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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Quasitriangular $+$ small compact $=$ strongly irreducible
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by You Qing Ji PDF
Trans. Amer. Math. Soc. 351 (1999), 4657-4673 Request permission

Abstract:

Let $T$ be a bounded linear operator acting on a separable infinite dimensional Hilbert space. Let $\epsilon$ be a positive number. In this article, we prove that the perturbation of $T$ by a compact operator $K$ with $\Vert K\Vert <\epsilon$ can be strongly irreducible if $T$ is a quasitriangular operator with the spectrum $\sigma (T)$ connected. The Main Theorem of this article nearly answers the question below posed by D. A. Herrero. Suppose that $T$ is a bounded linear operator acting on a separable infinite dimensional Hilbert space with $\sigma (T)$ connected. Let $\epsilon >0$ be given. Is there a compact operator $K$ with $\Vert K\Vert <\epsilon$ such that $T+K$ is strongly irreducible?
References
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Additional Information
  • You Qing Ji
  • Affiliation: Department of Mathematics, Jilin University, Changchun 130023, P.R. China
  • Received by editor(s): May 23, 1997
  • Published electronically: July 20, 1999
  • Additional Notes: This work is supported by MCSEC
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 351 (1999), 4657-4673
  • MSC (1991): Primary 47A10, 47A55, 47A58
  • DOI: https://doi.org/10.1090/S0002-9947-99-02307-7
  • MathSciNet review: 1603910