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Quasitriangular + small compact = strongly irreducible

Author(s): You Qing Ji
Journal: Trans. Amer. Math. Soc. 351 (1999), 4657-4673.
MSC (1991): Primary 47A10, 47A55, 47A58
Posted: July 20, 1999
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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

DOI: 10.1090/S0002-9947-99-02307-7
PII: S 0002-9947(99)02307-7
Keywords: Weyl spectrum, index, strongly irreducible, quasitriangular
Received by editor(s): May 23, 1997
Posted: July 20, 1999
Additional Notes: This work is supported by MCSEC
Copyright of article: Copyright 1999, American Mathematical Society

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