The invariant subspace problem on some Banach spaces with separable dual. (English) Zbl 0686.47010

Read, C. J.

Proc. Lond. Math. Soc., III. Ser. 58, No. 3, 583-607 (1989).

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

Keywords:

cyclic operator; \(\ell _ 2\)-sum of an infinite sequence of copies of the James space; nuclear perturbation of the sum of weighted shifts

Citations:

Zbl 0782.47002