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Roman Drnovsek - On reducibility of semigroups of compact quasinilpotent operators
ROMAN DRNOVSEK, Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia | |
On reducibility of semigroups of compact quasinilpotent operators |
Using the famous Lomonosov-Hilden technique we proved the following invariant subspace theorem. Let be a multiplicative semigroup of compact operators on a Banach space X of dimension greater than 1 such that for every finite subset of , where denotes the Rota-Strang spectral radius. Then is reducible, i.e., there exists a closed subspace of X, other than and X, which is invariant under every member of .
This result implies that the following assertions are equivalent:
(A) For each infinite-dimensional 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 (and independently by Shul'man) in 1984, and it seems to be still open.
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