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Invariant subspaces of \(X^{**}\) under the action of biconjugates. (English) Zbl 1164.47302
Summary: We study conditions on an infinite-dimensional separable Banach space  \(X\) implying that \(X\)  is the only nontrivial invariant subspace of  \(X^{**}\) under the action of the algebra  \(\mathbb A(X)\) of biconjugates of bounded operators on  \(X\: \mathbb A(X)=\{T^{**}\: T \in \mathcal {B}(X)\}\). Such a space is called simple. We characterize simple spaces among spaces which contain an isomorphic copy of  \(c_{0}\) and show in particular that any space which does not contain  \(\ell _1\) and has Pelczynski’s property (u) is simple.
47A15 Invariant subspaces of linear operators
46B25 Classical Banach spaces in the general theory
47L05 Linear spaces of operators
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