Coburn's lemma and the finite section method for random Jacobi operators
Coburn's lemma and the finite section method for random Jacobi operators
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Coburn 引理和随机雅可比算子的有限截面方法
DOI：
10.1016/j.jfa.2015.09.019
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发表时间：
2016
影响因子：
1.7
通讯作者：
Chandler-Wilde S
中科院分区：
文献类型：
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作者：
Chandler-Wilde S
本文研究了半无限和双无限三对角随机矩阵及其有限主子阵的谱和伪谱，其中三对角中的每一个都在一个单独的紧集上变化，比如U，V，W <$C.这样的矩阵有时被称为随机Toeplitz矩阵A+在半无限的情况下和随机Laurent矩阵A在双无限的情况下。只要A和A+是伪遍历的（在Davies（2001）[20]的意义上），它们的谱，= spec A和+= spec A+，与A和A+无关，这在随机情况下几乎肯定成立。这在Davies（2001）[20]中对A进行了证明;这对A+也成立是本文的一个主要结果。虽然用U、V和W来计算和+本质上是困难的，但我们给出了谱的上界和下界，并且我们明确地计算了一个集合G，它在G =+的意义上填补了和+之间的差距。我们还表明，一个（因此所有）运营商A+的可逆性意味着可逆性和一致有界的逆的所有有限三对角方阵的对角线变化的U，V和W。这意味着，只要A+是可逆的，那么用于系统A+ x= B的近似解的所谓有限截面法是适用的，并且用于估计A+的谱的有限截面法不遭受谱污染。这两个结果都表明三对角随机Toeplitz算子具有（经典）Toeplitz算子的重要性质。事实上，我们的主要工具之一是一个新的随机版本的Coburn引理经典Toeplitz运营商，说，一个随机三对角Toeplitz运营商，如果Fredholm，总是内射或满射。在本文的最后一部分，我们对U，V和W上的双无限，半无限和有限三对角矩阵的范数和逆矩阵的范数进行了界定和比较。这，特别是允许研究的预解规范，因此伪谱，这些运营商和矩阵。
We study the spectra and pseudospectra of semi-infinite and bi-infinite tridiagonal random matrices and their finite principal submatrices, in the case where each of the three diagonals varies over a separate compact set, say U, V, W⊂ C. Such matrices are sometimes termed stochastic Toeplitz matrices A+ in the semi-infinite case and stochastic Laurent matrices A in the bi-infinite case. Their spectra, Σ= spec A and Σ+= spec A+, are independent of A and A+ as long as A and A+ are pseudoergodic (in the sense of Davies (2001)[20]), which holds almost surely in the random case. This was shown in Davies (2001)[20] for A; that the same holds for A+ is one main result of this paper. Although the computation of Σ and Σ+ in terms of U, V and W is intrinsically difficult, we give upper and lower spectral bounds, and we explicitly compute a set G that fills the gap between Σ and Σ+ in the sense that Σ∪ G= Σ+. We also show that the invertibility of one (and hence all) operators A+ implies the invertibility–and uniform boundedness of the inverses–of all finite tridiagonal square matrices with diagonals varying over U, V and W. This implies that the so-called finite section method for the approximate solution of a system A+ x= b is applicable as soon as A+ is invertible, and that the finite section method for estimating the spectrum of A+ does not suffer from spectral pollution. Both results illustrate that tridiagonal stochastic Toeplitz operators share important properties of (classical) Toeplitz operators. Indeed, one of our main tools is a new stochastic version of the Coburn lemma for classical Toeplitz operators, saying that a stochastic tridiagonal Toeplitz operator, if Fredholm, is always injective or surjective. In the final part of the paper we bound and compare the norms, and the norms of inverses, of bi-infinite, semi-infinite and finite tridiagonal matrices over U, V and W. This, in particular, allows the study of the resolvent norms, and hence the pseudospectra, of these operators and matrices.