Common invariant subspaces for collections of operators |
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Authors: | Roman Drnov?ek |
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Institution: | (1) Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia |
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Abstract: | Let
be a collection of bounded operators on a Banach spaceX of dimension at least two. We say that
is finitely quasinilpotent at a vectorx
0 X whenever for any finite subset
of
the joint spectral radius of
atx
0 is equal 0. If such collection
contains a non-zero compact operator, then
and its commutant
have a common non-trivial invariant, subspace. If in addition,
is a collection of positive operators on a Banach lattice, then
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
be a multiplicative semigroup of quasinilpotent compact positive operators on a Banach lattice of dimension at least two. Then
has a common non-trivial invariant closed ideal.This work was supported by the Research Ministry of Slovenia. |
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Keywords: | Primary 47A15 47D03 |
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