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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On reducibility of semigroups
of compact quasinilpotent operators


Author: Roman Drnovsek
Journal: Proc. Amer. Math. Soc. 125 (1997), 2391-2394
MSC (1991): Primary 47A15, 47D03
MathSciNet review: 1422865
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Abstract: The following generalization of Lomonosov's invariant subspace theorem is proved. Let ${\mathcal S}$ be a multiplicative semigroup of compact operators on a Banach space such that $\hat {r} (S_1, \ldots , S_n) = 0$ for every finite subset $\{S_1, \ldots , S_n\}$ of ${\mathcal S}$, where $\hat {r}$ denotes the Rota-Strang spectral radius. Then ${\mathcal S}$ is reducible.

This result implies that the following assertions are equivalent:

(A) For each infinite-dimensional complex Hilbert space ${\mathcal H}$, every semigroup of compact quasinilpotent operators on ${\mathcal H}$ is reducible.

(B) For every complex Hilbert space ${\mathcal H}$, for every semigroup of compact quasinilpotent operators on ${\mathcal H}$, and for every finite subset $\{S_1, \ldots , S_n\}$ of ${\mathcal S}$ it holds that $\hat {r}(S_1, \ldots , S_n) = 0$.

The question whether the assertion (A) is true was considered by Nordgren, Radjavi and Rosenthal in 1984, and it seems to be still open.


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Additional Information

Roman Drnovsek
Affiliation: Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
Email: roman.drnovsek@fmf.uni-lj.si

DOI: http://dx.doi.org/10.1090/S0002-9939-97-04108-7
PII: S 0002-9939(97)04108-7
Keywords: Invariant subspaces, semigroups, reducibility, simultaneous triangularizability
Received by editor(s): March 1, 1996
Additional Notes: This work was supported in part by the Research Ministry of Slovenia.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society



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