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∉C1 and the vectors u and v have Fourier coefficients and with respect to an orthonormal basis that diagonalizes D that satisfy , then T has a nontrivial hyperinvariant subspace. This partially answers an open question of at least 30 years duration.