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Operators having no non-trivial closed invariant subspaces on \(\ell^1\): a step further. (English) Zbl 1480.47012

The paper under review deals with the invariant subspace problem, which asks whether a bounded operator \(T\) acting on a separable infinite-dimensional Banach space \(X\) admits a \(T\)-invariant closed subspace \(M\) which is non-trivial, i.e., distinct from \(\{0\}\) and \(X\). As shown by P. Enflo [Acta Math. 158, 213–313 (1987; Zbl 0663.47003)], the answer is negative in general. Read then constructed in the 1980s a series of counterexamples to the invariant subspace problem on classical Banach spaces such as the sequence space \(\ell^{1}(\mathbb{N})\) [C. J. Read, Bull. Lond. Math. Soc. 17, 305–317 (1985; Zbl 0574.47006)]. In [C. J. Read, J. Lond. Math. Soc., II. Ser. 34, 335–348 (1986; Zbl 0664.47006)], he conjectured the existence of an operator \(T\) on \(\ell^{1}(\mathbb{N})\) such that, for every non-constant polynomial \(p\), the operator \(p(T)\) has a non-trivial invariant subspace. It is the aim of the present work to construct such an operator \(T\), with the additional properties that it is quasi-nilpotent and such that, for every non-constant analytic analytic germ \(p\) at \(0\), the operator \(p(T)\) has no non-trivial invariant subspace. The proof relies on a subtle refinement of the classical Read construction of operators without invariant subspaces.

MSC:

47A15 Invariant subspaces of linear operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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