An inequality for eigenvalues of nuclear operators via traces and the generalized Hoffman–Wielandt theorem

IF 0.6 3区 数学 Q3 MATHEMATICS Analysis Mathematica Pub Date : 2024-10-17 DOI:10.1007/s10476-024-00040-x
M. Gil’
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Abstract

Let \(A\) be a Hilbert-Schmidt operator, whose eigenvalues are \(\lambda_k(A)(k=1,2 , \ldots )\). We derive a new inequality for the series \(\sum^{\infty}_{k=1}|\lambda_k(A)-z_k|^2\), where \(\{z_k\}\) is a sequence of numbers satisfying the condition \(\sum_k |z_k|^2<{\infty}\). That inequality is expressed via the self-commutator \(AA^*-A^*A\). If \(A\) is a nuclear operator, we obtain an inequality for the eigenvalues via the trace and self-commutator.

Our results are based on the generalization of the theorem of R. Bhatia and L. Elsner [1] which is an infinite-dimensional analog of the Hoffman–Wielandt theorem on perturbations of normal matrices.

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核算子经迹特征值的一个不等式及广义霍夫曼-维兰特定理
设 \(A\) 是一个希尔伯特-施密特算子,其特征值为 \(\lambda_k(A)(k=1,2 , \ldots )\).我们为数列 \(\sum^{\infty}_{k=1}|\lambda_k(A)-z_k|^2\)推导出一个新的不等式,其中 \(\{z_k\}\)是满足条件\(\sum_k |z_k|^2<{\infty}\) 的数列。这个不等式通过自交子 \(AA^*-A^*A\)来表示。 如果 \(A\) 是一个核算子,我们就可以通过迹和自换子得到特征值的不等式。我们的结果基于 R. Bhatia 和 L. Elsner [1] 的概括定理,它是关于正矩阵扰动的 Hoffman-Wielandttheorem 的无穷维类似定理。
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来源期刊
Analysis Mathematica
Analysis Mathematica MATHEMATICS-
CiteScore
1.00
自引率
14.30%
发文量
54
审稿时长
>12 weeks
期刊介绍: Traditionally the emphasis of Analysis Mathematica is classical analysis, including real functions (MSC 2010: 26xx), measure and integration (28xx), functions of a complex variable (30xx), special functions (33xx), sequences, series, summability (40xx), approximations and expansions (41xx). The scope also includes potential theory (31xx), several complex variables and analytic spaces (32xx), harmonic analysis on Euclidean spaces (42xx), abstract harmonic analysis (43xx). The journal willingly considers papers in difference and functional equations (39xx), functional analysis (46xx), operator theory (47xx), analysis on topological groups and metric spaces, matrix analysis, discrete versions of topics in analysis, convex and geometric analysis and the interplay between geometry and analysis.
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