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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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