Quasitriangular $+$ small compact $=$ strongly irreducible

Author:
You Qing Ji

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

Published electronically:
July 20, 1999

MathSciNet review:
1603910

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Abstract | References | Similar Articles | Additional Information

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?

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Additional Information

**You Qing Ji**

Affiliation:
Department of Mathematics, Jilin University, Changchun 130023, P.R. China

Keywords:
Weyl spectrum,
index,
strongly irreducible,
quasitriangular

Received by editor(s):
May 23, 1997

Published electronically:
July 20, 1999

Additional Notes:
This work is supported by MCSEC

Article copyright:
© Copyright 1999
American Mathematical Society