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Strictly singular operators and the invariant subspace problem. (English) Zbl 0929.47004
Recently W. T. Gowers and B. Maurey [Math. Ann. 307, No. 4, 543-568 (1997; Zbl 0876.46006)] proved that there exist Banach spaces on which every continuous operator is of the form $$\lambda I+S$$, where $$S$$ is strictly singular. So if strictly singular operators had invariant subspaces, such spaces would have the property that all operators on them had invariant subspaces. In the paper under review the author exhibits examples of strictly singular operators without nontrivial closed invariant subspaces.
Reviewer: Vladimir S.Pilidi

##### MSC:
 47A15 Invariant subspaces of linear operators 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
##### Keywords:
invariant subspace; strictly singular operator
Zbl 0876.46006
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