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Property (R) for Functions of Operators and its Perturbations

Property (R) for Functions of Operators and its Perturbations Let 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} be a complex separable infinite dimensional Hilbert space and B(H)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {B(H)}$$\end{document} be the algebra of all bounded linear operators 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}. T∈B(H)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$T\in \mathcal {B(H)}$$\end{document} is said to satisfy property (R) if σa(T)\σab(T)=π00(T)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma _{a}(T){\setminus }\sigma _{ab}(T)=\pi _{00}(T)$$\end{document}, where σa(T)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma _{a}(T)$$\end{document} and σab(T)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma _{ab}(T)$$\end{document} denote the approximate point spectrum and the Browder essential approximate point spectrum of T, respectively, and π00(T)={λ∈isoσ(T):0<dimN(T-λI)<∞}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\pi _{00}(T)=\{\lambda \in iso\sigma (T): 0<dim N(T-\lambda I)<\infty \}$$\end{document}. In this paper, using a new spectrum, we talk about the property (R) for functions of operators as well as its stability. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mediterranean Journal of Mathematics Springer Journals

Property (R) for Functions of Operators and its Perturbations

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Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022
ISSN
1660-5446
eISSN
1660-5454
DOI
10.1007/s00009-021-01942-y
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See Article on Publisher Site

Abstract

Let 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} be a complex separable infinite dimensional Hilbert space and B(H)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {B(H)}$$\end{document} be the algebra of all bounded linear operators 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}. T∈B(H)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$T\in \mathcal {B(H)}$$\end{document} is said to satisfy property (R) if σa(T)\σab(T)=π00(T)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma _{a}(T){\setminus }\sigma _{ab}(T)=\pi _{00}(T)$$\end{document}, where σa(T)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma _{a}(T)$$\end{document} and σab(T)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma _{ab}(T)$$\end{document} denote the approximate point spectrum and the Browder essential approximate point spectrum of T, respectively, and π00(T)={λ∈isoσ(T):0<dimN(T-λI)<∞}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\pi _{00}(T)=\{\lambda \in iso\sigma (T): 0<dim N(T-\lambda I)<\infty \}$$\end{document}. In this paper, using a new spectrum, we talk about the property (R) for functions of operators as well as its stability.

Journal

Mediterranean Journal of Mathematics – Springer Journals

Published: Feb 1, 2022

Keywords: Property (R); function of operator; stability; Primary 47A53; Secondary 47A10; 47A55

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