## The invariant subspace problem for rank-one perturbations.(English)Zbl 07080118

Summary: We show that for any bounded operator $$T$$ acting on an infinite-dimensional Banach space there exists an operator $$F$$ of rank at most one such that $$T+F$$ has an invariant subspace of infinite dimension and codimension. We also show that whenever the boundary of the spectrum of $$T$$ or $$T^\ast$$ does not consist entirely of eigenvalues, we can find such rank-one perturbations that have arbitrarily small norm. When this spectral condition is not satisfied, we can still find suitable finite-rank perturbations of arbitrarily small norm, but not necessarily of rank one.

### MSC:

 47A15 Invariant subspaces of linear operators 47A55 Perturbation theory of linear operators
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### References:

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