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Aluthge Operator Field and Its Numerical Range and Spectral Properties

Authors
  • Cassier, Gilles1
  • Perrin, Thomas2
  • 1 Université de Lyon 1, 43, boulevard du 11 Novembre 1918, Villeurbanne, 69622, France , Villeurbanne (France)
  • 2 École normale supérieure Paris-Saclay, 4 Avenue des Sciences, Gif-sur-Yvette, 91190, France , Gif-sur-Yvette (France)
Type
Published Article
Journal
Integral Equations and Operator Theory
Publisher
Springer International Publishing
Publication Date
Jun 25, 2021
Volume
93
Issue
4
Identifiers
DOI: 10.1007/s00020-021-02656-2
Source
Springer Nature
Keywords
Disciplines
  • Article
License
Yellow

Abstract

For an arbitrary operator T acting on a Hilbert space we consider a field of operators Δz(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \Delta _{z}(T)\right) $$\end{document} called the Aluthge operator field associated with T. After giving preliminary results, we establish that two fields (left and right), canonically linked to the Altuthge field Δz(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \Delta _{z}(T)\right) $$\end{document} and a support subspace, are constant on each horizontal segment where they are defined. This result leads to a positive solution of a conjecture stated by Jung-Ko-Pearcy in 2000. Then we do a detailed spectral study of Δz(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \Delta _{z}(T)\right) $$\end{document} and we give a Yamazaki type formula in this context.

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