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In many cases, some problems are either solved in an affirmative way, or in a negative way. However, in some cases it turns out that some logical axioms lead to a proof of a certain statement, while other axioms can lead to the construction of counterexamples. A well known example is the Whitehead problem: using the axiom of constructibility $V=L$, every Whitehead group can be shown to be free; using Martin's axiom and the negation of CH, examples of non-free Whitehead groups can be constructed. Now suppose that we can prove (maybe under ZFC?) that the invariant subspace problem (ISP) holds for all infinite dimensional complex Banach spaces. Then, this would mean that the construction of the counterexamples by, for instance, Enflo and Read, involves different axioms from the ones in the supposed proof. So my question is: which logical axioms have been used in the constructions by Enflo and Read that show the existence of operators with no nontrivial invariant subspaces for some non reflexive Banach spaces?

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    $\begingroup$ I am almost certain that Read's construction on $\ell_1$ works in ZFC and I am reasonably confident that Enflo's does. Note that the ISP is only interesting for separable Banach spaces, and is in some sense really a statement about separable subalgebras of B(E), so I do not see any reason to expect independence from ZFC $\endgroup$
    – Yemon Choi
    Commented Jul 12, 2020 at 22:51
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    $\begingroup$ Put it this way: if I want to "suppose that we can prove (maybe under ZFC?) that the invariant subspace problem (ISP) holds for all infinite dimensional complex Banach spaces" I think it would just be quicker for me to suppose 1=0 (ex falso quodlibet) $\endgroup$
    – Yemon Choi
    Commented Jul 12, 2020 at 22:52
  • $\begingroup$ Thank you very much! $\endgroup$
    – Manuel Norman
    Commented Jul 13, 2020 at 6:22

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