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 ifT /∈ C1 and the vectors u and v have Fourier coefficients {αn}∞n=1 and {βn}∞n=1 with respect to an orthonormal basis that diagonalizes D that satisfy ∞n=1(|αn|2/3 + |βn|2/3) <∞, then T has a nontrivial hyperinvariant subspace. This partially answers an open question of at least 30 years duration. © 2007 Elsevier Inc. All rights reserved