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 {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.