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

2006/06/21 by Roman Drnovšek, Drnovšek, Roman, Müller, Vladimir +2
Computer Science · Mathematics · #47A53 #47A65 #47A68 #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.math/0606516

openalex publication_date 2006/06/21 · openalex created_date 2022/09/27 · openalex updated_date 2026/07/28

Abstract

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

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