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An operator is a product of two quasi-nilpotent operators if and only if it is not semi-Fredholm

Published online by Cambridge University Press:  12 July 2007

Roman Drnovšek ,
Nika Novak  and
Vladimir Müller
Roman Drnovšek
Affiliation:
Department of Mathematics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Sloveniaroman.drnovsek@fmf.uni-lj.si
Nika Novak
Affiliation:
Department of Mathematics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenianika.novak@fmf.uni-lj.si
Vladimir Müller
Affiliation:
Mathematical Institute, Czech Academy of Sciences, Žitná 25, 11567 Prague 1, Czech Republic (muller@math.cas.cz)
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Abstract

We prove that a (bounded, linear) operator acting on an infinite-dimensional, separable, complex Hilbert space can be written as a product of two quasi-nilpotent operators if and only if it is not a semi-Fredholm operator. This solves the problem posed by Fong and Sourour in 1984. We also consider some closely related questions. In particular, we show that an operator can be expressed as a product of two nilpotent operators if and only if its kernel and co-kernel are both infinite dimensional. This answers the question implicitly posed by Wu in 1989.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 136 , Issue 5 , October 2006 , pp. 935 - 944
Copyright
Copyright © Royal Society of Edinburgh 2006

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