On rank-one perturbations of normal operators

Abstract

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.