Invariant subspaces of X * * under the action of biconjugates

Sophie Grivaux; Jan Rychtář

Czechoslovak Mathematical Journal (2006)

  • Volume: 56, Issue: 1, page 61-77
  • ISSN: 0011-4642

Abstract

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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 𝔸 ( X ) of biconjugates of bounded operators on X : 𝔸 ( X ) = { T * * T ∈ 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.

How to cite

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Grivaux, Sophie, and Rychtář, Jan. "Invariant subspaces of $X^{**}$ under the action of biconjugates." Czechoslovak Mathematical Journal 56.1 (2006): 61-77. <http://eudml.org/doc/31017>.

@article{Grivaux2006,
abstract = {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.},
author = {Grivaux, Sophie, Rychtář, Jan},
journal = {Czechoslovak Mathematical Journal},
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; algebras of operators with only one non-trivial invariant subspace; transitivity},
language = {eng},
number = {1},
pages = {61-77},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Invariant subspaces of $X^\{**\}$ under the action of biconjugates},
url = {http://eudml.org/doc/31017},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Grivaux, Sophie
AU - Rychtář, Jan
TI - Invariant subspaces of $X^{**}$ under the action of biconjugates
JO - Czechoslovak Mathematical Journal
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 1
SP - 61
EP - 77
AB - 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.
LA - eng
KW - 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; algebras of operators with only one non-trivial invariant subspace; transitivity
UR - http://eudml.org/doc/31017
ER -

References

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