Geodesic
Invariant subspaces of $X^{**}$ under the action of biconjugates
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 1, pp. 61-77
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We study conditions on an infinite dimensional separable Banach space $X$ implying that $X$ is the only non-trivial invariant subspace of $X^{**}$ under the action of the algebra $\mathbb{A}(X)$ of biconjugates of bounded operators on $X$: $\mathbb{A}(X)=\lbrace T^{**}\: T \in \mathcal {B}(X)\rbrace $. 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 property (u) of Pelczynski is simple.
We study conditions on an infinite dimensional separable Banach space $X$ implying that $X$ is the only non-trivial invariant subspace of $X^{**}$ under the action of the algebra $\mathbb{A}(X)$ of biconjugates of bounded operators on $X$: $\mathbb{A}(X)=\lbrace T^{**}\: T \in \mathcal {B}(X)\rbrace $. 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 property (u) of Pelczynski is simple.
Classification : 46B10, 46B25, 46B99, 47A15, 47L05
Keywords: algebras of operators with only one non-trivial invariant subspace; invariant subspaces under the action of the algebra of biconjugates operators; transitivity; property (u) of Pelczynski 
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Grivaux, Sophie; Rychtář, Jan. Invariant subspaces of $X^{**}$ under the action of biconjugates. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 1, pp. 61-77. http://geodesic.mathdoc.fr/item/CMJ_2006_56_1_a5/

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