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# A version of Lomonosov’s theorem for collections of positive operators

Proceedings of The American Mathematical Society - PROC AMER MATH SOC 01/2008; 137(05):1793-1800. DOI: 10.1090/S0002-9939-08-09775-X

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- Transactions of The American Mathematical Society - TRANS AMER MATH SOC. 01/1980; 261(1).
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**ABSTRACT:**Let C\mathcal{C} be a collection of bounded operators on a Banach spaceX of dimension at least two. We say that C\mathcal{C} is finitely quasinilpotent at a vectorx 0F\mathcal{F} of C\mathcal{C} the joint spectral radius of F\mathcal{F} atx 0 is equal 0. If such collection C\mathcal{C} contains a non-zero compact operator, then C\mathcal{C} and its commutant C¢\mathcal{C}' have a common non-trivial invariant, subspace. If in addition, C\mathcal{C} is a collection of positive operators on a Banach lattice, then C\mathcal{C} has a common non-trivial closed ideal. This result and a recent remarkable theorem of Turovskii imply the following extension of the famous result of de Pagter to semigroups. Let S\mathcal{S} be a multiplicative semigroup of quasinilpotent compact positive operators on a Banach lattice of dimension at least two. Then S\mathcal{S} has a common non-trivial invariant closed ideal.Integral Equations and Operator Theory 08/2001; 39(3):253-266. · 0.71 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We construct an irreducible multiplicative semigroup of non-negative square-zero operators acting onL p [0,1), for 1p.Integral Equations and Operator Theory 11/2002; 42(4):449-460. · 0.71 Impact Factor

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