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The Weyl–von Neumann theorem for skew-symmetric operators

The Weyl–von Neumann theorem for skew-symmetric operators We consider the diagonalization of self-adjoint operators in the infinite-dimensional orthogonal Lie algebra OC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal O_C$$\end{document}, which consists of all bounded linear operators T on a separable, infinite-dimensional, complex Hilbert space H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {H}$$\end{document} satisfying CTC=-T∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$CTC = -T^*$$\end{document}, where C is a conjugation on H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal H$$\end{document}. First, we establish the Weyl–von Neumann Theorem in OC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}_C$$\end{document}, showing that each self-adjoint operator in OC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}_C$$\end{document} can be diagonalized in OC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}_C$$\end{document} up to arbitrarily small perturbations with respect to the Schatten p-norm, p∈(1,∞)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p\in (1,\infty )$$\end{document}. Second, for any n≥2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n\ge 2$$\end{document}, we prove that n commuting self-adjoint operators in OC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}_C$$\end{document} can be diagonalized simultaneously in OC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}_C$$\end{document} up to arbitrarily small perturbations with respect to the Schatten n-norm. This is an OC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}_C$$\end{document}-analogue of Voiculescu’s result concerning the simultaneous diagonalization of commuting self-adjoint operators. Finally, as an application of the preceding results, we show that those irreducible ones constitute a norm dense subset of OC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}_C$$\end{document}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of Functional Analysis Springer Journals

The Weyl–von Neumann theorem for skew-symmetric operators

Bu, Qinggang; Zhu, Sen
Annals of Functional Analysis , Volume 14 (2) – Apr 1, 2023
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Springer Journals
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Copyright © Tusi Mathematical Research Group (TMRG) 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
2639-7390
eISSN
2008-8752
DOI
10.1007/s43034-023-00265-1
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Abstract

We consider the diagonalization of self-adjoint operators in the infinite-dimensional orthogonal Lie algebra OC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal O_C$$\end{document}, which consists of all bounded linear operators T on a separable, infinite-dimensional, complex Hilbert space H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {H}$$\end{document} satisfying CTC=-T∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$CTC = -T^*$$\end{document}, where C is a conjugation on H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal H$$\end{document}. First, we establish the Weyl–von Neumann Theorem in OC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}_C$$\end{document}, showing that each self-adjoint operator in OC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}_C$$\end{document} can be diagonalized in OC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}_C$$\end{document} up to arbitrarily small perturbations with respect to the Schatten p-norm, p∈(1,∞)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p\in (1,\infty )$$\end{document}. Second, for any n≥2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n\ge 2$$\end{document}, we prove that n commuting self-adjoint operators in OC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}_C$$\end{document} can be diagonalized simultaneously in OC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}_C$$\end{document} up to arbitrarily small perturbations with respect to the Schatten n-norm. This is an OC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}_C$$\end{document}-analogue of Voiculescu’s result concerning the simultaneous diagonalization of commuting self-adjoint operators. Finally, as an application of the preceding results, we show that those irreducible ones constitute a norm dense subset of OC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}_C$$\end{document}.

Journal

Annals of Functional Analysis – Springer Journals

Published: Apr 1, 2023

Keywords: Skew symmetric operators; Diagonal operators; Schatten p-class; Irreducible operators; Orthogonal Lie algebra; 47B15; 47A58; 47C10; 17B65

References

  • On maps preserving skew symmetric operators
    Amara, Z; Oudghiri, M; Souilah, K
  • Skew-symmetric operators and reflexivity
    Benhida, C; Kliś-Garlicka, K; Ptak, M
  • On the generalized mean transforms of complex symmetric operators
    Benhida, C; Chō, M; Ko, E; Lee, JE
  • An extension of the Weyl–von Neumann theorem to normal operators
    Berg, ID
  • The orthogonal Lie algebra of operators: ideals and derivations
    Bu, QG; Zhu, S
  • Ranks of complex skew symmetric operators and applications to Toeplitz operators
    Chen, Y; Koo, H; Lee, YJ
  • Mathematical and physical aspects of complex symmetric operators
    Garcia, SR; Prodan, E; Putinar, M
  • Ten problems in Hilbert space
    Halmos, PR
  • Irreducible operators
    Halmos, PR
  • Perturbation of continuous spectra by trace class operators
    Kato, T
  • On a theorem of Weyl–von Neumann
    Kuroda, ST
  • A generalization of the Voiculescu theorem for normal operators in semifinite von Neumann algebras
    Li, QH; Shen, JH; Shi, R
  • Skew symmetry of a class of operators
    Li, CG; Zhou, TT
  • Skew symmetric normal operators
    Li, CG; Zhu, S
  • The set of irreducible operators is dense
    Radjavi, H; Rosenthal, P
  • Some results on norm-ideal perturbations of Hilbert space operators
    Voiculescu, D
  • Über beschränkte quadratische Formen deren Differenz vollstetig ist
    Weyl, H
  • On a J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J$$\end{document}-polar decomposition of a bounded operator and matrices of J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J$$\end{document}-symmetric and J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J$$\end{document}-skew-symmetric operators
    Zagorodnyuk, SM
  • On the complex symmetric and skew-symmetric operators with a simple spectrum
    Zagorodnyuk, SM
  • Skew symmetric weighted shifts
    Zhu, S
  • Complex symmetric operators, skew symmetric operators and reflexivity
    Zhu, S