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Invariant subspace problem for positive operators on ℓ1

Determine whether every bounded positive operator on ℓ1 has a non-trivial closed invariant subspace.

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Background

Although Read constructed an operator on ℓ1 with no invariant subspace, this construction does not resolve the positive-operator case: the modulus of a Read operator on ℓ1 has a positive eigenvector. Hence the positive-operator version remains unresolved even on ℓ1.

The paper uses this to motivate studying typical properties of positive contractions, including the failure of certain criteria (AAB) to ensure invariant subspaces typically.

References

Moreover, even if Read's operators give a counter-example to the invariant subspace problem on Z = \ell_1, it was proved in [Troi] that the modulus of a Read's operator on \ell_1 has a positive eigenvector and so the invariant subspace problem for positive operators still remains open for the case Z = \ell_1.

— Typical properties of positive contractions and the invariant subspace problem (2409.14481 - Gillet, 22 Sep 2024) in Introduction