Invariant subspaces of $X^{**}$ under the action of biconjugates

Grivaux, Sophie; Rychtář, Jan

Geodesic

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Invariant subspaces of

X^{* *}

under the action of biconjugates

Grivaux, Sophie ; Rychtář, Jan

Czechoslovak Mathematical Journal, Tome 56 (2006) no. 1, pp. 61-77.

Voir la notice de l'article dans Czech Digital Mathematics Library

Résumé

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

A (X)

of biconjugates of bounded operators on

X

A (X) = {T^{* *} T \in 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

ℓ_{1}

and has property (u) of Pelczynski is simple.

MR Zbl

Classification : 46B10, 46B25, 46B99, 47A15, 47L05
Mots-clés : 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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     title = {Invariant subspaces of $X^{**}$ under the action of biconjugates},
     journal = {Czechoslovak Mathematical Journal},
     pages = {61--77},
     publisher = {mathdoc},
     volume = {56},
     number = {1},
     year = {2006},
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     zbl = {1164.47302},
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     url = {https://geodesic-test.mathdoc.fr/item/CMJ_2006__56_1_a5/}
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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. https://geodesic-test.mathdoc.fr/item/CMJ_2006__56_1_a5/