On rank-one perturbations of normal operators

This paper is concerned with operators on Hilbert space of the form T = D + u ⊗ v where D is a diagonalizable normal operator and u ⊗ v is a rank-one operator. It is shown that if T ∉ C 1 and the vectors u and v have Fourier coefficients {α_n}_{n = 1}^∞ and {β_n}_{n = 1}^∞ with respect to an orthonormal basis that diagonalizes D that satisfy ∑_{n = 1}^∞ (| α_n |^{2 / 3} + | β_n |^{2 / 3}) < ∞, then T has a nontrivial hyperinvariant subspace. This partially answers an open question of at least 30 years duration.