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The invariant subspace problem on some Banach spaces with separable dual. (English) Zbl 0686.47010
The existence of a cyclic operator T (i.e., an operator without nontrivial closed invariant subspaces) is demonstrated on Banach spaces X of the following type $(a)\quad X=c_ 0\oplus W,\quad (b)\quad X=J_{\infty}\oplus W,$ where W is an arbitrary separable Banach space and $$J_{\infty}=\ell_ 2\oplus \otimes^{\infty}_{i=1}J_ i$$, the $$\ell_ 2$$-sum of an infinite sequence of copies of the James space $J_ i=J=\{a\in c_ 0| \quad \| a\| =\sup \{(\sum^{n- 1}_{i=1}| a_{p_{i+1}}-a_{p_ i}|^ 2)^{1/2}| \quad n\in {\mathbb{N}},\quad p_ 1<p_ 2<...<p_ n\}<\infty \};$ observe that $$c_ 0$$ has a separable dual and $$J_{\infty}$$ has a separable bidual.
The required operator T is constructed as a nuclear perturbation of the sum of weighted shifts on $$c_ 0$$, resp. $$\ell_ 2$$ and $$\otimes^{\infty}_{i=1}J_ i$$, first on a dense subspace F of X, then, by continuation, on X. The proof is rather technical, as seems unavoidable for this type of results, but it is well-presented. Since the existence of a (hyper) cyclic operator T on $$\ell_ 1\oplus W$$, W separable, has already been shown [C. J. Read, Isr. J. Math. 63, 1–40 (1988; Zbl 0782.47002)], the author concludes by stating “that we cannot go much further until and unless we solve the invariant subspace problem for a reflexive Banach space”.
Reviewer: G.P.A.Thijsse

##### MSC:
 47A15 Invariant subspaces of linear operators 46B25 Classical Banach spaces in the general theory
Zbl 0782.47002
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