On reducibility of semigroups of compact quasinilpotent operators
Author:
Roman Drnovsek
Journal:
Proc. Amer. Math. Soc. 125 (1997), 23912394
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 be a multiplicative semigroup of compact operators on a Banach space such that for every finite subset of , where denotes the RotaStrang spectral radius. Then is reducible. This result implies that the following assertions are equivalent: (A) For each infinitedimensional complex Hilbert space , every semigroup of compact quasinilpotent operators on is reducible. (B) For every complex Hilbert space , for every semigroup of compact quasinilpotent operators on , and for every finite subset of it holds that . 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.unilj.si
DOI:
http://dx.doi.org/10.1090/S0002993997041087
PII:
S 00029939(97)041087
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
