Locally quasi-nilpotent elementary operators (original) (raw)
2013, arXiv: Rings and Algebras
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An operator is a product of two quasi-nilpotent operators if and only if it is not semi-Fredholm
Proceedings of The Royal Society of Edinburgh Section A-mathematics, 2006
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.
On the invertibility of elementary operators
Let mathscrX\mathscr{X}mathscrX be a complex Banach space and mathcalL(mathscrX)\mathcal{L}(\mathscr{X})mathcalL(mathscrX) be the algebra of all bounded linear operators on mathscrX\mathscr{X}mathscrX. For a given elementary operator Phi\PhiPhi of length 222 on mathcalL(mathscrX)\mathcal{L}(\mathscr{X})mathcalL(mathscrX), we determine necessary and sufficient conditions for the existence of a solution of the equation rmXPhi=0{\rm X} \Phi=0rmXPhi=0 in the algebra of all elementary operators on mathcalL(mathscrX)\mathcal{L}(\mathscr{X})mathcalL(mathscrX). Our approach allows us to characterize some invertible elementary operators of length 222 whose inverses are elementary operators.
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