An irreducible semigroup of non-negative square-zero operators
type: Item
instance of: scholarly article
title: An irreducible semigroup of non-negative square-zero operators
zbMATH Open document ID: 1011.47004
DOI: 10.1007/BF01270922
author: M. Omladič
author: Roman Drnovšek
author: Damjana Kokol Bukovšek
author: Leo Livshits
author: Gordon Macdonald
author: H. Radjavi
published in: Integral Equations and Operator Theory
publication date: 2003-06-01T00:00:00Z
review text: An irreducible multiplicative semigroup of non-negative square-zero operators acting on \(L^p[0,1)\) for \(1\leq p<\infty\) illustrates the complexity of the structure possible in such seemingly simple objects as semigroups of square-zero operators. Recently, a number of positive results have been obtained concerning reducibility of semigroups of (possibly non-negative) compact operators. In the authors' opinion, the semigroup constructed or a slightly modified version of it provides an almost universal counterexample to show that very few of these results can be extended to general (possibly non-negative) operator semigroups. The example proposed in the paper provides the first example of an irreducible semigroup of non-negative nilpotent operators where the order of nilpotency is bounded. This answers an open question of \textit{Y. Zhong} [Trans. Am. Math. Soc. 347, 3093-3100 (1995; Zbl 0846.47002)].
reviewed by: Pavel Dyshlovenko
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 1754750
zbMATH Keywords: irreducible semigroup
zbMATH Keywords: operator semigroup
zbMATH Keywords: reducibility of semigroups
zbMATH Keywords: square-zero operator
MaRDI profile type: Publication
cites work: Q4298654
cites work: Triangularizing semigroups of positive operators on an atomic normed Riesz Space
cites work: Common invariant subspaces for collections of operators
cites work: A nil algebra of bounded operators on Hilbert space with semisimple norm closure
cites work: Simultaneous triangularization
cites work: Q4264335
cites work: Volterra semigroups have invariant subspaces
cites work: Irreducible Semigroups of Functionally Positive Nilpotent Operators
description: scientific article; zbMATH DE number 1754750