{"title":"An inequality for eigenvalues of nuclear operators via traces and the generalized Hoffman–Wielandt theorem","authors":"M. Gil’","doi":"10.1007/s10476-024-00040-x","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span>\\(A\\)</span> be a Hilbert-Schmidt operator, \nwhose eigenvalues are <span>\\(\\lambda_k(A)(k=1,2 , \\ldots )\\)</span>.\nWe derive\na new inequality for the series \n<span>\\(\\sum^{\\infty}_{k=1}|\\lambda_k(A)-z_k|^2\\)</span>, \nwhere <span>\\(\\{z_k\\}\\)</span> is a sequence of numbers\nsatisfying the condition\n<span>\\(\\sum_k |z_k|^2<{\\infty}\\)</span>. That inequality is expressed\nvia the self-commutator <span>\\(AA^*-A^*A\\)</span>. \nIf <span>\\(A\\)</span> is a nuclear operator, we \nobtain an inequality for the eigenvalues via the \ntrace and self-commutator.</p><p>\nOur results are based on the generalization of the theorem of R. Bhatia and\nL. Elsner [1] which is an infinite-dimensional analog of the Hoffman–Wielandt\ntheorem on perturbations of normal matrices.\n</p></div>","PeriodicalId":55518,"journal":{"name":"Analysis Mathematica","volume":"50 4","pages":"1033 - 1043"},"PeriodicalIF":0.6000,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Analysis Mathematica","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10476-024-00040-x","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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.
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.