Sequences of variable-coefficient Toeplitz matrices and their singular values

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• 作者：H. Mascarenhas^a ; ^{hmasc@math.ist.utl.pt" class="auth_mail" title="E-mail the corresponding author} ; B. Silbermann^b ; ^{bernd.silbermann@mathematik.tu-chemnitz.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Singular values ; Splitting property ; Generalized Toeplitz matrices ; Fredholm sequences
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：270
• 期：4
• 页码：1479-1500
• 全文大小：433 K

文摘

The purpose of this paper is to describe asymptotic spectral properties of sequences of variable-coefficient Toeplitz matrices. These sequences, A_N(a)$A_N(a)$, with a   being in a Wiener type algebra and defined on an annular cylinder ([0,1]²×T)$({[0,1]}^2\times \mathbb{T})$, widely generalize the sequences of finite sections of a Toeplitz operator. We prove that if a(x,x,t)$a(x,x,t)$ does not vanish for every (x,t)∈[0,1]×T$(x,t)\in [0,1]\times \mathbb{T}$ then the singular values of A_N(a)$A_N(a)$ have the k-splitting property, which means that, there exists an integer k such that, for N large enough, the first k  -singular values of A_N(a)$A_N(a)$ converge to zero as N→∞$N\to \infty$, while the others are bounded away from zero, with k=dim⁡ker⁡T(a(0,0,t))+dim⁡ker⁡T(a(1,1,t⁻¹))$k=\dim \ker T(a(0,0,t))+\dim \ker T(a(1,1,t^{-1}))$, the sum of the kernel dimensions of two Toeplitz operators. In the end of the paper we discuss Fredholm properties of the mentioned sequences and describe them completely.