Invariant subspaces for certain finite-rank perturbations of diagonal operators

Suppose that {e _k} is an orthonormal basis for a separable, infinite-dimensional Hilbert space H. Let D be a diagonal operator with respect to the orthonormal basis {e _k}. That is, D=∑ _k=1 ^∞λ _ke _k⊗e _k, where {λ _k} is a bounded sequence of complex numbers. LetT=D+u1⊗v1+...+u _n⊗v _n. Improving a result of Foias et al. (2007) [3], we show that if the vectors u ₁,..., u _n and v1,...,vn satisfy an ℓ ¹-condition with respect to the orthonormal basis {e _k}, and if T is not a scalar multiple of the identity operator, then T has a non-trivial hyperinvariant subspace.