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The invariant subspace problem for a class of Banach spaces. II: Hypercyclic operators. (English) Zbl 0782.47002
Summary: We continue here the line of investigation begun in [Lect. Notes Math. 1317, 1-20 (1988; Zbl 0663.47004)], where we showed that on every Banach space \(X=\ell_ 1\oplus W\) (where \(W\) is separable) there is an operator \(T\) with no nontrivial invariant subspaces. Here, we work on the same class of Banach spaces, and produce operators which not only have no invariant subspaces, but are also hypercyclic. This means that for every nonzero vector \(x\) in \(X\), the translates \(T^ r x\) \((r=1,2,3,\dots)\) are dense in \(X\). This is an interesting result even if stated in a form which disregards the linearity of \(T\): it tells us that there is a continuous map of \(X\backslash\{0\}\) into itself such that the orbit \(\{T^ r x: r\geq 0\}\) of any \(x\in X\backslash\{0\}\) is dense in \(X\backslash\{0\}\). The methods used to construct the new operator \(T\) are similar to those in [loc. cit.], but we need to have somewhat greater complexity in order to obtain a hypercyclic operator.

MSC:
47A15 Invariant subspaces of linear operators
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References:
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