在最后一章中，我们将结果应用于上的积分算子。
In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator (its operator spectrum). In the second half of this memoir we study bounded linear operators on the generalised sequence space , where and is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of and . Our tools in this study are the results from the first half of the memoir and an exploitation of the partial duality between and and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the memoir. Results in this second half of the memoir include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrodinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on .