We show that for any bounded operator T acting on an infinite dimensional complex Banach space, and for any epsilon > 0, there exists an operator F of rank at most one and norm smaller than epsilon such that T + F has an invariant subspace of infinite dimension and codimension. A version of this result was proved in [15] under additional spectral conditions for T or T*. This solves in full generality the quantitative version of the invariant subspace problem for rank-one perturbations. (c) 2022 Elsevier Inc. All rights reserved.
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