Invariant subspaces for certain finite-rank perturbations of diagonal operators

Suppose that {ek} is an orthonormal basis for a separable, infinite-dimensional Hilbert space H. Let D be a diagonal operator with respect to the orthonormal basis {ek}. That is, D = ∞k =1 λkek ⊗ ek, where {λk} is a bounded sequence of complex numbers. Let T = D + u1 ⊗v1 +···+un ⊗vn. Improving a result of Foias et al. (2007) [3], we show that if the vectors u1, . . . , un and v1, . . . , vn satisfy an 1-condition with respect to the orthonormal basis {ek}, and if T is not a scalar multiple of the identity operator, then T has a non-trivial hyperinvariant subspace. © 2012 Elsevier Inc. All rights reserved.